Question

A mass m₁ = 6.00 kg is connected by a light string that passes over a pulley of mass m₃ = 7.5 kg to a mass m₂ = 6.00 kg sliding on a frictionless inclined surface that makes an angle of 23° with the horizontal (see figure). The coefficient of kinetic friction between the mass m₁ and the horizontal surface is 0.16. There is no slippage between the string and the pulley. What is the magnitude of the acceleration of the system? (The moment of inertia of the pulley is ½Mr².)

Answer 1

Let tension on m1 side of string be T1, and that on m2 side be T2, acceleration be a,

Force equation on m1: Fnet = ma

or T1- um1g = m1a

Force equation on m2: Fnet = ma

or m2g sin theta - T2 = m2a

Torque equation on pully: (T2-T1)*r = I*alpha where alpha = a/r as there is no slipping.

(T2-T1)*r = 0.5 m3*r^2*a/r

T2-T1 = 0.5*m3*a

Adding this to two force equations,

  m2g sin theta - um1g = [m1+m2+0.5m3]*a

a= [m2g sin theta - um1g]/(m1+m2+0.5m3)

= [6*9.8*sin 23 degree - 0.16*6*9.8]/(6+6+0.5*7.5)

= 0.8613 m/s^2 answer