Mechanics Topic D (Rotation)  1 David Apsley


 Cornelia Goodwin
 1 years ago
 Views:
Transcription
1 TOPIC D: ROTATION SPRING Angular kinematics 1.1 Angular velocity and angular acceleration 1.2 Constantangularacceleration formulae 1.3 Displacement, velocity and acceleration in circular motion 2. Angular dynamics 2.1 Torque 2.2 Angular momentum 2.3 The angularmomentum principle for motion in a circle 2.4 The angularmomentum principle for arbitrary motion 3. Rigidbody rotation 3.1 Moment of inertia 3.2 Second moments and the radius of gyration 3.3 The equations of rotational motion 3.4 Comparing translation and rotation 3.5 Examples 4. Calculation of moments of inertia 4.1 Methods of calculation 4.2 Fundamental shapes 4.3 Stretching parallel to an axis 4.4 Volumes of revolution 4.5 Change of axis 5. General motion of a rigid body (optional) 5.1 Rolling without slipping 5.2 Rolling with slipping Appendix 1: Moments of inertia of simple shapes Appendix 2: Second moment of area Mechanics Topic D (Rotation)  1 David Apsley
2 1. ANGULAR KINEMATICS 1.1 Angular Velocity and Angular Acceleration P For a particle moving in a circular arc, or for a rigid body rotating about a fixed axis, the instantaneous position is defined by the angle between a radius vector and a fixed line. O r s If s is length of arc and r is radius then the angle θ in radians is defined such that (1) Angular velocity ω is the rate of change of angle: (2) Angular acceleration α is the rate of change of angular velocity: (3) It is common to use a dot to indicate differentiation w.r.t. time; e.g. means dθ/dt. 1.2 ConstantAngularAcceleration Formulae There is a direct correspondence between linear and angular motion. Linear Angular Displacement s θ Velocity Acceleration Constantacceleration formulae Mechanics Topic D (Rotation)  2 David Apsley
3 For nonconstant acceleration: distance is the area under a v t graph; angle is the area under an ω t graph; v acceleration is the gradient of a v t graph; angular acceleration is the gradient of a ω t graph. s t t Example 1. What is the angular velocity in radians per second of the minute hand of a clock? Example 2. A turbine starts from rest and has a constant angular acceleration of 0.1 rad s 2. How many revolutions does it make in reaching a rotation rate of 50 rpm? 1.3 Displacement, Velocity and Acceleration in Circular Motion Consider a particle moving at a fixed radius r. The following have already been derived in Topic A (Kinematics), as a special case of motion in general polar coordinates. v Velocity r Since s = rθ and r is constant, the velocity is tangential and its magnitude (speed) is (4) Acceleration dv dt Because it is not moving in a straight line, the particle has two components of acceleration: tangential, if its speed is changing: O r v 2 r or (5a) radially inward, because its direction is changing: or (5b) The latter is called the centripetal acceleration. A centripetal force is necessary to maintain this and keep the particle moving in a circular path. This force can be provided in many ways: for example, the tension in a cable, a normal reaction from an outer boundary or friction. Mechanics Topic D (Rotation)  3 David Apsley
4 Example 3. Find the minimum coefficient of friction necessary to prevent slipping for a particle which is placed: (a) (b) 100 mm from the rotation axis on a turntable rotating at 78 rpm; on the inside of a cylindrical drum, radius 0.3 m, rotating about a vertical axis at 200 rpm. Example 4. (Exam 2017) A particle of mass 3 kg is whirled around in a horizontal circle by a light elastic string of unstretched length 1.5 m and stiffness 90 N m 1 attached to a fixed point O. At a particular speed, the cable makes an angle of 15º with the horizontal. Find: (a) the tension in the cable; (b) the extension of the cable; (c) the speed of the particle. O 15 o 3 kg Example 5. (Exam 2010) A building s roof consists of a smooth hemispherical dome with outside radius 20 m. A brief gust of wind dislodges a small object at the top of the dome and it slides down the roof. 20 m (a) (b) (c) (d) (e) Find an expression for the velocity v of the object when its position vector makes an angle θ with the vertical through the centre of the dome (see the figure). While it is in contact with the roof the object is undergoing motion in a circular arc. Write down an expression for its centripetal acceleration as a function of angle θ. Find an expression for the normal contact force as a function of angle θ and the mass m of the object. Hence determine the angle θ at which the object leaves the roof, as well as its height and speed at this point. Find the distance from the outside wall of the dome at which the object hits the ground. Mechanics Topic D (Rotation)  4 David Apsley
5 2. ANGULAR DYNAMICS 2.1 Torque The torque 1 (or moment of force) T about an axis is given by: axis r F torque = force perpendicular distance from axis (6) Torque measures the turning effect of a force. When the force is not perpendicular to the radius vector then only the component perpendicular to the radius vector contributes torque. v 2.2 Angular Momentum Angular momentum (or moment of momentum) h is given by: axis r m angular momentum = momentum perpendicular distance from axis (7) For noncircular motion, v is the transverse component of velocity see Section The AngularMomentum Principle For Motion in a Circle Forcemomentum principle: If F is the tangential component of force and r is constant (i.e. circular motion) then torque = rate of change of angular momentum (8) In fact, (8) holds for noncircular motion, but the proof requires more effort; see Section 2.4. Equation (8) is the rotational analogue of the momentum principle for translational motion: force = rate of change of momentum For single particles the angularmomentum equation offers no advantage over the momentum equation. However, it is invaluable for rigidbody rotation, in which it is applied by summing over all masses in the system. The torque is then the sum of the moments of the external forces only, since internal forces between particles are equal and opposite and cancel in pairs. 1 Whilst one can have a moment of any physical quantity, torque is used almost exclusively for moment of force. Mechanics Topic D (Rotation)  5 David Apsley
