# Dynamics of Rotational Motion Rotational Motion & Torque Angular Displacement Angular displacement is a measure of the angle a line from the center to a point on the outside edge sweeps through as the object rotates We use the greek letter theta to represent angular displacement Angular displacement is measured in radians Arc Length

Arc length is represented by the letter s it is the linear distance a point on the edge of the object rotates through measured in meters r s

r s r Angular Velocity Angular velocity is a measure of the rate of change of the angular position or the spin rate We use the greek letter lowercase omega () to represent angular velocity mathematically Angular velocity is measured in radians per second (rads/sec)

Angular Velocity t r May also be expressed in rpm r

(revolutions per minute) but must be converted to rad/sec for calculations rev 2rad rad 1 min 60 sec 30 sec Angular Acceleration Angular acceleration is the rate of change of angular velocity We use the greek letter alpha to represent angular acceleration

Angular acceleration is measured in radians per second per second (rads/sec2) Angular Acceleration t r r

Rotational Motion Relationships f i t 2 2 f i 2 2 1 it 2 t What is Torque? Torque is a measure of how much a

force acting on an object causes that object to rotate. Torque is dependent on force and lever arm. Torque is measured in Newtonmeters (Nm) Lever Arm Distance measured perpendicularly from the line of force to the pivot point. Measured in meters Lever arm 1 F1

Lever arm 2 pivot F2 Calculating Torque Torque = Force * lever-arm Fl The symbol for torque is the greek letter tau pivot

Note: Force and lever arm must be perpendicular to each other F Calculating Torque Fl Fr sin n i s r =rsin

pivot r r is the shortest distance from where the force is applied to the pivot point is the angle between r and the line of F F Calculating Torque By finding components of F F - the perpendicular component

of the force will cause a the door F to pivot about its hinges (torque) F pivot r F F// - the parallel component of the force acts through the pivot point, so it does not cause the door to pivot (no torque)

F Fl ( F sin )r Net Torque & Rotational Equilibrium The net torque is the sum of the individual torques. net

In rotational equilibrium, the sum of the torques is equal to zero. In other words, there is no net torque on the object. net 0 OR ccw cw Note: ccw = counterclockwise and cw = clockwise EXAMPLE PROBLEM ON TORQUE: The Swinging Door Question

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)? Hints 1. Where is the pivot point? 2. What was the force applied? 3. How far from the pivot point was the force applied? 4. What was the angle between the door and the direction of force? Solution The pivot point is at the hinges of the door, opposite to

where you were pushing the door. The force you used was 50N, at a distance 1.0m from the pivot point. You hit the door perpendicular to its plane, so the angle between the door and the direction of force was 90 degrees. Since = r x F = r F sin( ) then the torque on the door was: = (1.0m) (50N) sin(90) = 50 N m Rotational Kinetic Energy Rotational energy that an object has because it is rotating about an axis 2 1 K rot I

2 I is rotational inertia measured in kgm2 Angular Momentum If momentum can be described as inertia in motion then Angular momentum can be described as inertia rotating L I Like momentum, angular momentum must be conserved if there are no outside torques acting on the body

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