Rotational Motion If a body is rotating such that every point on the body moves in a circular path, and all of the centers of those circles lie along the.

Rotational Motion

If a body is rotating such that every point on the body moves in a circular path, and all of the centers of those circles lie along the same line *, then it is undergoing pure rotation (uniform rotational motion).

* that line is referred to as the axis of rotation, and it is either internal or tangent to the body.

For now we will only consider pure rotational motion not objects in both translational and rotational or special cases (swirling waters)

As with translational motion, measurements of angular motion are based upon angular displacement:

Quantity Linear Angular

displacement x, y, z (m) Ø (rad)

av. speed v = ∆d/t (m/s) = ∆ø/t (rad/s)

inst. speed v = dx/dt = dø /dt

av. accel. a = ∆v/t = ∆ / t

inst. accel. a = dv/dt = d / dt

These equations all represent how to find the magnitudes! They do not reflect directions, which will be described later.

The fundamental unit upon which angular motion is based is the radian (radius length).

1 revolution = 360˚ = 2π rad

A convenient relationship between angular and translational motion involves the radian:

ø = s

r

s would be represent the linear distance traveled while ø

represents angular displ. in rad!

Rotation with constant angular acceleration

If angular acceleration is constant, then x, v, a, in previous motion equations can be replaced with ø, , and !

A wheel is accelerated from rest for 2.70 s at a rate of 3.20 rad/s2. The power is then turned off and the slight amount of friction between the wheel and the shaft halts the wheel 192 s after the power is turned off. Find A) the maximum angular speed and B)the total angular displacement of the wheel.

while accelerating:

0 = 0

= 3.2 rad/s2

t1 = 2.70 s

= o + t = 8.64 rad/s

ø = øo + o t + .5 t2 = 11.6 rad

while decelerating:

t2 = 192 s = - o

t2

= 00 = 8.64 rad/s

= - .045 rad/s2

ø = øo + o t + .5 t2 = 829 rad

ø = 11.6 + 829 = 841 rad

Rotational Quantities as Vectors

The motion of rotational vector quantities (ø, , and ) is defined in terms of the axis of rotation, and therefore, so is the direction:

By convention, the motion vector is determined to lie along the axis of rotation following the right hand rule:

Curl the fingers of the right hand in the direction of the rotation and the extended thumb points gives the direction of the angular vector!

axis of rotation

r s = ør

v = r

a = r

This is when the angular quantity is measured in radians!

A CD player is designed so that as the read head moves out from the center the angular speed of the disc changes so that the linear speed of the of the disc under the head will be 1.3 m/s. What is the angular speed of the disc when the head is 2.0 cm and 5.6 cm from the center?

= v/r 1.3 m/s

.020 m

= 65 rad/s

1.3 m/s

.056 m

= 23 rad/s

If it takes an hour to play a CD, what is the length of the CD track?

d = v• t = (1.3 m/s)(3600 s) = 4680 m

If the CD has a diameter of 12 cm, how many times does it turn during the one hour?

ø = s / r = 4680 m

.06 m/rad

= 78, 000 rad (1 rev/ 2π rad)

= 12,400 rev