6 Example 6. (Ohanian) The original Ferris wheel built by George Ferris had radius 38 m and mass kg. Assuming that all of its mass was uniformly distributed along the rim of the wheel, if the wheel was initially rotating at 0.05 rev min 1, what constant torque would stop it in 30 s? What force exerted on the rim of the wheel would have given such a torque? In the absence of an external torque, a direct corollary of the angularmomentum principle is: The Principle of Conservation of Angular Momentum The angular momentum of an isolated system remains constant. 2.4 The AngularMomentum Principle For Arbitrary Motion For a particle of mass m, the angular momentum is O r sin r P v i.e. only the transverse component of velocity, v = v sin α, contributes to the angular momentum. The radial component, v r = v cos α, has no moment about the axis. A similar decomposition applies for the torque. O r v sin P v v cos Using a vector cross product (denoted by ), both angular momentum and torque may be represented by vectors oriented along the rotation axis (in the sense of a righthand screw): angular momentum: or (9a) torque: or (9b) Differentiating the vector expression for angular momentum, using the product rule: Hence, which is, in vector form, the angularmomentum principle: rate of change of angular momentum = torque By summation this can be applied to the whole, with T the torque due to external forces only. (10) Mechanics Topic D (Rotation)  6 David Apsley
7 3. RIGIDBODY ROTATION 3.1 Moment of Inertia Example. A bicycle wheel and a flat disk have the same mass, the same radius and are spinning at the same rate. Which has the greater angular momentum and kinetic energy? For rotating rigid bodies, different particles lie at different radii and hence have different speeds. Particles at greater radius move faster and contribute more to the body s angular momentum and kinetic energy. Thus, the angular momentum and kinetic energy depend on the distribution of mass relative to the axis of rotation. The total angular momentum and kinetic energy may be obtained by summing over individual particles of mass m at radius r. Most importantly, although particles at different radii have different speeds v, they all have the same angular velocity ω. Angular Momentum r m r Kinetic Energy The quantity (11) is the moment of inertia (or second moment of mass) of the body about the specified axis. angular momentum (12) kinetic energy (13) (c.f. momentum P = Mv and kinetic energy for translation). Mechanics Topic D (Rotation)  7 David Apsley
8 Moment describes the distribution of mass relative to the selected axis. (It gives higher weighting to masses at greater radii.) Inertia refers to a resistance to a change in motion (acceleration). In this sense, the moment of inertia fulfils the same role for rotation as the mass of a body in translation. Example revisited. For a hoop (a close approximation to the bicycle wheel) all the mass is concentrated at the same radius R. Hence For a flat disk it turns out (see later) that the moment of inertia is ½MR 2. Other things being equal, the disk will have half the angular momentum and half the kinetic energy of the hoop. This is because some of its mass is at a smaller radius and is moving more slowly. 3.2 Second Moments and the Radius of Gyration The moment (strictly, the first moment) of any quantity is defined by first moment = quantity distance Similarly, second moment = quantity (distance) 2 For an extended body the distance varies, so we must sum over constituent parts; e.g. (14) (In Hydraulics and Structures courses a similar quantity second moment of area appears in connection with hydrostatic forces on and resistance to bending, respectively.) The centre of mass is where the same concentrated mass would have the same first moment: (15) The radius of gyration k is that at which the same mass would have the same second moment: (16) Distributed mass Concentrated mass R k Same total mass and moment of inertia Mechanics Topic D (Rotation)  8 David Apsley
9 Examples. Hoop of mass M and radius R (axis through centre, perpendicular to plane) Moment of inertia, radius of gyration, k = R Here, radius of gyration is geometric radius as all mass is concentrated at the circumference. Circular disc of mass M and radius R (axis through centre, perpendicular to plane) Moment of inertia radius of gyration The radius of gyration is less than the geometric radius because mass is distributed over a range of distances from the axis. 3.3 The Equations of Rotational Motion (Angular) Momentum For rigidbody rotation the equation of motion is the angular momentum equation: torque = rate of change of angular momentum (17) This is the rotational equivalent of Newton s Second Law: force = rate of change of momentum For solid bodies, where I and M are constant we can write these in terms of acceleration: (rotation) (translation) (18) (Angular) Impulse If we integrate (17) with respect to time we obtain an impulse equation: torque time = change in angular momentum The LHS is called the angular impulse. (19) Mechanics Topic D (Rotation)  9 David Apsley
10 Energy Alternatively, integrate (17) wrt angle to obtain an energy equation. First rewrite it as Integrating with respect to angle θ gives the Mechanical Energy Principle: work done (torque angle) = change in kinetic energy (20) 3.4 Comparing Translation and Rotation Translation Rotation Displacement x θ Velocity v ω Acceleration a α Inertia m I Effective location of mass centre of mass radius of gyration Cause of motion force torque Translation Rotation Momentum Kinetic energy Power Fv Tω Equation of motion rate form force = rate of change of momentum torque = rate of change of angular momentum Equation of motion impulse form impulse (force time) = change of momentum angular impulse (torque time) = change of angular momentum Equation of motion energy form work done (force distance) = change of kinetic energy work done (torque angle) = change of kinetic energy Mechanics Topic D (Rotation)  10 David Apsley
11 3.5 Examples Example 7. A bucket of mass M is fastened to one end of a light inextensible rope. The rope is coiled round a windlass in the form of a circular cylinder (radius r, moment of inertia I) which is left free to rotate about its axis. Prove that the bucket descends with acceleration r M Mg Example 8. A flywheel whose axial moment of inertia is 1000 kg m 2 rotates with an angular velocity of 300 rpm. Find the angular impulse which would be required to bring the flywheel to rest. Hence, find the frictional torque at the bearings if the flywheel comes to rest in 10 min under friction alone. Example 9. A flywheel of radius 500 mm is attached to a shaft of radius 100 mm, the combined assembly having a moment of inertia of 500 kg m 2. Long cables are wrapped around flywheel and shaft in opposite directions and are attached to masses of 10 kg and 20 kg respectively, which are initially at rest as shown. Calculate: (a) how far the 10 kg mass must drop in order to raise the 20 kg mass by 1 m; (b) the angular velocity of the shaft at this point. 500 mm 100 mm 10 kg 20 kg Mechanics Topic D (Rotation)  11 David Apsley
