Question

Two masses M₁ and M₂ are connected by a light rod and the system is slipping down a rough incline of angle θ with the horizontal. The friction coefficient at both the contacts is μ. Find the acceleration of the system and the force by the rod on one of the blocks.

Answer 1

From the free body diagram


R₁ = M₁g cos θ (1)
R₂ = M₂g cos θ (2)
T + M₁g sin θ − M₁a − μR₁ = 0 (3)
T − M₂g + M₂a + μR₂ = 0 (4)

From Equation (3),
T + M₁g sin θ − M₁ a − μM₁g cos θ = 0 (5)

From Equation (4),
T − M₂ g sin θ + M₂ a + μM₂ g cos θ = 0 (6)

From Equations (5) and (6),
g sin θ(M₁ + M₂) − a(M₁ + M₂) − μg cos θ(M₁ + M₂)
⇒ a(M₁ + M₂) = g sin θ(M₁ + M₂) = μg cos θ(M₁ + M₂)
⇒ a = g(sin θ − μ cos θ)a − g(sin θ − μ cos θ)
∴ The acceleration of the block (system) = g(sin θ − μcos θ)

The force exerted by the rod on one of the blocks is tension, T.
T = −M₁g sin θ + M₁a + μM₁g cos θ
T = −M₁g sin θ + M₁(g sin θ − μg cos θ) + μM₁g cos θ = 0