Conceptual Physics Rotational Motion

Transcription

1 Conceptual Physics Rotational Motion Lana Sheridan De Anza College July 11, 2015

2 Last time energy, mechanical advantage, efficiency energy sources discussion collisions

3 Overview circular motion and rotation centripetal force fictitious forces torque moment of inertia center of mass angular momentum introduce gravity (?)

4 Rotational Motion Objects can move through space, but they can have another kind of motion too: They can rotate about some axis. Examples of rotating objects: the Earth, makes a complete rotation once per day merry-go-rounds records / cds on a player

5 defining kinem To define angular position for the disc, a fixed reference here for rotatio line is chosen. A particle at P Figure 10.1 i Consider a marked point is located P onat the a distance disk. As r from time passesdisc it moves: rotates abo the rotation axis through O. Rotating disk a O r P Reference line The distance it moves As the is s, disc if θrotates, is measured a particle in at radians. P moves through an arc length s on a circular s = path rθ of radius r. The angular position of P is u. through the cen ticle at P is at a radius r. (In fac convenient to re the distance fro ence line fixed i changes in tim cle from the re length s as in F relationship 1 Figures from Serway & Jewett, 9th ed.

6 Rotating disk length s as in F relationship As the disc rotates, a particle at P moves through an arc length Consider a marked point s on a Pcircular on the path disk. of radius As time r. passes it moves: The angular position of P is u. b O r P u s Reference line The distance it moves Figure is s, 10.1 if θa iscompact measured disc in radians. rotating about a fixed axis through O perpendicular s = rθ to the plane of the figure. Pitfall Prevention Figures from Serway & Jewett, 9th ed. Because u is the ber. Usually, ho the angle subten cumference of a to an angle of ( angle in degree For example, 60 Because the

7 P moves through an arc length s on a circular path of radius r. The angular position of P is u. The angle that the disk rotates by is θ, in some amount of time t Angular speed O r P u s Reference line b the angular speed ( rotational speed ) of the disk is Figure 10.1 A compact disc rotating about a fixed axis angular through speed O perpendicular = to the plane of the figure. change in angle change in time Pitfall Prevention 10.1 Remember the Radian In rotational equations, you must use Because u is the ber. Usually, ho the angle subten cumference of a to an angle of ( angle in degree For example, 60 Because the angle u from th the same angle

8 P moves through an arc length s on a circular path of radius r. The angular position of P is u. The angle that the disk rotates by is θ, in some amount of time t Angular speed O r P u s Reference line b the angular speed ( rotational speed ) of the disk is Figure 10.1 A compact disc rotating about a fixed axis angular through speed O perpendicular = to the plane of the figure. change in angle change in time In notation: Pitfall Prevention 10.1 ω = θ Remember the Radian t In rotational equations, angular you speed. must where we let ω represent use Because u is the ber. Usually, ho the angle subten cumference of a to an angle of ( angle in degree For example, 60 Because the angle u from th the same angle

9 Angular speed The angular speed of the Earth s rotation is 2π per day or ω E = 2π 86, 400 s The units are radians per second. (Or just s 1.) We can also measure rotational speed in terms of the number of complete rotations in some amount of time. Records speeds are a good example of this. Typical angular speeds: RPM (called a 33 ) 45 RPM 78 RPM where RPM means rotations per minute.

10 Angular speed and Tangential speed The tangential speed of point P is its instantaneous speed. We write it as v because it is fundamentally the same thing we called speed before: speed = distance traveled change in time For the point P it travels a distance s in time t v = s t

11 Angular speed and Tangential speed But remember: s = rθ. We can write v = s t = rθ t

12 Angular speed and Tangential speed But remember: s = rθ. We can write v = s t = rθ t However, ω = θ t so we can make a relation between tangential speed v and angular speed ω: v = rω ( tangential speed = distance to axis angular speed )

13 Henry Lea Rolling Motion A rolling object moves along a surface as it rotates. Consider a wheel: anslational speed of (10.28) olds whenever a cylure rolling motion. ass for pure rolling R u s s Ru (10.29) Figure For pure rolling If the outside edge of the wheel does not slip on the surface, then motion, as the cylinder rotates there is a relation between through the an angle angular u its center speed of the wheel s t speed rotation v CM, staying and the speed moves thata linear the wheel distance itselfs 5 Ru. moves along. ass of the object. As Interestingly, it is also: n around its center Pitfall Prevention 10.6, center, and bottom v wheel = rω

14 Centripetal Force Now consider an object that is rotating about an axis. For example a puck on a string: 6.1 Extending the Particle in S A force F r, directed toward the center of the circle, keeps the puck moving in its circular path. S F r r m When the string breaks, the puck moves in the direction tangent to the circle. r S Fr Figure 6.1 An overhead view of a puck moving in a circular path in a horizontal plane. Figure 6.2 The string holding th puck in its circular path breaks.