12 Example 10. A 15 kg mass hangs in the loop of a light inextensible cable, one end of the cable being fixed and the other wound round a wheel of radius 0.3 m and moment of inertia 0.9 kg m 2 so that the lengths of cable are vertical (see the figure). The mass is released from rest and falls, turning the wheel. Neglecting friction between the mass and the loop of cable and between the wheel and its bearings, find: (a) a relationship between the downward velocity of the mass, v, and the angular velocity of the wheel, ω; (b) the downward acceleration of the mass; (c) the speed of the mass when it has fallen a distance 2 m; (d) the number of turns of the wheel before it reaches a rotation rate of 300 rpm. 15 kg Example 11. A square plate of mass 6 kg and side 0.2 m is suspended vertically from a frictionless hinge along one side. It is struck dead centre by a lump of clay of mass 1 kg which is moving at 10 m s 1 horizontally and remains stuck (totally inelastic collision). To what height will the bottom of the plate rise after impact? (The moment of inertia of a square lamina, side a and mass M, about one side, is ) Mechanics Topic D (Rotation)  12 David Apsley
13 4. CALCULATION OF MOMENTS OF INERTIA 4.1 Methods of Calculation The moment of inertia I depends on: the distribution of mass; the axis of rotation. Some common methods of calculating I are as follows. Method 1. First Principles For isolated particles this can be done by direct summation. For continuous bodies integration is necessary. Method 2. Combination of Fundamental Elements (Hoop, Disk, Rod) hoop surface of revolution First principles disc solid of revolution rod rectangular lamina Method 3. Stretching Parallel to the Axis If a shape is simply stretched parallel to an axis then the moment of inertia is unchanged since the relative disposition of mass about the axis is not changed. e.g. hoop cylindrical shell disc solid cylinder rod rectangular lamina Method 4. Change of Axis hoop/disc rod cylinder rectangle Calculations may be performed first about some convenient (typically symmetry) axis; the moment of inertia about the actual axis is then determined by one of two techniques for changing axes: the parallelaxis theorem and the perpendicularaxes theorem. Mechanics Topic D (Rotation)  13 David Apsley
14 4.2 Fundamental Shapes Hoop For a hoop (an infinitesimallythin circular arc) of mass M and radius R, rotating about its symmetry axis, all the mass is concentrated at the single distance R from the axis. Hence, R For a hoop of mass M and radius R, about the symmetry axis perpendicular to its plane: (21) Disc Consider the moment of inertia of a uniform circular disc (an infinitesimallythin, circular plane lamina) of mass M and radius R, about the axis of symmetry perpendicular to its plane. The disc can be broken down into subelements which are hoops of radius r and thickness r. Let ρ be the mass per unit area. Sum over all elements: r r R For a hoop of mass M and radius R, about the symmetry axis perpendicular to its plane: (22) Rod Consider the moment of inertia of a rod (an infinitesimallythin line segment) of mass M and length L, about its axis of symmetry. The rod can be broken down into subelements of length δx, distance x from the axis. Let ρ be the mass per unit length. L x x Summing: Mechanics Topic D (Rotation)  14 David Apsley
15 For a rod of mass M and length L, about its axis of symmetry: (23) 4.3 Stretching Parallel to an Axis The distribution of mass about the axis and hence the moment of inertia is not changed by stretching parallel to the axis of rotation without change of mass. Hence, for the axes shown: hoop cylindrical shell: R disc solid cylinder: rod rectangular lamina: hoop/disc cylinder b (In the last case the axis is in the plane of the lamina.) a The only dependence on the dimension parallel to the axis of rotation is via its effect on the total mass M. rod rectangle Mechanics Topic D (Rotation)  15 David Apsley
16 Example 12. outer steel rim (part (b)) 40 cm 6 cm flywheel shaft 30 cm (a) (b) (c) 10 cm A flywheel consists of an aluminium disc of diameter 40 cm and thickness 6 cm, mounted on an aluminium shaft of diameter 10 cm and length 30 cm as shown. Calculate the moment of inertia of flywheel + shaft. To increase the moment of inertia a steel rim is fixed to the outside of the flywheel. Calculate the outer radius of the steel rim required to double the moment of inertia of the assembly. If the flywheel is initially rotating at 100 rpm, calculate the constant frictional braking force which needs to be applied to the outside of the steel rim in part (b) if the flywheel is to be brought to rest in 30 seconds. For this question you may require the following information. Density of aluminium: 2650 kg m 3 ; steel: 7850 kg m 3. Moment of inertia of a solid cylinder of radius R and mass M about its axis:. Mechanics Topic D (Rotation)  16 David Apsley
17 4.4 Volumes of Revolution Moments of inertia for volumes of revolution may be deduced by summing over infinitesimal discs (or very thin cylinders) of radius y and length δx. Let ρ be the mass per unit volume. Then the elemental mass and moment of inertia are, respectively: mass: moment of inertia: y x x Summing over all elemental masses and moments of inertia: (24) (25) Example. Find the moment of inertia of a solid sphere, mass M and radius R about an axis of symmetry. For a solid sphere,. Hence, between. Thus, Hence, R axis whence Mechanics Topic D (Rotation)  17 David Apsley
18 4.5 Change of Axis ParallelAxis Rule If the moment of inertia of a body of mass M about an axis through its centre of mass is I G, then the moment of inertia about a parallel axis A is given by where M is the mass of the body and d is the distance between axes. (26) Proof. Take (x,y,z) coordinates relative to the centre of mass, with the z direction parallel to the axes of rotation. By Pythagoras, G (0,0) d P(x,y) A(x A, y A) Expanding the second of these: The last two terms vanish because there are no moments about the centre of mass. Corollary 1. The corresponding radii of gyration are related by (27) Corollary 2. For a set of parallel axes, the smallest moment of inertia is about an axis through the centre of mass. Example. The moment of inertia of a rod of mass M and length L about an axis through its centre and normal to the rod is the end of the rod is. Hence the moment of inertia about a parallel axis through A G 1 2 L 1 2 L Mechanics Topic D (Rotation)  18 David Apsley
