Chapter 11 Rotational Dynamics and Static Equilibrium · b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with

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Chapter 11

Rotational Dynamics and Static Equilibrium


Units of Chapter 11• Torque

• Torque and Angular Acceleration

• Zero Torque and Static Equilibrium

• Center of Mass and Balance

• Dynamic Applications of Torque

• Angular Momentum

• Conservation of Angular Momentum

• Rotational Work and Power

• The Vector Nature of Rotational Motion


Review: 11-1 Torque

Only the tangential component of force causes a torque:


Review: 11-1 Torque

This leads to a more general definition of torque:


Review: 11-2 Torque and Angular Acceleration

Once again, we have analogies between linear and angular motion:


11-3 Zero Torque and Static Equilibrium

Static equilibrium occurs when an object is at rest – neither rotating nor translating.


11-3 Zero Torque and Static Equilibrium

If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; we are free to choose the one that makes our calculations easiest.


11-4 Center of Mass and BalanceIf an extended object is to be balanced, it must be supported through its center of mass.


11-4 Center of Mass and Balance

This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.

Question 11.10Question 11.10 Balancing RodBalancing Rod

1kg

1m

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

a) ¼ kg

b) ½ kg

c) 1 kg

d) 2 kg

e) 4 kg

1 kg

XCM of rod

same distance mROD = 1 kg

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

The total torque about the pivot The total torque about the pivot must be zero !!must be zero !! The CM of the rod is at its center, 0.25 m to the 0.25 m to the right of the pivotright of the pivot. Because this must balance the ball, which is the same distance to the left of same distance to the left of the pivotthe pivot, the masses must be the same !!

a) ¼ kg

b) ½ kg

c) 1 kg

d) 2 kg

e) 4 kg

Question 11.10Question 11.10 Balancing RodBalancing Rod

Question 11.12aQuestion 11.12a Tipping Over ITipping Over I

1 2 3

a) all

b) 1 only

c) 2 only

d) 3 only

e) 2 and 3

A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dotin each case. In which case(s) does the box tip over?

Question 11.12aQuestion 11.12a Tipping Over ITipping Over I

1 2 3

a) all

b) 1 only

c) 2 only

d) 3 only

e) 2 and 3

A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dotin each case. In which case(s) does the box tip over?

The torque due to gravity acts like all the mass of an object is concentrated at the CM. Consider the bottom right corner of the box to be a pivot point. If the box can rotate such that If the box can rotate such that the CM is lowered, it will!!the CM is lowered, it will!!


11-5 Dynamic Applications of TorqueWhen dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly.


Key: free body diagrams




11-6 Angular Momentum

Using a bit of algebra, we find for a particle moving in a circle of radius r,



For more general motion,





Looking at the rate at which angular momentum changes,




11-7 Conservation of Angular Momentum

If the net external torque on a system is zero, the angular momentum is conserved.

The most interesting consequences occur in systems that are able to change shape:


11-7 Conservation of Angular Momentum

As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change.

Angular momentum is also conserved in rotational collisions:

Question 11.7Question 11.7 Figure SkaterFigure Skater

a) the samea) the sameb) larger because sheb) larger because she’’s rotating s rotating

fasterfaster

c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller

A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be

Question 11.7Question 11.7 Figure SkaterFigure Skater

a)a) the samethe same

b)b) larger because shelarger because she’’s rotating s rotating fasterfaster

c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller

A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

KErot = I ω2 = L ω (used L = Iω ). Because L is conserved, larger ωmeans larger KErot. The “extra”energy comes from the work she does on her arms.

FollowFollow--upup:: Where does the extra energy come from?Where does the extra energy come from?

12

12




11-8 Rotational Work and Power

A torque acting through an angular displacement does work, just as a force acting through a distance does.

The work-energy theorem applies as usual.


11-8 Rotational Work and Power

Power is the rate at which work is done, for rotational motion as well as for translational motion.

Again, note the analogy to the linear form:


11-9 The Vector Nature of Rotational Motion

The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign.



A similar right-hand rule gives the direction of the torque.



Conservation of angular momentum means that the total angular momentum around any axis must be constant. This is why gyroscopes are so stable.


Summary of Chapter 11

•In order for an object to be in static equilibrium, the total force and the total torque acting on the object must be zero.

• An object balances when it is supported at its center of mass.



• In systems with both rotational and linear motion, Newton’s second law must be applied separately to each.

• Angular momentum:

• For tangential motion,

• In general,

• Newton’s second law:

• In systems with no external torque, angular momentum is conserved.



• Work done by a torque:

• Power:

• Rotational quantities are vectors that point along the axis of rotation, with the direction given by the right-hand rule.

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Chapter 11 Rotational Dynamics and Static Equilibrium · b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with