Rotational Dynamics

Rotational Dynamics Torque & Rotational Kinematics Presentation 2003 R. McDermott

Torque • Torque is the rotational equivalent of force • Force tends to cause linear acceleration • Torque tends to cause rotational acceleration • Occurs when forces do not meet at a point (non-concurrent forces)

Example: Lever • A simple lever is the easiest example of non-concurrent forces:

The Lever • The up and down forces balance, so no linear acceleration occurs • The support force causes no rotation • The force on one side is larger than on the other, so rotation will occur (and there will be acceleration as well)

See-Saw • But what about the situation below? • We all recognize that rotation may not occur in the case below, even though the forces on the ends are not equal. • Why?

Force and Positioning • Linear acceleration is only concerned with the size of competing forces • Torque is concerned with the size of competing forces AND their position with respect to the pivot point • A smaller force offset by a greater distance can have the same effect as a large force at a smaller offset

F  d Definition: • T = Fdsin • Like the work definition, but with a different trig function • Cross-product, not dot-product • Another way of multiplying vectors • Multiplying only perpendicular components Fsin

15N 5N 10N 4m 1m Sample #1: • Each force is perpendicular to the measured distance from the pivot point • The 15N force is located 0m from the pivot point, so its torque is 0m x 15N = 0 m-N

15N 5N 10N 4m 1m Sample: • The 10N force is located 1m from the pivot point, so its torque is 1m x 10N = 10m-N • The 5N force is located 4m from the pivot point, so its torque is 4m x 15N = 20m-N • But these act in opposite directions

15N 5N 10N 4m 1m Sample (cont): • The 10N force would rotate the board clockwise • The 5N force would rotate the board counterclockwise • Overall then, we have 20m-N counterclockwise and 10m-N clockwise, which equals 10m-N counterclockwise

Rotation “Speed” • Most people would agree that the objects above are spinning at the same “speed”. • But what is meant by that?

Rotation and Period • When we say two objects are rotating at the same speed, we usually mean that they complete one rotation in the same amount of time – Have the same period. • A point on the rim of the larger object, however, has farther to travel and must actually have a greater velocity than a point on the rim of the smaller object.

Then What is the Same? • When we refer to rotational speed, we are actually referring to an angular speed. • We are rotating through some angle each second, not some distance each second. • The “natural” way to measure angles is by using the radian – an arc length equal to the radius of rotation.

Linear: x in meters v in meters/sec a in meters/sec2 F in Newtons m in kilograms Angular:  in radians  in radians/sec  in radians/sec2  in meter-Newtons  in kilogram-meters2 Linear Versus Angular

Conversions: • x = r A point on the rim of an object of radius 2m that rotates through an angle of 2 radians travels a distance of 4 meters. • v = r If the same object has an angular velocity of 3 rad/s, a point on the rim has a velocity of 6 m/s. • a = r If this object is increasing its angular velocity by 1 rad/s every second, then the acceleration of that same point would be 2 m/s2

Linear: Vavg = x/t a = v/t Vf = Vi + at Vf2 = Vi2 + 2ax x = Vit + ½at2 Angular: avg = /t  = /t f =i + t f2=i2 + 2  = it + ½t2 Kinematics Equations:

Rotational Dynamics