Chapter 11 Rotational Dynamics and Static Equilibrium

Home , Acceleration, Angular acceleration, Moment (physics), Power (physics), Torque

• 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

Copyright Weining Man 11-1 Torque From 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.

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

r is moment arm length

Only the tangential component of force causes a torque:

This leads to a more general definition of torque:

is called moment arm length, it’s the perpendicular

distance between the rotation center to the force line.7

Copyright Weining Man r is the moment arm length. To use longer r, can saves force. Examples: Level, bottle opener, dolly, wrench, screw driver….

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

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

Total torque is zero respect to any rotational axis.

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.

When you consider total torque = 0, Ccw torque are positive Cw torque are negative. 12

Copyright Weining Man 1, label forces, 2, choose rotational axis 3, 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=0 4, 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:

Copyright Weining Man 11-4 Center of Mass and Balance If an extended object is to be balanced, it must be supported through its center of mass.

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.

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,

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

=∑mr2

= at/r = Fr s θ =s /r

v ω=vt /r p=mv L =I ω KE=1/2 mv2 KE =1/2 I ω2

Copyright Weining Man 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? The length of the whole arm = L; Total arm mass = m Choose the shoulder to be the rotation axis, Total torque exerted on the arm: L Assume the whole arm as a long rod of block, When rotating around its end(the shoulder) Arm’s rotational inertia: mg

At that moment,the angular acceleration : Fingertip linear acceleration: Arm center linear acceleration: 23 Fingertip fall (rotates) faster than free fall. Upper arm falls slower than free fall. Copyright Weining Man 11-5 Dynamic Applications of Torque When dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly.

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

Copyright Weining Man 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:

If < Ii So that, ωf > ωi

As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change. 26 Copyright Weining Man 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

=∑mr2

= at/r = Fr s θ =s /r

v ω=vt /r p=mv L =I ω KE=1/2 mv2 KE =1/2 I ω2

Copyright Weining Man 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. (most important) • 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: •In systems with no external torque, total

angular momentum is conserved. 29

Planet’s elliptical orbit: Rotating around the sun

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. 30 Copyright Weining Man 11-7 Conservation of Angular Momentum Optional Angular momentum is also conserved in rotational collisions:

Looking at the rate at which angular momentum changes,

Copyright Weining Man 11-8 Rotational Work and Power Optional Not Require 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.

Copyright Weining Man 11-8 Rotational Work and Power Optional Not Require 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:

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

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