15 Centripetal Force If an object moves on a circular path, its velocity must always be changing. It is accelerating. 1 Figures from Serway & Jewett.

16 Centripetal Force If an object moves on a circular path, its velocity must always be changing. It is accelerating. F net = ma F net 0 1 Figures from Serway & Jewett.

17 Centripetal Force If an object moves on a circular path, its velocity must always be changing. It is accelerating. F net = ma F net 0 s Laws Any object moving in a circular (or curved) path must be experiencing a force. sion) We call this the centripetal force. irled in a circle he Sun in a perfectly pter 13) 1 Figures from Serway & Jewett. F r S S a c v S

18 Uniform Circular Motion For an object moving in a circle at constant speed v, a = a c = v 2 = rω 2 r

19 Uniform Circular Motion For an object moving in a circle at constant speed v, a = a c = v 2 r = rω 2 This gives the expression for centripetal force! so, F = ma F c = mv 2 r

20 S Centripetal Force Something must provide this force: 6.1 Extending the Particle in S A force F r, directed toward the center of the circle, keeps the puck moving in its circular path. S F r r m When the string breaks, the puck moves in the direction tangent to the circle. r S Fr It could be tension in a rope. Figure 6.1 An overhead view of a puck moving in a circular path in a horizontal plane. Figure 6.2 The string holding th puck in its circular path breaks.

21 tion of the velocity vector will be smaller in a given time Centripetal Force ed tension in the string is smaller. As a result, the string radius. Something must provide this force: he Car? AM curve as shown fficient of static aximum speed S f s large circle so It could be friction. e model the car n. The car is not m in the vertical a S n

22 Centripetal Force Consider the example of a string constraining the motion 6.1 of Extending a puck: the Parti S A force F r, directed toward the center of the circle, keeps the puck moving in its circular path. S F r r m When the string breaks, the puck moves in the direction tangent to the circle. r S Fr Figure 6.1 An overhead view of a puck moving in a circular path in a horizontal plane. Figure 6.2 The string hold puck in its circular path brea

23 Centripetal Force Question. What will the puck do if the string breaks? (A) Fly radially outward. (B) Continue along the circle. (C) Move tangentially to the circle.

24 Centripetal Force Question. What will the puck do if the string breaks? (A) Fly radially outward. (B) Continue along the circle. (C) Move tangentially to the circle. 6.1 Extending the Particle in Uniform Circular Motion Model 15 S A force F r, directed toward the center of the circle, keeps the puck moving in its circular path. S F r r m When the string breaks, the puck moves in the direction tangent to the circle. r S Fr S v Figure 6.1 An overhead view of a puck moving in a circular path in a horizontal plane. Figure 6.2 The string holding the puck in its circular path breaks.

25 Orbits A centripetal force also holds Earth in orbit around the Sun. What is the force due to? 1 Figure from EarthSky.org.

26 A Fictitious Force: Centrifugal force fictitious fictional. otion in Accelerated Frames 159 The centrifugal force is the force that makes you feel sucked to the outside in a turn: rce such observaact on an between us force. ) In gene accelernd there he train. a

27 nd car s or on ove ard cirhe ion ent ure A Fictitious her toward Force: the Centrifugal right door, but force it is a fictitious force. Fictitious force ers b

28 nsion in the string is smaller. As a result, the string us. A Fictitious Force: Centrifugal force ar? AM e as shown nt of static um speed S f s e circle so a del the car e car is not S n

29 to change direction along with The real forcethe is Centripetal rest of the car. force of gh, the ight-line f view of he seat. enable nt force system. circular on page server is Real force c

30 Rotation and Force Question Two pennies are place on a circular rotating platform, one near to the center, the other, towards the outside rim. The platform starts at rest and is slowly spun faster and faster (increasing angular speed). Which penny slides off the platform first? (A) The one near the center. (B) The one near the rim.

31 Rotating reference frame If you are in a rotating frame, you can describe your world as if it is at rest by adding a fictitious outward centrifugal force to your physics. You can use this to simulate gravity: for example, in rotating space stations, eg. in the films 2001, Elysium, Interstellar.

