Chapter 11 – Rotational Dynamics & Static Equilibrium

Torque Increased Force = Increased Torque Increased Radius = Increased Torque

Torque Only 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 11.1Using 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

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.

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

Torque Right Hand Rule for Torque 1.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.

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 LinearRotational Force Mass Acceleration 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.

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.

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 Balance If 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.

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.

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).

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: