First slide

Connected bodies

Question

Two blocks of masses $M_{1} and M_{2}$ , are connected with a string passing over a pulley as shown in figure. The block $M_{1}$ lies on a horizontal surface. The coefficient of friction between the block $M_{1}$ and the horizontal surface is $μ$ . The system accelerates. What additional mass m should be placed on the block $M_{1}$ so that the system does not accelerate?

Moderate

A

$\frac{M_{2} - M_{1}}{μ}$
B

$\frac{M_{2}}{μ} - M_{1}$
C

$M_{2} - \frac{M_{1}}{μ}$
D

$(M_{2} - M_{1}) μ$

Solution

For the equilibrium of block of mass $M_{1}$ :
Friction force, f = tension in the string T
where T = f = $μ (m + M_{1}) gt$ ----------(i)
For the equilibrium of block of mass $M_{2}$ :
T = $M_{2} g$ ------------(ii)
From (i) and (ii) $μ (m + M_{1}) g = M_{3} g$
$m = \frac{M_{2}}{μ} - M_{1}$

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