A 400 m length of light nylon cord is wound around a uniform

A 4.00 m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg. The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude 2.60 m/s².

(a) How much work has been done on the spool when it reaches an angular speed of 7.40 rad/s?

(b) Assuming that there is enough cord on the spool, how long does it take the spool to reach this angular speed?

(c) What length of cord is pulled from the spool when the angular speed is reached?

Solution

a) Work done = energy gained = 1/2 . 1/2 . 1 . 0.5^2 . 7.6^2 = 3.61 J
b) time = velcity/acc.

linear velocity = angular velocity*radius = 7.4*0.5 m/sec

So, time = 7.4*0.5/2.60 = 1.288 sec

c) d= (1/2)*a*t^2 = (1/2)(2.60)(1.288)^2 = 2.45 metre