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.
From figure (a)
R1 = m1g cos θ
T + m1sin θ = m1a+ μR1
T + m1g sin θ = m1a + μm1g cos θ (i)
From figure (b)
R2 = m2g cos θ
T − m2g sin θ = m2a -μR2
T − m2g sin θ + m2a + μm2 gcos θ =0 (ii)
From Equations (i) and (ii),
g sin θ(m1 +m2) − a(m1 + m2) − μg cos θ(m1 + m2)=0
⇒ a(m1 + m2) = gsin θ(m1 + m2) - μg cos θ(m1 + m2)
⇒ a = g(sin θ − μ cos θ)
Hence, the acceleration of the 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
⇒ T=0
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