Awheel with radius 36 cm is rotating at a rate of 19 rev/s.(a) what is the angular speed in radians per second? rad/s(b) in a time interval of 5 s, what is the angle in radians through which the wheel rotates? rad(c) at t=10 s the angular speed begins to increase at a rate of 1.6 rad/s/s. at t=15 s, what is the angular speed in radians per second? rad/s(d) through what angle in radians did the wheel rotate during the time between t=10 s and t=15 s? rad(e) if the wheel rolls along the ground without slipping, the instantaneous velocity of the atoms of the object that are momentarily in contact with the ground is zero. this zero-velocity condition implies that vcm=ω⁢r, where ω is the angular speed of the object, since the instantaneous speed of the contact point is vcm-ω⁢r. during the time between t=10 s and t=15 s, how far did the center of the wheel move, in meters? m

(a) 119.3 rad/s

The angular speed of the wheel is

we need to convert it into radiands per second. We know that

Therefore, we just need to multiply the angular speed of the wheel by this factor, to get the angular speed in rad/s:

(b) 596.5 rad

The angular displacement of the wheel in a time interval t is given by

where

and

t = 5 s is the time interval

Substituting numbers into the equation, we find

(c) 127.3 rad/s

At t=10 s, the angular speed begins to increase with an angular acceleration of

So the final angular speed will be given by

where

is the initial angular speed

is the angular acceleration

is the time interval

Solving the equation,

(d) 616.5 rad

The angle through which the wheel has rotated during this time interval is given by

Substituting the numbers into the equation, we find

(e) 222 m

The instantaneous speed of the center of the wheel is given by

(1)

where

is the average angular velocity of the wheel during the time t=10 s and t=15 s, and it is given by

and

R = 36 cm = 0.36 m is the radius of the wheel

Substituting into (1),

And so the displacement of the center of the wheel will be

(75°f − 32) × 5/9 = 23.889°c