Question

A green drum with a moment of inertia of 19.24kgm^2 and radius 0.95m rotates freely about a vertical axis. A rope is wound around it and strung over a red wheel with a moment of inertia of 16.11kgm2 and radius 0.71m that rotates freely about a horizontal axis. The rope is then attached to a cube dangling below with a mass of 13.47kg. The cube falls 1.8m1.8m from rest. Take the acceleration of gravity to be 9.81ms^2. Assume that the rope does not slip and has negligible mass. What is its velocity at that point?

(a) what is the acceleration?

(b) what is the tension in the string right above the cube?

Answer 1

let I1 = 19.24 kg.m^2, R1 = 0.95 m

I2 = 16.11 kg.m^2, R2 = 0.71 m

m = 13.47 kg

h = 1.8 m

a) Apply conservation of energy

loss of potential energy = gain in kinetic energy of wheels and cube

m*g*h = (1/2)*m*v^2 + (1/2)*I1*w1^2 + (1/2)*I2*w2^2

m*g*h = (1/2)*m*v^2 + (1/2)*I1*(v/R1)^2 + (1/2)*I2*(v/R2)^2

13.47*9.81*1.8 = (1/2)*13.47*v^2 + (1/2)*19.24*(v/0.95)^2 + (1/2)*16.11*(v/0.71)^2

==> v = 2.67 m/s

now use,

v^2 - vo^2 = 2*a*d

a = (v^2 - vo^2)/(2*d)

= (2.67^2 - 0^2)/(2*1.8)

= 1.98 m/s^2 <<<<<<<<---------------Answer

B) Tension in the string, T = m*(g - a)

= 13.47*(9.81 - 1.98)

= 105 N <<<<<<<<---------------Answer