32 Rotating reference frame If you are in a rotating frame, you can describe your world as if it is at rest by adding a fictitious outward centrifugal force to your physics. You can use this to simulate gravity: for example, in rotating space stations, eg. in the films 2001, Elysium, Interstellar. Water in a bucket...

33 Chapter 6 Circular Motion and Other Applications of Newton s Laws Rotating Frames: Coriolis Force By the time t f that the ball arrives at the other side of the platform, your friend is no longer there to catch it. According to this observer, the ball follows a straight-line path, consistent with Newton s laws. From your friend s point of view, the ball veers to There is another fictitious force thatone non-inertial side during its flight. observers Your friend see introduces in a rotating frame. fictitious force to explain this deviation from the expected path. Friend at t 0 Friend at t t f Ball at t t f a You at t t f You at t 0 b Ball at t 0 The Coriolis force appears as a fictitious sideways force to a non-inertial observer. Figure 6.11 You and your friend stand at the edge of a rotating circular platform. You throw t ball at t 5 0 in the direction of your friend. (a) Overhead view observed by someone in an inerti erence frame attached to the Earth. The ground appears stationary, and the platform rotates cl wise. (b) Overhead view observed by someone in an inertial reference frame attached to the pla

34 Rotating Frames: Coriolis Force We can detect the effect of Earth s rotation: they manifest as Coriolis effects. eg. ocean surface currents: 1 Figure from

35 Torque Torque is a forcethe causing component a rotation. F sin f tends to rotate the wrench about Torque an axis = lever through arm O. force O d r S r f F sin f Figure 10.7 The force F S has a f S F F cos f Line of action Torque is written τ ( tau ) greater rotating tendency about an axis through τ = d O F as = F (r increases sin φ)f and 10.4 Tor In our study o we studied the What is the ca Imagine try ular to the do hinges. You wi force near the When a for to rotate abou axis is measur but we will co Chapter 11. Consider th

36 Torque Torque = lever arm force τ = (r sin φ)f = d F The units of torque are N m (Newton meters).

37 Torque is not Work O The component F sin f tends to rotate the wrench about an axis through O. d r S r f F sin f f S F F cos f Line of action 10.4 Torque In our study of translation we studied the cause of c What is the cause of chang Imagine trying to rotat ular to the door surface n hinges. You will achieve a force near the doorknob t When a force is exerted to rotate about that axis. axis is measured by a quan but we will consider only Chapter 11. Consider the wrench in perpendicular to the page If we take the lever Figure arm to10.7 be The d, as force in S Fd has = a r sin φ, then we can greater rotating tendency about an write: axis through O as F increases and as the moment τ arm = d increases. F This looks a lot like the formula for work: W = Fd cos θ. Work is measured in Joules, and 1 J = 1 N 1 m. But torque is not the same thing at all!

38 Torque is not Work O The component F sin f tends to rotate the wrench about an axis through O. d r S r f F sin f f S F F cos f Line of action 10.4 Torque In our study of transla we studied the cause What is the cause of c Imagine trying to r ular to the door surf hinges. You will achie force near the doorkn When a force is exe to rotate about that a axis is measured by a but we will consider o Chapter 11. Consider the wrenc perpendicular to the Figure 10.7 The force S F has a In the diagram, the greater forcerotating does not tendency moveabout the an end of wrench in the direction of r. axis through O as F increases and as the moment arm d increases. It moves the end of the wrench perpendicular to r. This causes a rotation. 1 Figures from Serway & Jewett.

39 Torque is a Force causing a Rotation Technically, torque is a vector: the direction of the vector tells us whether 296 the rotation Chapter will 10 be Rotation clockwise of a Rigid or Object counterclockwise. About a Fixed Axis So, to keep it separate: units of torque: N m v S units of work or energy: J v S direct is cou wise. T strated direct direct v S if th direction of v S speed for the particle is out of the plane of the is counterclockwise and into the plane of the diagram wise. To illustrate this convention, it is convenient to us strated in Figure When the four fingers of the rig direction of rotation, the extended right thumb points direction of S a follows v S from its definition S a ; dv S In /dt. ouri v S if the angular speed is increasing in time, and it consid is an speed is decreasing Figure in 10.3 time. The right-hand rule for determining the direction of the angular velocity vector same p Ima and h 10.2 Analysis Model: Rigid Object partic Constant Angular Acceleratio tional In our study of translational motion, after introducing develo considered the special case of a particle under constant 10.5 in Figure 10.3 The right-hand rule same procedure here for a rigid object under constant for determining the direction of the Rotational kinematic Imagine a rigid object such as the CD in Figure 10. angular velocity vector. equations and has a constant angular acceleration. In parallel where wit 1 Figures from Serway & Jewett. particle under constant acceleration, we generate us to a nf v S

