# Unit 8 Notetaking Guide Torque and Rotational Motion

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1 Unit 8 Notetaking Guide Torque and Rotational Motion

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3 Rotational Motion Until now, we have been concerned mainly with translational motion. We discussed the kinematics and dynamics of translational motion (the role of force). We also discussed the energy and momentum for translational motion. In this unit we will deal with rotational motion. We will consider mainly the rotation of rigid objects about a fixed axis. A rigid object is an object with a definite shape that doesn t change, so that the particles composing it stay in fixed positions relative to one another. 8-1 Angular Quantities Purely rotational motion means that all points in the object move in circles and that the centers of these circles all lie on one line called the axis of rotation. To indicate the angular position of a rotating object, or how far it has rotated, we specify the angle Ɵ of some particular line in the object with respect to a reference line. Angles are commonly measured in degrees, but the mathematics of circular motion is much simpler if we use the radian for angular measure. Where r is the radius of the circle, and l is the arc length subtended by the angle Ɵ specified in radians. Ɵ = l r Radians can be related to degrees in the following way. In a complete circle there are 360º, which must correspond to an arc length equal to the circumference of the circle, l=2πr. For a full circle, Ɵ= l = 2πr/r=2πrad. Thus, 360º = 2πrad=1 rev. r Since the object makes one complete revolution (rev) has rotated through 360º, or 2πradians. Example 8-1: Bike wheel. A bike wheel rotates 4.50 revolutions. How many radians has it rotated?

4 Example 8-2: Birds of prey in radians. A particular bird s eye can just distinguish objects that subtend an angle no smaller than about 3 x 10-4 rad. A) How many degrees is this? B) How small an object can the bird just distinguish when flying at a height of 100m? To describe rotational motion, we make use of angular quantities, such as angular velocity and angular acceleration. These are defined in analogy to the corresponding quantities in linear motion, and are chosen to describe the rotating object as a whole, so they are the same for each point in the rotating object. Instead of linear displacement, we use the angular displacement. Thus, the average angular velocity of an object rotating about a fixed axis is defined as Average angular velocity The instantaneous angular velocity is the limit of this ratio as Δt approaches zero: Instantaneous angular velocity Note that all points in a rigid object rotate with the same angular velocity, since every position in the object moves through the same angle in the same time interval.

5 Angular acceleration is defined as the change in angular velcoitu divided by the time required to make this change. The average angular acceleration is defined as Average angular acceleration Instantaneous angular velocity Instantaneous angular acceleration is defined as the limit of this ratio as Δt approaches zero: Instantaneous angular acceleration Since ώ is the same for all points of a rotating object, ά also will be the same for all points. Thus ά and ώ are properties of the rotating object as a whole. Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related: Therefore, objects farther from the axis of rotation will move faster. If the angular velocity of a rotating object changes, it has a tangential acceleration:

6 Even if the angular velocity is constant, each point on the object has a centripetal acceleration: Conceptual Example 8-3 Is the lion faster than the horse? On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center. A) Which child has the greater linear velocity? B) Which child has the greater angular velocity?

7 Example 8-4 Angular and linear velocities. A carousel is initially at rest. At t=0 it is given a constant angular acceleration ά=0.060rad/s 2, which increases its angular velocity for 8.0s. At t=8.0s, determine a) the angular velocity of the carousel, and b) the linear velocity of a child coated 2.5m from the center, point P.

8 Example 8-5 Angular and linear accelerations. For the child on the rotating carousel of Example 8-4, determine that child s a) tangential (linear) acceleration, b) centripetal acceleration, c) total acceleration. a) b) c) The frequency is the number of complete revolutions per second: f = ω/2π Frequencies are measured in hertz. 1 Hz = 1 s 1 The period is the time one revolution takes:

9 In example 8-4 we found that the carousel, after 8.0s, rotates at an angular velocity ω=0.48 rad/s, and continues to do so after t=8.0s because the acceleration ceased. What are the frequency and period of the carousel when rotating at this constant angular velocity ω=0.48 rad/s? 8-2 Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

10 Example 8-6 Centrifuge acceleration. A centrifuge rotor is accelerated for 30s from rest to 20,000rpm (revolutions per minute). A) What is the average angular acceleration? B) Through how may revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration? 8-3 Rolling Motion (Without Slipping) In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v. In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity v. Relationship between linear and angular speeds: v = rω

11 Example 8-7 Bicycle. A bicycle slows down uniformly form v o =8.40m/s to rest over a distance of 115m. Each wheel and tire has an overall diameter of 68.0cm. Determine a) the angular velocity of the wheels at the initial instant (t=0); b) the total number of revolutions each wheel rotates before coming to rest; c) the angular acceleration of the wheel; and d) the time it took to come to a stop.

