Question

Two blocks A and B of masses m and M are connected to two ends of of a string passing over a pulley of B lies on plane inclined at an angle theta with the horizontal and A is hanging freely as shown the coefficient of static friction between B and the plane is mus find the maximum and minimum values of m so that the system is at rest

Answer 1

Given that,

Mass of block A = M

Mass  of block B =m

Angle = θ

Coefficient static friction =μ

We need to calculate the maximum and minimum values of m

Using Fbd diagram,

When M tending to move up the plane,

For block A,

The normal force is

$N = mg\cos\theta$

The tension is

$T=Mg\sin\theta+f_{\mu}$

We know that,

Friction force is

$f_{\mu}=\mu N$

$T =Mg\sin\theta+\mu Mg\cos\theta$

$T=Mg(\sin\theta+\mu\cos\theta)$....(I)

For block B,

The tension is

$T = mg$

Put the value of T in equation (I)

$m=M(\sin\theta+\mu\cos\theta)$

When M tending to move down the plane,

For block A,

The normal force is

$N = mg\cos\theta$

The tension is

$T+f_{r}=Mg\sin\theta$

$T+\mu Mg\cos\theta=Mg\sin\theta$

$T=Mg(\sin\theta-\mu\cos\theta)$...(II)

For block B.

$T = mg$

Put the value of T in equation (II)

$m=M(\sin\theta-\mu\cos\theta)$

We can say that,

M will move to up then m will move to down

M will move to down then m will move to up.

For m to be at rest,

$M(\sin\theta-\mu\cos\theta)\leq m\leq M(\sin\theta+\mu\cos\theta)$

This equation is necessary for system when the system is at rest

Hence,  This is the required solution.