40 Question B A A torque is supplied by applying a force at point A. To produce the same torque, the force applied at point B must be: (A) greater (B) less (C) the same 1 Image from Harbor Freight Tools,

41 Moment of Inertia A solid, rigid object like a ball, a record (disk), a brick, etc. has a moment of inertia. Moment of inertia, I, is similar to mass. A net torque causes an object to rotate, and moment of inertia measures the object s resistance to rotation. Mass behaves the same way! A net force causes the motion (acceleration) of an object and the mass measures the object s resistance to changes in its motion.

42 Moment of Inertia If the object s mass is far from the point of rotation, more torque is needed to rotate the object (with some angular acceleration). The barbell on the right has a greater moment of inertia. 1 Diagram from Dr. Hunter s page at (by Hewitt?)

43 Table 10.2 Moments of Inertia of Homogeneous Rigid Objects Different shapes have different Moments of Inertia with Different Geometries Hoop or thin cylindrical shell I CM MR 2 R Hollow cylinder 1 I CM M(R R 2 2 ) R 1 R 2 Solid cylinder or disk 1 2 I CM MR 2 R Rectangular plate 1 I CM M(a 2 b 2 ) 12 b a Long, thin rod with rotation axis through center 1 I CM ML 2 12 L Long, thin rod with rotation axis through end 1 I 3 ML2 L Solid sphere 2 I CM MR 2 5 R Thin spherical shell 2 I CM MR 2 3 R

44 Different shapes have different Moments of Inertia If an object changes shape, its moment of inertia can change also.

45 Center of Mass For a solid, rigid object: center of mass the point on an object where we can model all the mass as being, in order to find the object s trajectory; a freely moving object rotates about this point

46 Center of Mass For a solid, rigid object: center of mass the point on an object where we can model all the mass as being, in order to find the object s trajectory; a freely moving object rotates about this point The center of mass of the wrench follows a straight line as the wrench rotates about that point.

47 Center of Mass 1 Figure from

48 Center of Mass Questions Where is the center of mass of this pencil? (A) Location A. (B) Location B. (C) Location C. (D) Location D. 1 Pencil picture from kingofwallpapers.com.

49 Center of Mass Questions Where is the center of mass of this hammer? (A) (B) (C) (D) 1 Location Location Location Location A. B. C. D. Hammer picture from pngimg.com.

50 Center of Mass Questions Where is the center of mass of this boomerang? (A) Location A. (B) Location B. (C) Location C. (D) Location D. 1 Boomerang picture from

51 Center of Gravity The center of gravity is the single point on an object where we can model the force of gravity as acting on the object. Near the surface of the Earth, the Earth s gravitational field is uniform, so this is the same as the center of mass. The center of gravity is the point at which you can balance an object on a single point of support. 1 Figure from

52 Center of Mass vs Center of Gravity For a solid, rigid object: center of mass the point on an object where we can model all the mass as being, in order to find the object s trajectory; a freely moving object rotates about this point center of gravity the single point on an object where we can model the force of gravity as acting on the object; the point at which you can balance the object.

53 Angular Momentum Angular momentum is a rotational version of momentum. Remember momentum = mass velocity or p = mv. Angular momentum, L, can be defined in a similar way: angular momentum = moment of inertia angular velocity L = Iω

54 he were skating directly at the instant she is a disng a straight The angular path that momentum is of a small object, with mass m, moving in line has angular momentum about any axis displaced from mvd (c) a circle mva (d) is: impos- the path of the particle. L = mvr lar momentum even if the particle is not moving in a circular path. A particle moving in a straight determine Angular (ii) Momentum: What special case This is the linear momentum mv times the distance from the icle in center Circular of the Motion circle r. ius r as shown in Figure entum relative to an axis y S v re 11.5 (Example 11.3) A cle moving in a circle of radius r n angular momentum about an through O that has magnitude The vector L S 5 r S 3 p S points f the page. O S r m x 1 Figure from Serway & Jewett, 9th ed. continued

55 Angular Momentum is Conserved The momentum of a system is constant (does not change) unless the system is acted upon by an external force. A similar rule holds for angular momentum: The angular momentum of a system does not change unless it acted upon by an external torque.

56 Angular Momentum is Conserved The angular momentum of a system does not change unless it acted upon by an external torque. This means L = 0 and so L i = L f.

