Chapter 11 Rotational Dynamics and Static Equilibriumphysics.sfsu.edu/~wman/phy111hw/lecture notes/chapter11allnew.pdf · Rotational Dynamics and Static Equilibrium Copyright Weining

Chapter 11

Rotational Dynamics and Static Equilibrium

Copyright Weining Man

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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


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Units of Chapter 11

Optional

Not required

• Conservation of Angular Momentum

• Rotational Work and Power

• The Vector Nature of Rotational Motion


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11-1 TorqueFrom experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.

This is why we have long-handled wrenches, and why doorknobs are not next to hinges.


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11-1 TorqueWe define a quantity called torque:

The torque increases as the force increases, and also as the distance increases.

r is moment arm length


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11-1 Torque

Only the tangential component of force causes a torque:


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11-1 Torque

This leads to a more general definition of torque:


is called moment arm length, it’s the perpendiculardistance between the rotation center to the force line.7


r is the moment arm length. To use longer r, can saves force.

Examples: Level, bottle opener, dolly, wrench, screw driver….

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11-1 Torque

If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.


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11-3 Zero Torque and Static Equilibrium

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


Total torque is zero respect to any rotational axis. 10

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.


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|τccw| = |τcw| When you consider SIZE only, use the absolute value. CCW torque=CW torque

When you considertotal torque = 0, Ccw torque are positiveCw torque are negative.

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1, label forces, 2, choose rotational axis3, write out torque created by each forces.Use the correct moment arm length for each force. If a force is going through the rotational axis, its torque=04, use equation total torque = 0, To solve bicep force, 5, Finally use equation total forces =0, to solve the unknown bone force. Will the bicep force be more or less

than Mg+mg?Will the bone force point upward or downward?What if the bone becomes a soft rope?

Equations:

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


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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.


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11-2 Torque and Angular Acceleration

Newton’s second law:

If we consider a mass m rotating around an axis a distance r away, we can reformat Newton’s second law to read:

Or equivalently,


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11-2 Torque and Angular Acceleration

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


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svp=mvKE=1/2 mv2

θ =s /rω =vt /rL =I ωKE =1/2 I ω2

=∑mr2

= at/r= Fr


Example: open your arm and let it fall without applying any muscle force.What will be the angular acceleration for your arm due to its gravity at that moment? What will be the linear acceleration of your finger tip at that moment? Will your finger tip fall faster, or slower than free fall? put a small object at your finger tip to compare. Will your upper arm fall faster or slower than free fall?

mg

L

The length of the whole arm = L; Total arm mass = mChoose the shoulder to be the rotation axis,Total torque exerted on the arm:

Assume the whole arm as a long rod of block, When rotating around its end(the shoulder)Arm’s rotational inertia:

At that moment,the angular acceleration :

Fingertip linear acceleration:Arm center linear acceleration:Fingertip fall (rotates) faster than free fall. Upper arm falls slower than free fall.

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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.


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11-6 Angular Momentum

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


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11-7 Conservation of Angular MomentumIf 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:


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If < Ii So that, ωf > ωi

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

Summary of Chapter 11• A force applied so as to cause an angular

acceleration is said to exert a torque.

• Torque due to a tangential force:

• Torque in general:

r is the moment arm length. Longer r, saves force.

(but longer r means to apply force across longer distance. It saves forces, but nothing can save work.

• Newton’s second law for rotation:

Angular acceleration = total torque / rotational inertia


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


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svp=mvKE=1/2 mv2

θ =s /rω =vt /rL =I ωKE =1/2 I ω2

=∑mr2

= at/r= Fr

Summary of Chapter 11

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

• Angular momentum:

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


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

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

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11-7 Conservation of Angular Momentum


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Optional

Gravity of the force going through the rotational center (the sun) hence doesn’t create torque on the planet to change its angular momentum. Iω will be constant, hence mr2ω will be constant.After the same amount of time t, the area swept by the line between a planet and the sun Area=r2∆θ=r2ωt, will be the same.

Planet’s elliptical orbit:Rotating around the sun

11-7 Conservation of Angular Momentum

OptionalAngular momentum is also conserved in rotational collisions:


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11-6 Angular Momentum

Looking at the rate at which angular momentum changes,


Optional

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11-8 Rotational Work and Power

Optional Not RequireA torque acting through an angular displacement does work, just as a force acting through a distance does.

The work-energy theorem applies as usual.


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11-8 Rotational Work and Power

Optional Not RequirePower 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:


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11-9 The Vector Nature of Rotational Motion, Optional Not Require

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


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11-9 The Vector Nature of Rotational Motion, Optional Not Require

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


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Chapter 11 Rotational Dynamics and Static Equilibriumphysics.sfsu.edu/~wman/phy111hw/lecture notes/chapter11allnew.pdf · Rotational Dynamics and Static Equilibrium Copyright Weining