Kinetic Energy of Rolling

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Annabel McDaniel
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1 Kinetic Energy of Rolling A solid disk and a hoop (with the same mass and radius) are released from rest and roll down a ramp from a height h. Which one is moving faster at the bottom of the ramp? A. they have the same speed B. disk C. hoop

2 Kinetic Energy of Rolling A solid disk and a hoop (with the same mass and radius) are released from rest and roll down a ramp from a height h. Which one is moving faster at the bottom of the ramp? B. disk How much faster?

3 Moment of Inertia

4 Kinetic Energy of Rolling particle: disk: hoop: note: it doesn t matter if they have the same mass and radius

5 Vectors of Rotational Motion 1D motion uses scalar velocity, force, etc. more generally, we use vectors,, etc. for rotational motion the +/- signs on and really just components of vectors and direction defined by the right-hand rule: are

6 Angular Velocity Vector The magnitude of the angular velocity vector is ω. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated.

7 Angular Velocity Vector The Earth s angular velocity vector points A. out the north pole B. out the south pole C. something else

8 Torque Vector τ = rfsinϕ. r and F are the magnitudes of vectors This is a really a cross product:

9 Angular Momentum Angular momentum is the rotational analog of linear momentum Symbol:

10 Angular Momentum of a Particle A particle of mass m is moving. The particle s momentum vector makes an angle β with the position vector.

11 Angular Momentum of a Particle A particle of mass m is moving. The particle s momentum vector makes an angle β with the position vector.

12 Angular Momentum of a Particle Why this definition? take the time derivative, you find Torque causes angular momentum to change. This is the rotational equivalent of Newton s second law for rotation: (planar rotation) (generally)

13 Angular Momentum For a rigid body, add the angular momenta of all the particles forming the object. Then it can be shown that (for a symmetrical object) And it s still the case that

14 Analogy with Linear Dynamics

15 Conservation of Angular Momentum An isolated system that experiences no net torque has Then the angular momentum vector is a constant: where or

16 Conservation of Angular Momentum An isolated system that experiences no net torque has A spinning object will keep spinning until an outside torque affects it.

17 Conservation of Angular Momentum Consider a particle moving through space. It s momentum is constant. It s angular momentum about the origin is given by Is conserved in this case? A. yes B. no C. depends

18 Conservation of Angular Momentum As an ice skater spins, external torque is small, so her angular momentum is almost constant. bringing mass closer to the axis of rotation reduces her moment of inertia I. To conserve angular momentum, her angular speed ω must increase. = constant

19 Conservation of Angular Momentum

20 Conservation of Angular Momentum

21 Rotational Dynamics A rectangular object of uniform density ρ has dimensions It is made to spin at a rate ω. About which axis (through the CM) should it spin to have the greatest angular momentum? A. x B. y C. z

22 Rotational Dynamics A rectangular object of uniform density ρ has dimensions It is made to spin at a rate ω. About which axis (through the CM) should it spin to have the greatest angular momentum? B. y Iy is greatest.

23 Rotational Dynamics A rectangular object of uniform density ρ has dimensions It is made to spin at a rate ω. About which axis (through the CM) should it spin to have the least angular momentum? A. x B. y C. z

24 Rotational Dynamics A rectangular object of uniform density ρ has dimensions It is made to spin at a rate ω. About which axis (through the CM) should it spin to have the least angular momentum? A. x Ix is smallest

25 Moment of Inertia - example A rectangular object of uniform density ρ has dimensions Compute the moments of inertia about each axis,.

26 Rotational Dynamics Intermediate Axis Theorem: If angular momentum is conserved, rotation about an axis of intermediate (not largest or smallest) moment of inertia is unstable. (a.k.a. the tennis racket theorem ) This shape cannot rotate stably about the z axis. Try this with your phone!

27 Rotational Dynamics Intermediate Axis Theorem:

28 Rotational Dynamics Intermediate Axis Theorem: If angular momentum is conserved, rotation about an axis of intermediate (not largest or smallest) moment of inertia is unstable.

29 Conservation of Angular Momentum In the absence of external torques, angular momentum is conserved.

30 Conservation of Angular Momentum In the absence of external torques, angular momentum is conserved.

31 Conservation of Angular Momentum In the absence of external torques, angular momentum is conserved. Maneuvering an object in microgravity:

32 HW08

33 HW08

34 Gyroscopes

35 Gyroscopes Newton s 2nd Law: momentum changes in the direction of the applied force

36 Gyroscopes Newton s 2nd Law: momentum changes in the direction of the applied force changing momentum direction, not magnitude

37 Gyroscopes Newton s 2nd Law: momentum changes in the direction of the applied force Newton s 2nd Law for rotational motion: angular momentum changes in the direction of the applied torque

38 Gyroscopes Newton s 2nd Law for rotational motion: angular momentum changes in the direction of the applied torque

39 Gyroscopes Consider initial state at rest. The gravitational force acts on the center of mass of the wheel. What is the direction of the gravitational torque about the pivot point? A. z B. + z C. x D. + x E. + y

40 Gyroscopes Consider initial state at rest. The gravitational force acts on the center of mass of the wheel. What is the direction of the gravitational torque about the pivot point? A. z B. + z C. x D. + x E. + y

41 Gyroscopes Consider initial state at rest. The gravitational force acts on the center of mass of the wheel. The wheel falls, acquiring angular momentum in the direction of the torque.

42 Gyroscopes Now consider an initial state with fast spin. What is the direction of the initial angular momentum? A. + x B. x C. + y D. y E. + z

43 Gyroscopes Now consider an initial state with fast spin. What is the direction of the initial angular momentum? A. + x B. x C. + y D. y E. + z

44 Gyroscopes Initial state with fast spin. The gravitational torque about the pivot is the same as before. changing momentum direction, not magnitude

45 Gyroscopes Initial state with fast spin. The gravitational torque about the pivot is the same as before.

46 Gyroscopes Initial state with fast spin. The gravitational torque about the pivot is the same as before. Result is precession.

47 Gyroscopes A gyroscope is set up as shown. Which way will it precess? A. + z B. z Need to identify: due to gravity

48 Gyroscopes A gyroscope is set up as shown. Which way will it precess? A. + z B. z Need to identify: due to gravity

49 Gyroscope Precession This steady change in the orientation of the rotation axis is called precession, and we say that the gyroscope precesses about its point of support. The precession frequency Ω is much less that the disk s rotation frequency ω.

50 Gyroscope Precession View from above: precession frequency :

51 Gyroscope Precession precession frequency : where d is the moment arm of the gravitational torque

52 Gyroscope Precession precession frequency : where d is the moment arm of the gravitational torque

53 Gyroscope Precession A gyroscope can be made to precess when starting horizontally (1), or at an angle (2). Which will precess faster (for the same ω)? A. 1 B C. the same 2.

5/6/2018. Rolling Without Slipping. Rolling Without Slipping. QuickCheck 12.10

5/6/2018. Rolling Without Slipping. Rolling Without Slipping. QuickCheck 12.10 Rolling Without Slipping Rolling is a combination of rotation and translation. For an object that rolls without slipping, the translation of the center of mass is related to the angular velocity by Slide