Question

A block of mass m can slide freely along the vertical surface of a bigger block of mass M as shown in figure. There is no friction anywhere in the system. The block m is connected to one end of a string whose other end is fixed at point P. The string between $P_1$ and P is horizontal and other parts of the string are vertical. System is released from rest. Find the accelerations of m during their subsequent motion.

• A. g
• B. mg (m+2m)
• C.
• D.

Answer 1

The correct option is C

Let us divide the string in three segments as shown in figure. As we release the system, the block m will try to move downwards due to which length of segment I will increase. Then length of segment III should decrease equally because length of segment II is fixed. This will make M to move towards right. Let acceleration of m downwards be A then acceleration of M towards right will be A. Then in horizontal direction acceleration of m will also be A, because m is constrained to move horizontally with M.

If we make free body drawings of M and m separately, then

From free body drawing of M: T - N = MA ...(ii)

From free body drawing of m: N = mA ...(iii)

and mg - T = ma ...(iv)

From (ii) and (iii), we get T = (m + M)A ...(v)

Equation (v) is same as Eqn. (ii). Now solve to get the answers.

From (ii) (iii) and (iv)

m(g - A) = (M + m)A $\Rightarrow A=\frac{mg}{M+2m}=a$

Hence net acceleration of m $a_{net}=\sqrt{a^2+A^2}=\sqrt{2}A=\sqrt{2}\frac{mg}{(M+2m)}$