19 4.3.2 PerpendicularAxis Rule Important note. This is applicable to plane laminae only. However, it can often be combined with stretching parallel to the axis to give 3d shapes. If a plane body has moments of inertia I x and I y about perpendicular axes Ox and Oy in the plane of the body, then its moment of inertia about an axis Oz, perpendicular to the plane, is: (28) Proof. By Pythagoras, z y Hence r x y x Example. Find the moment of inertia of a rectangular lamina, mass M and sides a and b, about an axis through the centre, perpendicular to the lamina. Solution. From the earlier examples, the moments of inertia about axes in the plane of the lamina are b z y x a Example. Find the moment of inertia of a circular disc, radius R, about a diameter. Solution. In this case we use the perpendicularaxis theorem in reverse, because we already know the moment of inertia about an axis through the centre, perpendicular to the plane of the disc:. By rotational symmetry the unknown moment of inertia I about a diameter is the same for both x and y axes. Hence, 1 I MR 2 z 2 I I Mechanics Topic D (Rotation)  19 David Apsley
20 Example 13. Find the radius of gyration of the squareframe lamina shown about an axis along one side. axis axis 0.1 m 0.5 m 0.1 m 0.5 m Example 14. A rigid framework consists of four rods, each of length L and mass M, connected in the form of a square ABCD as shown. Find expressions, in terms of L and M, for the moment of inertia of the framework about: (a) the axis of symmetry SS; (b) the side AB; (c) an axis perpendicular to the framework and passing through centre O; (d) an axis perpendicular to the framework and passing through vertex A; (e) the diagonal AC. Data: the moment of inertia of a rod, length L and mass M, about an axis through its centre and perpendicular to the rod is. B S C O A S D Mechanics Topic D (Rotation)  20 David Apsley
21 5. GENERAL MOTION OF A RIGID BODY (Optional) The motion of a rigid body which is allowed to rotate as well as translate (e.g. a rolling body) can be decomposed into: It may be shown (optional exercise) that, for a system of particles (e.g. a rigid body): (1) The centre of mass moves like a single particle of mass M under the resultant of the external forces: where (29) (2) The relationship torque = rate of change of angular momentum : holds for the torque of all external forces about a point which is either: fixed, or moving with the centre of mass. (3) The total kinetic energy can be written as the sum: kinetic energy of the centre of mass ( ) + kinetic energy of motion relative to the centre of mass For a rigid body, motion relative to the centre of mass must be rotation and hence: (30) 5.1 Rolling Without Slipping Consider a body with circular crosssection rolling along a plane surface. If the body rolls without slipping then the distance moved by the point of contact must equal the length of arc swept out: v r r Hence the linear and angular velocities are related by: Mechanics Topic D (Rotation)  21 David Apsley
22 The instantaneous point of contact with the plane has zero velocity; hence the friction force does no work but it is responsible for rotating the body! The total kinetic energy is given by Example 15. (Synge and Griffiths) A wheel consists of a thin rim of mass M and n spokes each of mass m, which may be considered as thin rods terminating at the centre of the wheel. If the wheel is rolling with linear velocity v, express its kinetic energy in terms of M, m, n, v. A common example is of a spherical or cylindrical body rolling down an inclined plane. The forces on the body are its weight Mg, the normal reaction force R and the friction force F. R v mg F Consider the linear motion of the centre of mass and the rotational motion about it. force = mass acceleration for translation of the centre of mass: (along slope) (normal to slope) torque = rate of change of angular momentum for rotation about the centre of mass: v and ω are related, if not slipping, by v = rω Mechanics Topic D (Rotation)  22 David Apsley
23 Example. (Ohanian) A piece of steel pipe, mass 360 kg, rolls down a ramp inclined at 30 to the horizontal. What is the acceleration if the pipe rolls without slipping? What is the magnitude of the friction force that acts at the point of contact between the pipe and ramp? Solution. Linear motion: Rotation about centre of mass: (i) (ii) Eliminate F by (i) r + (ii), noting that : Hence, But for a hoop, and hence (by stretching parallel to the axis) a pipe,. Thus, This is the linear acceleration. For the friction force use either linear or rotational equation of motion; e.g. from (i): Answer: 2.45 m s 2 ; 883 N. Mechanics Topic D (Rotation)  23 David Apsley
24 5.2 Rolling With Slipping For a body which is rolling along a surface the condition for no slipping is that the instantaneous point of contact is not moving; that is, the linear velocity of the centre of mass (v) must be equal and opposite to that of the relative velocity of a point on the rim which is rotating (rω). Hence, slipping occurs whilst v rω (31) v r r R In this case, friction will act to oppose slipping. If a spinning body is placed on a surface then it is the friction force which initiates its forward motion. Note that, while slipping occurs, there is relative motion and so friction is maximal: F mg Example. (Synge and Griffiths) A hollow spherical ball of radius 5 cm is set spinning about a horizontal axis with an angular velocity of 10 rad s 1. It is then gently placed on a horizontal plane and released. If the coefficient of friction between the ball and the plane is 0.34, find the distance traversed by the ball before slipping ceases. [The moment of inertia of a spherical shell of mass m and radius r is ]. Solution. Initially slipping must occur, because the ball is not moving forward but it is rotating. Whilst slipping it is friction which (a) accelerates the translational motion from rest; (b) decelerates the rotation. Slipping ceases when v = rω, but until this point friction is maximal and given by. Linear motion Whilst slipping,. Hence, with v = 0 at t = 0. (i) Mechanics Topic D (Rotation)  24 David Apsley
25 Rotational motion Whilst slipping,. Also,. Hence, with ω = ω 0 = 10 rad s 1 at t = 0. (ii) Slipping stops when v = rω. From (i) and (ii), this occurs when This distance travelled may be determined from the linear constantacceleration formula, with Hence, u = 0, a = μg, Using consistent length units of metres: Answer: 6.0 mm. Mechanics Topic D (Rotation)  25 David Apsley
