Chapter 11 – Rotational Dynamics & Static Equilibrium

11.1 - Torque

•Increased Force = Increased Torque•Increased Radius = Increased Torque

11.1 - TorqueOnly the tangential component of force causes a torque:

11-1 Torque

This leads to a more general definition of torque:

**********r is also referred to as the “moment arm”************

Question 11.1 Using a Wrench

You are using a wrench to

loosen a rusty nut. Which

arrangement will be the most

effective in loosening the nut?

a

c d

b

e) all are equally effective

11.1 - Torque

• Is torque a vector?– YES! Why?• Because FORCE is a vector!

• What is the torque direction?• If the torque in question causes – Counterclockwise (CCW) angular acceleration• Torque is positive.

– Clockwise (CW) angular acceleration• Torque is negative.

11.1 - Torque

The Right Hand Rule in Physics• Coordinate systems• Moving charges in magnetic fields• Magnetic fields produced by current• Torque• Angular Momentum

11.1 - Torque

Right Hand Rule for Torque1. Make a “backwards c”

with your right hand.2. Turn hand so your fingers

curl in the direction of rotation that particular torque would cause.

3. Direction of thumb dictates “direction” of torque.

• Positive torque points out of the page.

• Negative torque points into the page.

http://electron9.phys.utk.edu/Collisions/rotational_motiondetails.htm

11.2 - Torque & Angular Acceleration

Linear Dynamics• Newtons’s Second Law for

Linear Dynamics:

• Reads if we apply a FORCE to an object with some mass, the object undergoes an acceleration.

Rotational Dynamics• Newton’s Second Law for

Rotational Dynamics:

• Reads, if we apply a TORQUE to some object with some moment of inertia, the object undergoes an angular acceleration.

Newton’s Second Law

Linear Rotational

Force

MassAcceleration Torque

Moment of Inertia

Angular Acceleration

A person holds his outstretched arm at rest in a horizontal position. The mass of the arm is m, and its length is .740 m. When the person allows their arm to drop freely, it begins to rotate about the shoulder joint. Find (a) the initial angular acceleration of the arm, and (b) the initial linear acceleration of the hand.

(a) α = ?(b) a = ?

Notice anything interesting about the acceleration of the hand?

11.2 – Torque & Angular Acceleration

• We found that the acceleration of the hand was:a= (3/2)g

• This means for points on the arm > (2/3)L away from the axle have an acceleration 1.5g!!

11.3 – Static Equilibrium

• Static Equilibrium occurs when– An object has no

translational motion.

AND– An object has no

rotational motion.

• Conditions for static equilibrium.– Net force in the x-

direction is zero.

– Net force in the y-direction is zero.

– Net torque is zero.

11.3 - Static EquilibriumIf 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.

11.3 - Static Equilibrium

When forces have both vertical and horizontal components, in order to be in equilibrium an object must have no net torque, and no net force in either the x- or y-direction.

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 BalanceThis 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.

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.

11.6 – Angular Momentum

Linear Momentum• An object with mass (m)

moving linearly at velocity (v) has a certain amount of linear momentum (p).

Angular Momentum• A rotating object with

moment of inertia (I) rotating at some angular velocity (ω) has a certain amount of angular momentum (L).

11.6 - Angular Momentum

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

11.6 – Angular Momentum

Linear Momentum• We were able to relate the

linear momentum of an object to the linear version of Newton’s Second Law.

Angular Momentum• We can do the same by

relating the angular momentum of an object to the rotational version of Newton’s Second Law.

11.7 – Conservation of Angular Momentum

Linear Momentum• If the net external force on

a system is zero, the linear momentum is conserved.

Angular Momentum• If the net external torque

on a system is zero, the angular momentum is conserved.

11.8 – Rotational Power & Work

Linear Work• A force acting through a

distance does work on an object to move it.

Rotational Work• A torque acting through an

angular displacement does work on an object to rotate it.

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: