Question

A small block on a frictionless, horizontal surface has a mass of 2.50×10−2 kg . It is attached to a massless cord passing through a hole in the surface (Figure 1) . The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 2.33 rad/s . The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 m. Model the block as a particle.

a.What is the new angular speed?

b.Find the change in kinetic energy of the block

c.How much work was done in pulling the cord?

Answer 1

here

mass of block, m = 0.0250 kg

distance, d = 0.3 m

angular speed, w1 = 2.33 rad/s

radius of path , r = 0.150 m

Part a:
From conservation of momentum :
initial = final
r1^2 * w1 = r2^2 * w2

angular speed, w2 = r1^2 * w1 / r2^2
angular speed, w2 = 0.3^2 * 2.33 / 0.150^2
angular speed, w2 = 9.32 rad/s

Part b:
change in KE = final - initial

change in KE = 0.5 ( I1 * w2^2 - I2*w2^2 )
( I1 and I2 are initial anf final moment of inertia)

change in KE = 0.5 ( m * d^2 * w2^2 - m * r^2 * w1^2 )

change in KE = 0.5 ( 0.0250 * 0.3^2 * 9.32^2 - 0.0250 * 0.150^2 * 2.33^2 )

change in KE = 0.096 J

Part c:

From work energy theoram,

Work done = change in KE

Work done = 0.096 J