Question

A massless string is wound around a solid cylinder that has a radius of 0.27 m and a mass of 28.8 kg. The free end of the string is tied to a block of mass 4 kg which hangs straight down. At t=0, the cylinder is allowed to spin about an axis through its center and the block falls, unwinding the string. Assume that the cylinder spins without friction. What is the acceleration of the block?

Answer 1

Gravitational acceleration = g = 9.81 m/s²

Mass of the cylinder = M = 28.8 kg

Radius of the cylinder = R = 0.27 m

Moment of inertia of the cylinder = I

I = MR²/2

I = (28.8)(0.27)²/2

I = 1.05 kg.m²

Mass of the block = m = 4 kg

Tension in the string = T

Acceleration of the block = a

Angular acceleration of the cylinder =

= a/R

From the free body diagram of the block,

ma = mg - T

T = mg - ma

For the cylinder,

I = TR

I(a/R) = (mg - ma)R

Ia = mgR² - maR²

(I + mR²)a = mgR²

[1.05 + (4)(0.27)²]a = (4)(9.81)(0.27)²

a = 2.13 m/s²

Acceleration of the block = 2.13 m/s²