26 Appendix 1: Moments of Inertia for Simple Shapes Many formulae are given in the textbooks of Meriam and Kraige or Gere and Timoshenko. Only some of the more common ones are summarised here. Geometric figures are assumed to have a uniform density and have a total mass M. Geometry Axis I Rod, length L (1) Through centre (2) End Rectangular lamina, sides L (perpendicular to axis) and W (1) Inplane; symmetry (2) Side (3) Perpendicular to plane; symmetry ) Triangular lamina, base B, altitude H Base Circular ring, radius R (1) Perpendicular to plane; symmetry (2) Diameter Circular disc, radius R (1) Perpendicular to plane; symmetry (2) Diameter Circular cylinder, radius R, height H. Symmetry axis Solid sphere (or hemisphere), radius R Any diameter Spherical (or hemispherical) shell, radius R Any diameter Moments of inertia for many different shapes or axes can be constructed from these by: use of the parallelaxis or perpendicularaxes rules; stretching parallel to an axis (without change of mass distribution); combination of fundamental elements. Mechanics Topic D (Rotation)  26 David Apsley
27 Appendix 2: Second Moment of Area Second moment of area rather than second moment of mass appears in structural engineering (resistance to beam bending) and hydrostatics (pressure force). The formulae for second moments of area of plane figures are exactly the same as those in the table above except that mass M is replaced by area A. The same symbol (I) is used. Dependence on a length dimension parallel to the axis is often hidden inside M or A; e.g. second moment of area of a rectangular lamina about an inplane symmetry axis: (since ) second moment of area of a triangular lamina about a side of length B: (since ) You will meet second moments of area a great deal in your Structures courses. Mechanics Topic D (Rotation)  27 David Apsley
28 Numerical Answers to Examples in the Text Full worked answers are given in a separate document online. Example rad s 1 Example Example 3. (a) 0.680; (b) Example 4. (a) 114 N; (b) 1.26 m; (c) 9.88 m s 1 Example 5. (a) ; (b) (c) ; (d) 48.2 ; 13.3 m; 11.4 m s 1 ; (e) 2.49 m Example N m; N Example 7. No numerical answer Example N m s; 52.4 N m Example 9. (a) 5 m; (b) 1.08 rad s 1 Example 10. (a) ; (b) 2.68 m s 2 ; (c) 3.27 m s 1 ; (d) 4.41 rev Example m Example 12. (a) kg m 2 ; (b) 215 mm; (c) 1.32 N Example m Example 14. (a) ; (b) ; (c) ; (d) ; (b) Example 15. Mechanics Topic D (Rotation)  28 David Apsley
TOPIC D: ROTATION EXAMPLES SPRING 2018
TOPIC D: ROTATION EXAMPLES SPRING 018 Q1. A car accelerates uniformly from rest to 80 km hr 1 in 6 s. The wheels have a radius of 30 cm. What is the angular acceleration of the wheels? Q. The University
More informationRotation. PHYS 101 Previous Exam Problems CHAPTER
PHYS 101 Previous Exam Problems CHAPTER 10 Rotation Rotational kinematics Rotational inertia (moment of inertia) Kinetic energy Torque Newton s 2 nd law Work, power & energy conservation 1. Assume that
More informationName: Date: Period: AP Physics C Rotational Motion HO19
1.) A wheel turns with constant acceleration 0.450 rad/s 2. (99) Rotational Motion H19 How much time does it take to reach an angular velocity of 8.00 rad/s, starting from rest? Through how many revolutions
More informationTutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning?
1. rpm is about rad/s. 7. ROTATIONAL MOTION 2. A wheel rotates with constant angular acceleration of π rad/s 2. During the time interval from t 1 to t 2, its angular displacement is π rad. At time t 2
More informationAdvanced Higher Physics. Rotational motion
Wallace Hall Academy Physics Department Advanced Higher Physics Rotational motion Problems AH Physics: Rotational Motion 1 2013 Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration
More informationCHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque
7 1. Define Torque 2. State the conditions for equilibrium of rigid body (Hint: 2 conditions) 3. Define angular displacement 4. Define average angular velocity 5. Define instantaneous angular velocity
More information1 MR SAMPLE EXAM 3 FALL 2013
SAMPLE EXAM 3 FALL 013 1. A merrygoround rotates from rest with an angular acceleration of 1.56 rad/s. How long does it take to rotate through the first rev? A) s B) 4 s C) 6 s D) 8 s E) 10 s. A wheel,
More information= o + t = ot + ½ t 2 = o + 2
Chapters 89 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the
More informationChapter Rotational Motion
26 Chapter Rotational Motion 1. Initial angular velocity of a circular disc of mass M is ω 1. Then two small spheres of mass m are attached gently to diametrically opposite points on the edge of the disc.
More informationTOPIC E: OSCILLATIONS EXAMPLES SPRING Q1. Find general solutions for the following differential equations:
TOPIC E: OSCILLATIONS EXAMPLES SPRING 2019 Mathematics of Oscillating Systems Q1. Find general solutions for the following differential equations: Undamped Free Vibration Q2. A 4 g mass is suspended by
More informationAP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems
AP Physics C: Rotation II (Torque and Rotational Dynamics, Rolling Motion) Problems 1980M3. A billiard ball has mass M, radius R, and moment of inertia about the center of mass I c = 2 MR²/5 The ball is
More information1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t 4 t 2
CFE Advanced Higher Physics Unit 1 Rotational Motion and Astrophysics Kinematic relationships 1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t 4 t 2 a) Find
More informationWebreview Torque and Rotation Practice Test
Please do not write on test. ID A Webreview  8.2 Torque and Rotation Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 0.30mradius automobile
More informationSuggested Problems. Chapter 1
Suggested Problems Ch1: 49, 51, 86, 89, 93, 95, 96, 102. Ch2: 9, 18, 20, 44, 51, 74, 75, 93. Ch3: 4, 14, 46, 54, 56, 75, 91, 80, 82, 83. Ch4: 15, 59, 60, 62. Ch5: 14, 52, 54, 65, 67, 83, 87, 88, 91, 93,
More informationPhys 106 Practice Problems Common Quiz 1 Spring 2003
Phys 106 Practice Problems Common Quiz 1 Spring 2003 1. For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the rim to the speed
More informationMechanics Topic B (Momentum)  1 David Apsley
TOPIC B: MOMENTUM SPRING 2019 1. Newton s laws of motion 2. Equivalent forms of the equation of motion 2.1 orce, impulse and energy 2.2 Derivation of the equations of motion for particles 2.3 Examples
More informationSlide 1 / 133. Slide 2 / 133. Slide 3 / How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m?
