Question

A solid cylinder of radius 0.5 m and mass 5.0 kg, initially at rest, starts to rotate about an axis through its center, with an angular acceleration of 0.2 rad/s^2.

(a) Assuming a piece of string is wrapped around the cylinder, in such a way that the turning cylinder pulls the string onto itself, what is the total length of string wrapped on the cylinder at t= 10s?

(b) What is the linear acceleration of a knot in the string as it travels towards the cylinder ?

(c) What is the velocity of the string at t= 10s?

(d) What is the rotational kinetic energy of the cylinder at t=10 s?

Answer 1

here,

mass of cyclinder , m = 5 kg

radius , r = 0.5 m

angular accelration , alpha = 0.2 rad/s^2

a)

t = 10 s

the angle covered ,theta = 0 + 0.5 * alpha * t^2

theta = 0 + 0.5 * 0.2 * 10^2 = 10 rad

the length of string wrapped , l = r * theta = 5 m

b)

the linear accelration of knot , a = r * alpha

a = 0.1 m/s^2

c)

the velocity of string , v = (w) * r

v = ( alpha * t) * r

v = 0.2 * 10 * 0.5 = 1 m/s

d)

the rotational kinetic energy , KEr = 0.5 * I * w^2

KEr = 0.5 * ( 0.5 * m * r^2) * ( alpha * t)^2

KEr = 0.5 * ( 0.5 * 5 * 0.5^2) * ( 0.2 * 10)^2

KEr = 1.25 J