12 8-4 Torque To make an object start rotating, a force is needed; the position and direction of the force matter as well. The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm. A longer lever arm is very helpful in rotating objects. Here, the lever arm for F A is the distance from the knob to the hinge; the lever arm for F D is zero; and the lever arm for F C is as shown. The torque is defined as:

13 Example 8-8 Biceps torque. The biceps muscle exerts a vertical force on the lower arm, bent as shown in Figs a and b. For each case, calculate the torque about the axis of rotation through the elbow joint, assuming the muscle is attached 5.0cm from the elbow as shown. Two forces (F A = 20N and F B = 30N) are applied to a meterstick which can rotate about its left end. Force F B is applied perpendicularly at the midpoint. Which force exerts the greater torque: F A, F B, or both the same?

14 8-5 Rotational Dynamics; Torque and Rotational Inertia Knowing that F = ma, we see that τ = mr2α This is for a single point mass; what about an extended object? As the angular acceleration is the same for the whole object, we can write: The quantity I = Σmr2 is called the rotational inertia of an object (also called the moment of inertia). The distribution of mass matters here these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation. Example 8-9 Two weights on a bar: different axis, different I. Two small weights, of mass 5.0kg and 7.0kg, are mounted 4.0m apart on a light rod (whose mass can be ignored), as shown in figure. Calculate the moment of inertia of the system a) when rotated about an axis halfway between the weights (a), and b) when rotated about an axis 0.50m to the left of the 5.0kg mass(b).

15 The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation compare (f) and (g), for example. 8-6 Solving Problems in Rotational Dynamics 1. Draw a diagram. 2. Decide what the system comprises. 3. Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act. 4. Find the axis of rotation; calculate the torques around it. 5. Apply Newton s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step. 6. Apply Newton s second law for translation and other laws and principles as needed. 7. Solve. 8. Check your answer for units and correct order of magnitude.

16 Example 8-10 A heavy pulley. A 15.0N force (represented by F T ) is applied to a cord wrapped around a pulley of mass M=4.00kg and radius R=33.0cm. The pulley accelerates uniformly form rest to an angular speed of 30.0rad/s in 3.00s. If there is a frictional torque τ fr =1.10m*N at the axle, determine the moment of inertia of the pulley. The pulley rotates about its center. Example 8-11 Pulley and bucket. Consider again the pulley in Example But instead of a constant 15.0N force beign exerted on the cord, we now have a bucket of weight w=15.0n (mass m=w/g=1.53kg) hanging from the cord. We assume the cord has negligible mass and does not stretch or slip on the pulley. Calculate the angular acceleration ά of the pulley and the linear acceleration a of the bucket. Assume the same frictional torque τ fr =1.10m*N acts.

17 8-7 Rotational Kinetic Energy The kinetic energy of a rotating object is given by KE = Σ(½ mv2) By substituting the rotational quantities, we find that the rotational kinetic energy can be written: A object that has both translational and rotational motion also has both translational and rotational kinetic energy: When using conservation of energy, both rotational and translational kinetic energy must be taken into account. All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have. Work done by torque: The torque does work as it moves the wheel through an angle θ:

18 Example 8-12 Sphere rolling down an incline. What will be the speed of a solid sphere of mass M and radius R when it reaches the bottom of an incline if it starts from rest at a vertical height H and rolls without slipping? (Assume sufficient static friction so no slipping occurs: we will see shortly that static friction does no work.) Compare your result to that for an object sliding down a frictionless isncline. Example 8-13 Which is fastest? Several objects roll without slipping down an incline of vertical height H, all starting from rest at the same moment. The objects are a thin hoop or a plain wedding band), a spherical marble, a solid cylinder (a D-cell battery), and an empty soup can. In addition, a greased box slides down without friction. In what order do they reach the bottom of the incline?

19 8-8 Angular Momentum and Its Conservation In analogy with linear momentum, we can define angular momentum L: We can then write the total torque as being the rate of change of angular momentum. If the net torque on an object is zero, the total angular momentum is constant. Iω = I0ω0 = constant Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation: When a spinning figure skater pulls in her arms, her moment of inertia decreases; to conserve angular momentum, her angular velocity increases. Does her rotational kinetic energy also increase? If so, where does the energy come from?

20 Example 8-14 Clutch. A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed, in a piece of machinery. The two plates have masses M A =6.0kg and M B =9.0kg, with equal radii R=0.60m. They are initially separated. Plate M A is accelerated from rest to an angular velocity ώ 1 =7.2rad/s in time Δt=2.0s. Calculate a) the angular momentum of M A, and b) the torque required to accelerate M A from rest to ώ 1. c) Next, plate M B, initially at rest but free to rotate without friction, is placed in firm contact with freely rotating plate M A, and the two plates then both rotate at a constant angular velocity ώ 2, which is considerably less than ώ 1. Why does this happen, and what is ώ 2?

21 Example 8-15 Neutron star. Astronomers detect stars that are rotating extremely rapidly, known as neutron stars. A neutron star is believed to form from the inner core of a larger star that collapsed, under its own gravitation, to a star of very small radius and very high density. Before collapse, suppose the core of such a star is the size of our Sun (R 7 x 10 5 km) with mass 2.0 times as great as the Sun, and is rotating at a frequency of 1.0 revolution every 100 days. If it were to undergo gravitational collapse to a neutron star of radius 10 km, what would its rotation frequency be? Assume the star is a uniform sphere at all times, and loses no mass.

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