1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 1 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 2 / 133 3 A ball rotates
More informationSlide 2 / 133. Slide 1 / 133. Slide 3 / 133. Slide 4 / 133. Slide 5 / 133. Slide 6 / 133
Slide 1 / 133 1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 2 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 3 / 133
More informationRotational Mechanics Part III Dynamics. Pre AP Physics
Rotational Mechanics Part III Dynamics Pre AP Physics We have so far discussed rotational kinematics the description of rotational motion in terms of angle, angular velocity and angular acceleration and
More informationCircular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics
Circular Motion, Pt 2: Angular Dynamics Mr. Velazquez AP/Honors Physics Formulas: Angular Kinematics (θ must be in radians): s = rθ Arc Length 360 = 2π rads = 1 rev ω = θ t = v t r Angular Velocity α av
More informationMechanics Answers to Examples B (Momentum)  1 David Apsley
TOPIC B: MOMENTUM ANSWERS SPRING 2019 (Full worked answers follow on later pages) Q1. (a) 2.26 m s 2 (b) 5.89 m s 2 Q2. 8.41 m s 2 and 4.20 m s 2 ; 841 N Q3. (a) 1.70 m s 1 (b) 1.86 s Q4. (a) 1 s (b) 1.5
More informationTest 7 wersja angielska
Test 7 wersja angielska 7.1A One revolution is the same as: A) 1 rad B) 57 rad C) π/2 rad D) π rad E) 2π rad 7.2A. If a wheel turns with constant angular speed then: A) each point on its rim moves with
More information6. Find the net torque on the wheel in Figure about the axle through O if a = 10.0 cm and b = 25.0 cm.
1. During a certain period of time, the angular position of a swinging door is described by θ = 5.00 + 10.0t + 2.00t 2, where θ is in radians and t is in seconds. Determine the angular position, angular
More informationExercise Torque Magnitude Ranking Task. Part A
Exercise 10.2 Calculate the net torque about point O for the two forces applied as in the figure. The rod and both forces are in the plane of the page. Take positive torques to be counterclockwise. τ 28.0
More informationUnit 8 Notetaking Guide Torque and Rotational Motion
Unit 8 Notetaking Guide Torque and Rotational Motion Rotational Motion Until now, we have been concerned mainly with translational motion. We discussed the kinematics and dynamics of translational motion
More informationRotational Dynamics Smart Pulley
Rotational Dynamics Smart Pulley The motion of the flywheel of a steam engine, an airplane propeller, and any rotating wheel are examples of a very important type of motion called rotational motion. If
More informationTextbook Reference: Wilson, Buffa, Lou: Chapter 8 Glencoe Physics: Chapter 8
AP Physics Rotational Motion Introduction: Which moves with greater speed on a merrygoround  a horse near the center or one near the outside? Your answer probably depends on whether you are considering
More informationBig Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular
Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.F.1: Only
More informationPHYSICS 221 SPRING 2014
PHYSICS 221 SPRING 2014 EXAM 2: April 3, 2014 8:1510:15pm Name (printed): Recitation Instructor: Section # INSTRUCTIONS: This exam contains 25 multiplechoice questions plus 2 extra credit questions,
More informationRotational Kinematics and Dynamics. UCVTS AIT Physics
Rotational Kinematics and Dynamics UCVTS AIT Physics Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position,
More informationWe define angular displacement, θ, and angular velocity, ω. What's a radian?
We define angular displacement, θ, and angular velocity, ω Units: θ = rad ω = rad/s What's a radian? Radian is the ratio between the length of an arc and its radius note: counterclockwise is + clockwise
More informationAP Physics 1: Rotational Motion & Dynamics: Problem Set
AP Physics 1: Rotational Motion & Dynamics: Problem Set I. Axis of Rotation and Angular Properties 1. How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? 2. How many degrees are
More informationRolling, Torque & Angular Momentum
PHYS 101 Previous Exam Problems CHAPTER 11 Rolling, Torque & Angular Momentum Rolling motion Torque Angular momentum Conservation of angular momentum 1. A uniform hoop (ring) is rolling smoothly from the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Common Quiz Mistakes / Practice for Final Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A ball is thrown directly upward and experiences
More informationEndofChapter Exercises
EndofChapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. Figure 11.21 shows four different cases involving a
More informationCentripetal acceleration ac = to2r Kinetic energy of rotation KE, = \lto2. Moment of inertia. / = mr2 Newton's second law for rotational motion t = la
The Language of Physics Angular displacement The angle that a body rotates through while in rotational motion (p. 241). Angular velocity The change in the angular displacement of a rotating body about
More informationTOPIC B: MOMENTUM EXAMPLES SPRING 2019
TOPIC B: MOMENTUM EXAMPLES SPRING 2019 (Take g = 9.81 m s 2 ). ForceMomentum Q1. (Meriam and Kraige) Calculate the vertical acceleration of the 50 cylinder for each of the two cases illustrated. Neglect
More informationTwoDimensional Rotational Kinematics
TwoDimensional Rotational Kinematics Rigid Bodies A rigid body is an extended object in which the distance between any two points in the object is constant in time. Springs or human bodies are nonrigid
More information31 ROTATIONAL KINEMATICS
31 ROTATIONAL KINEMATICS 1. Compare and contrast circular motion and rotation? Address the following Which involves an object and which involves a system? Does an object/system in circular motion have
More informationRolling, Torque, and Angular Momentum
AP Physics C Rolling, Torque, and Angular Momentum Introduction: Rolling: In the last unit we studied the rotation of a rigid body about a fixed axis. We will now extend our study to include cases where
More informationChapter 8 Lecture Notes
Chapter 8 Lecture Notes Physics 2414  Strauss Formulas: v = l / t = r θ / t = rω a T = v / t = r ω / t =rα a C = v 2 /r = ω 2 r ω = ω 0 + αt θ = ω 0 t +(1/2)αt 2 θ = (1/2)(ω 0 +ω)t ω 2 = ω 0 2 +2αθ τ
More informationPhysics 12. Unit 5 Circular Motion and Gravitation Part 1
Physics 12 Unit 5 Circular Motion and Gravitation Part 1 1. Nonlinear motions According to the Newton s first law, an object remains its tendency of motion as long as there is no external force acting
More informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from
More informationDYNAMICS ME HOMEWORK PROBLEM SETS
DYNAMICS ME 34010 HOMEWORK PROBLEM SETS Mahmoud M. Safadi 1, M.B. Rubin 2 1 safadi@technion.ac.il, 2 mbrubin@technion.ac.il Faculty of Mechanical Engineering Technion Israel Institute of Technology Spring
More informationChapter 910 Test Review
Chapter 910 Test Review Chapter Summary 9.2. The Second Condition for Equilibrium Explain torque and the factors on which it depends. Describe the role of torque in rotational mechanics. 10.1. Angular
More informationAP Physics QUIZ Chapters 10
Name: 1. Torque is the rotational analogue of (A) Kinetic Energy (B) Linear Momentum (C) Acceleration (D) Force (E) Mass A 5kilogram sphere is connected to a 10kilogram sphere by a rigid rod of negligible
More informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from
More information5. Plane Kinetics of Rigid Bodies
5. Plane Kinetics of Rigid Bodies 5.1 Mass moments of inertia 5.2 General equations of motion 5.3 Translation 5.4 Fixed axis rotation 5.5 General plane motion 5.6 Work and energy relations 5.7 Impulse
More informationChapter 9 [ Edit ] Ladybugs on a Rotating Disk. v = ωr, where r is the distance between the object and the axis of rotation. Chapter 9. Part A.