57 Angular Momentum is Conserved The angular momentum of a system does not change unless it acted upon by an external torque. This means L = 0 and so L i = L f. Suppose an object is changing shape, so that its moment of inertia gets smaller: I f < I i. L i = L f I i ω i = I f ω f That means the angular speed increases! ω f > ω i

58 direc in em out th The conservation of angular momentum can be used to orient setup spacecraft. Le 352 Chapter 11 Angular Momentum a tion o as in The angular When the speed gyroscope v p is called the cente prec turns counterclockwise, only when v p,, v. Otherwise, a much scope m the spacecraft turns you can see from Equation 11.20, the system clockwise. con that is, when the wheel spins rapidly. rema Fu frequency decreases as v increases, that zero i of symmetry. of the As an example of the usefulness of space gyro deep space and you need to alter your rotat traj inclu direction, you need to turn the spacecraft tion i in empty space? One way is to have small Th out the side of the spacecraft, providing a its fli setup is desirable, and many spacecraft ha Each Let us consider another method, howe a space tion of rocket b fuel. Suppose the spacecraf had t as in Figure 11.15a. In this case, the angul 1 Figures from Serway & Jewett, 9th ed. Figure (a) A spacecraft the ro Conservation of Angular Momentum

59 Conservation of Angular Momentum The conservation of angular momentum also makes tops and gyroscopes stable when rotating. 1 From

60 Gravity

61 Motion of the stars, planets, and falling apples Part of Newton s genius was to realize that the same force that dictates the motion of the stars and planets is what holds us on the Earth. This realization is called the Newtonian synthesis. The planets are also falling, but they are constantly falling around the Sun.

62 Law of Gravitation Newton s Universal Law of Gravitation states that any two massive object in the universe interact with each other according to the same rule. They attract each other with a force that depends on the two masses and the distance between their centers: Force mass 1 mass 2 distance distance As an equation: (where G is a constant.) F = G m 1m 2 d 2

63 The Universal Gravitational Constant, G F = G m 1m 2 d 2 G sets the scale of the force due to gravity (and makes the units come out correctly). G = N m 2 /kg 2

64 The Universal Gravitational Constant, G F = G m 1m 2 d 2 G sets the scale of the force due to gravity (and makes the units come out correctly). This could also be written: so it is quite a small number. G = N m 2 /kg 2 G = N m 2 /kg 2

65 The Universal Gravitational Constant, G The fact that G is so small indicates that gravity is a weak force. Force Rel. strength Range (m) Attract/Repel Gravitational attractive Electromagnetic 10 2 attr. & rep. Weak Nuclear < attr. & rep. Strong Nuclear 1 < attr. & rep.

66 The Universal Gravitational Constant, G The fact that G is so small indicates that gravity is a weak force. Force Rel. strength Range (m) Attract/Repel Gravitational attractive Electromagnetic 10 2 attr. & rep. Weak Nuclear < attr. & rep. Strong Nuclear 1 < attr. & rep.

67 The Universal Gravitational Constant, G The fact that G is so small indicates that gravity is a weak force. Force Rel. strength Range (m) Attract/Repel Gravitational attractive Electromagnetic 10 2 attr. & rep. Weak Nuclear < attr. & rep. Strong Nuclear 1 < attr. & rep. Despite the fact that gravity is a weak force, it is the only one that (typically) matters on large scales.

68 Measurement of G G was first measured by Henry Cavendish using a torsion balance. 1 Diagram from Wikipedia by Chris Burks.

69 Mass of the Earth Determining G allowed for a the mass of the Earth M E to be calculated. The weight of an object is the force on the object due to gravity, when the distance is the radius of the Earth. F = G mm E d 2 = mg G M E R 2 E = g M E = gr2 E G M E = kg

70 Mass of the Earth This is a very large number! M E = kg And yet, despite the fact that the force on you from the Earth depends on: F your mass the mass of the Earth you still have the strength to jump off the surface! So G must be very small, and the force of gravity very weak.

71 Value of g mg = G mm E d 2 g = G M E d 2 g takes the value 9.8 m s 2 at the Earth s surface, but it can vary with height: as d increases, g decreases.

72 Variation of g Center of the Earth Surface of the Earth Outside of the Earth 1 Figure from

73 Summary Circular motion / rotation torque moment of inertia center of mass angular momentum introducing gravity Essay Homework due July 13th. Homework Hewitt, Ch 8, onward from page 146. Exercises: 1, 3; Problems: 5. NEW: Ch 8, onward from page 145. Plug and Chug: 1, 3, 5. Exercises: 19, 39, 41