Question

Two blocks of weights  W₁ = m₁g and W₂ = m₂g are suspended by a pin-supported rope drum (moment of inertia Θ_A) as shown in Fig.Determine the angular acceleration of the drum and the force in rope ① using dynamic equilibrium conditions. Neglect the mass of the ropes. 

Answer 1

We first introduce the coordinates x₁ and x₂ describing the motion of the blocks. The inertial forces −m_ix_i point in the negative xi-directions (see the freebody diagram). In addition, we have to consider the pseudo moment −Θ_Aϕ¨ which acts in the negative ϕdirection. Moment equilibrium about point A then yields

Using the kinematic relations

we obtain the angular acceleration of the drum:

In order to determine the force in rope ① we cut the rope. Force equilibrium (see the free-body diagram) yields

↑ : S₁ − m₁g − m₁x₁ = 0

or

Note: For r₂m₂ > r₁m₁ the drum rotates clockwise, for r₂m₂ < r₁m₁ it rotates counterclockwise. In the special case r₂m₂ = r₁m₁ the system is in static equilibrium ( ¨ϕ = 0).