Chapter 9 [ Edit ] Chapter 9 Overview Summary View Diagnostics View Print View with Answers Due: 11:59pm on Sunday, October 30, 2016 To understand how points are awarded, read the Grading Policy for this
More informationConcept Question: Normal Force
Concept Question: Normal Force Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is 1. larger than 2. identical
More informationPhys101 Third Major161 Zero Version Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1
Coordinator: Dr. Ayman S. ElSaid Monday, December 19, 2016 Page: 1 Q1. A water molecule (H 2 O) consists of an oxygen (O) atom of mass 16m and two hydrogen (H) atoms, each of mass m, bound to it (see
More informationChapter 10. Rotation of a Rigid Object about a Fixed Axis
Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small
More informationb) 2/3 MR 2 c) 3/4MR 2 d) 2/5MR 2
Rotational Motion 1) The diameter of a flywheel increases by 1%. What will be percentage increase in moment of inertia about axis of symmetry a) 2% b) 4% c) 1% d) 0.5% 2) Two rings of the same radius and
More informationPHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011
PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a righthanded Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this
More informationRotational Motion and Torque
Rotational Motion and Torque Introduction to Angular Quantities Sections 8 to 82 Introduction Rotational motion deals with spinning objects, or objects rotating around some point. Rotational motion is
More informationForce, Energy & Periodic Motion. Preparation for unit test
Force, Energy & Periodic Motion Preparation for unit test Summary of assessment standards (Unit assessment standard only) In the unit test you can expect to be asked at least one question on each subskill.
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Assignment 8 Textbook (Giancoli, 6 th edition), Chapter 78: Due on Thursday, November 13, 2008  Problem 28  page 189 of the textbook  Problem 40  page 190 of the textbook
More information第 1 頁, 共 7 頁 Chap10 1. Test Bank, Question 3 One revolution per minute is about: 0.0524 rad/s 0.105 rad/s 0.95 rad/s 1.57 rad/s 6.28 rad/s 2. *Chapter 10, Problem 8 The angular acceleration of a wheel
More informationAngular Displacement. θ i. 1rev = 360 = 2π rads. = "angular displacement" Δθ = θ f. π = circumference. diameter
Rotational Motion Angular Displacement π = circumference diameter π = circumference 2 radius circumference = 2πr Arc length s = rθ, (where θ in radians) θ 1rev = 360 = 2π rads Δθ = θ f θ i = "angular displacement"
More informationMOTION OF SYSTEM OF PARTICLES AND RIGID BODY CONCEPTS..Centre of mass of a body is a point where the entire mass of the body can be supposed to be concentrated For a system of nparticles, the centre of
More informationRotational Motion About a Fixed Axis
Rotational Motion About a Fixed Axis Vocabulary rigid body axis of rotation radian average angular velocity instantaneous angular average angular Instantaneous angular frequency velocity acceleration acceleration
More informationHandout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum
Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion Torque and angular momentum In Figure, in order to turn a rod about a fixed hinge at one end, a force F is applied at a
More informationPhysics for Scientist and Engineers third edition Rotational Motion About a Fixed Axis Problems
A particular bird s eye can just distinguish objects that subtend an angle no smaller than about 3 E 4 rad, A) How many degrees is this B) How small an object can the bird just distinguish when flying
More informationWiley Plus. Final Assignment (5) Is Due Today: Before 11 pm!
Wiley Plus Final Assignment (5) Is Due Today: Before 11 pm! Final Exam Review December 9, 009 3 What about vector subtraction? Suppose you are given the vector relation A B C RULE: The resultant vector
More informationChapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:
linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)
More informationCHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY
CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY OUTLINE 1. Angular Position, Velocity, and Acceleration 2. Rotational
More information3. Kinetics of Particles
3. Kinetics of Particles 3.1 Force, Mass and Acceleration 3.3 Impulse and Momentum 3.4 Impact 1 3.1 Force, Mass and Acceleration We draw two important conclusions from the results of the experiments. First,
More informationThe University of Melbourne Engineering Mechanics
The University of Melbourne 436291 Engineering Mechanics Tutorial Eleven Instantaneous Centre and General Motion Part A (Introductory) 1. (Problem 5/93 from Meriam and Kraige  Dynamics) For the instant
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 4 MOMENT OF INERTIA. On completion of this tutorial you should be able to
ENGINEEING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D5 TUTOIAL 4 MOMENT OF INETIA On completion of this tutorial you should be able to evise angular motion. Define and derive the moment of inertia of a body.
More informationJNTU World. Subject Code: R13110/R13
Set No  1 I B. Tech I Semester Regular Examinations Feb./Mar.  2014 ENGINEERING MECHANICS (Common to CE, ME, CSE, PCE, IT, Chem E, Aero E, AME, Min E, PE, Metal E) Time: 3 hours Max. Marks: 70 Question
More informationUse the following to answer question 1:
Use the following to answer question 1: On an amusement park ride, passengers are seated in a horizontal circle of radius 7.5 m. The seats begin from rest and are uniformly accelerated for 21 seconds to
More informationAP practice ch 78 Multiple Choice
AP practice ch 78 Multiple Choice 1. A spool of thread has an average radius of 1.00 cm. If the spool contains 62.8 m of thread, how many turns of thread are on the spool? "Average radius" allows us to
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More information1.1. Rotational Kinematics Description Of Motion Of A Rotating Body
PHY 19 PHYSICS III 1. Moment Of Inertia 1.1. Rotational Kinematics Description Of Motion Of A Rotating Body 1.1.1. Linear Kinematics Consider the case of linear kinematics; it concerns the description
More information16. Rotational Dynamics
6. Rotational Dynamics A Overview In this unit we will address examples that combine both translational and rotational motion. We will find that we will need both Newton s second law and the rotational
More informationPhysics 201 Exam 3 (Monday, November 5) Fall 2012 (Saslow)
Physics 201 Exam 3 (Monday, November 5) Fall 2012 (Saslow) Name (printed) Lab Section(+2 pts) Name (signed as on ID) Multiple choice Section. Circle the correct answer. No work need be shown and no partial
More informationENGINEERING COUNCIL CERTIFICATE LEVEL MECHANICAL AND STRUCTURAL ENGINEERING C105 TUTORIAL 13  MOMENT OF INERTIA
ENGINEERING COUNCIL CERTIFICATE LEVEL MECHANICAL AND STRUCTURAL ENGINEERING C15 TUTORIAL 1  MOMENT OF INERTIA This tutorial covers essential material for this exam. On completion of this tutorial you
More informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003
FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003 NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 14 pages. Make sure none are missing 2. There is
More informationReview questions. Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right.
