Problem 157 - A Complete Resource Book in Physics for JEE Main

Question

A block of mass \(m\) is connected to another block of mass \(M\) by a spring (massless) of spring constant \(k\). The blocks are kept on a smooth horizontal plane. Initially the blocks are at rest and the spring is unstretched. Then a constant force \(F\) starts acting on the block of mass \(M\) to pull it. Find the force on the block of mass \(m .\) (A) \(\frac{m F}{M}\) (B) \(\frac{(M+m) F}{m}\) (C) \(\frac{m F}{(m+M)}\) (D) \(\frac{M F}{(m+M)}\)

Step-by-Step Solution

Answer

The force on the block of mass m is given by \(\frac{m F}{(m+M)}\).

Step 1: Analyze the Motion of the Masses

Since the surface is smooth and the force is only applied to mass M, the only horizontal forces acting on the masses are the tension in the spring and the force F. Mass M will get accelerated towards the right and mass m, due to the stretching of the spring, will also get accelerated towards the right.

Step 2: Apply Newton's Laws to the Masses

Let's denote the acceleration of mass M as \(a_M\) and the acceleration of mass m as \(a_m\). By using Newton's second law, we can set up the following equations: \(M a_M = F - kx\) (for mass M) \(m a_m = kx\) (for mass m) where x is the extension of the spring.

Step 3: Use the Equations to Find the Force on Mass m

Since the spring is the only force acting on mass m and it is connected to the spring extension x, we can write the force acting on mass m as: Force on mass m = \(m a_m\) So, we first need to determine the acceleration of mass m (\(a_m\)). To do that, we can rewrite the first equation as: \(a_M = \frac{F - kx}{M}\) Using the second equation, we can express the spring extension x in terms of \(a_m\): \(x = \frac{m a_m}{k}\) Replacing the x in the first equation, we get: \(a_M = \frac{F - k(\frac{m a_m}{k})}{M}\) After simplifying, we get: \(a_M = \frac{F}{M} - \frac{m a_m}{M}\) Now we can solve for \(a_m\) in terms of \(a_M\) and use this relationship to determine the force on mass m: \(\frac{m a_m}{M} = \frac{F}{M} - a_M\) Solving for \(a_m\), we get: \(a_m = \frac{F}{m + M}\) Now, we can find the force acting on mass m using Force on mass m = \(m a_m = \frac{m F}{m + M}\) Therefore, the force on the block of mass m is given by (C) \(\frac{m F}{(m+M)}\).