Review questions Before the collision, 70 kg ball is stationary. Afterward, the 30 kg ball is stationary and 70 kg ball is moving to the right. 30 kg 70 kg v (a) Is this collision elastic? (b) Find the
More informationHonors Physics Review
Honors Physics Review Work, Power, & Energy (Chapter 5) o Free Body [Force] Diagrams Energy Work Kinetic energy Gravitational Potential Energy (using g = 9.81 m/s 2 ) Elastic Potential Energy Hooke s Law
More informationName:. Set:. Don: Physics: PreU Revision Toytime Rotational and Circular Motion
Name:. Set:. Don: Physics: PreU Revision Toytime 201516 Rotational and Circular Motion 1 19 (ii) Place ticks in the table below to identify the effect on waves of light as they refract from diamond into
More informationGeneral Physics 1. School of Science, University of Tehran Fall Exercises (set 07)
General Physics 1 School of Science, University of Tehran Fall 139697 Exercises (set 07) 1. In Fig., wheel A of radius r A 10cm is coupled by belt B to wheel C of radius r C 25 cm. The angular speed of
More informationPLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)
PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when
More informationUNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics
UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics Physics 111.6 MIDTERM TEST #2 November 15, 2001 Time: 90 minutes NAME: STUDENT NO.: (Last) Please Print (Given) LECTURE SECTION
More informationSYSTEM OF PARTICLES AND ROTATIONAL MOTION
Chapter Seven SYSTEM OF PARTICLES AND ROTATIONAL MOTION MCQ I 7.1 For which of the following does the centre of mass lie outside the body? (a) A pencil (b) A shotput (c) A dice (d) A bangle 7. Which of
More informationAdvanced Higher Physics. Rotational Motion
Wallace Hall Academy Physics Department Advanced Higher Physics Rotational Motion Solutions AH Physics: Rotational Motion Problems Solutions Page 1 013 TUTORIAL 1.0 Equations of motion 1. (a) v = ds, ds
More informationω avg [between t 1 and t 2 ] = ω(t 1) + ω(t 2 ) 2
PHY 302 K. Solutions for problem set #9. Textbook problem 7.10: For linear motion at constant acceleration a, average velocity during some time interval from t 1 to t 2 is the average of the velocities
More informationChapter 10.A. Rotation of Rigid Bodies
Chapter 10.A Rotation of Rigid Bodies P. Lam 7_23_2018 Learning Goals for Chapter 10.1 Understand the equations govern rotational kinematics, and know how to apply them. Understand the physical meanings
More informationA) 4.0 m/s B) 5.0 m/s C) 0 m/s D) 3.0 m/s E) 2.0 m/s. Ans: Q2.
Coordinator: Dr. W. AlBasheer Thursday, July 30, 2015 Page: 1 Q1. A constant force F ( 7.0ˆ i 2.0 ˆj ) N acts on a 2.0 kg block, initially at rest, on a frictionless horizontal surface. If the force causes
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Two men, Joel and Jerry, push against a wall. Jerry stops after 10 min, while Joel is
More informationPhysics 201 Midterm Exam 3
Physics 201 Midterm Exam 3 Information and Instructions Student ID Number: Section Number: TA Name: Please fill in all the information above. Please write and bubble your Name and Student Id number on
More informationPHYS 1303 Final Exam Example Questions
PHYS 1303 Final Exam Example Questions 1.Which quantity can be converted from the English system to the metric system by the conversion factor 5280 mi f 12 f in 2.54 cm 1 in 1 m 100 cm 1 3600 h? s a. feet
More informationClass XI Chapter 7 System of Particles and Rotational Motion Physics
Page 178 Question 7.1: Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie
More informationDescription: Using conservation of energy, find the final velocity of a "yo yo" as it unwinds under the influence of gravity.
Chapter 10 [ Edit ] Overview Summary View Diagnostics View Print View with Answers Chapter 10 Due: 11:59pm on Sunday, November 6, 2016 To understand how points are awarded, read the Grading Policy for
More informationPhysics for Scientists and Engineers 4th Edition, 2017
A Correlation of Physics for Scientists and Engineers 4th Edition, 2017 To the AP Physics C: Mechanics Course Descriptions AP is a trademark registered and/or owned by the College Board, which was not
More informationif the initial displacement and velocities are zero each. [ ] PARTB
Set No  1 I. Tech II Semester Regular Examinations ugust  2014 ENGINEERING MECHNICS (Common to ECE, EEE, EIE, iotech, E Com.E, gri. E) Time: 3 hours Max. Marks: 70 Question Paper Consists of Part and
More informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym
FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 13 pages. Make sure none are missing 2.
More informationPhysics 211 Sample Questions for Exam IV Spring 2013
Each Exam usually consists of 10 Multiple choice questions which are conceptual in nature. They are often based upon the assigned thought questions from the homework. There are also 4 problems in each
More informationMoment of Inertia Race
Review Two points, A and B, are on a disk that rotates with a uniform speed about an axis. Point A is closer to the axis than point B. Which of the following is NOT true? 1. Point B has the greater tangential
More informationDynamics of Rotation
Dynamics of Rotation 1 Dynamic of Rotation Angular velocity and acceleration are denoted ω and α respectively and have units of rad/s and rad/s. Relationship between Linear and Angular Motions We can show
More information