A block of mass m is connected to a another block of mass M by a massless spring of spring constant k.The blocks are kept on a smooth horizontal plane.Initailly the blocks are at rest and the spring is unstretched when a constant force F starts acting on the block of mass M to pull it.Find the maximum extension of the spring.

To find the maximum extension of the spring when a constant force F is applied to the block of mass M, we need to analyze the forces acting on both blocks and how they interact through the spring. This situation involves concepts from Newton's laws of motion and the properties of springs. Let's break it down step by step.

Understanding the System

We have two blocks: one with mass m (the block connected to the spring) and another with mass M (the block being pulled by the force F). The spring, which connects these two blocks, has a spring constant k and is initially unstretched. When the force F is applied, it will accelerate the block of mass M, and as a result, the spring will stretch due to the relative motion between the two blocks.

Applying Newton's Second Law

First, let's apply Newton's second law to both blocks. For the block of mass M, the net force acting on it is the applied force F minus the force exerted by the spring when it stretches. The force exerted by the spring can be described by Hooke's Law, which states that the force exerted by a spring is proportional to its extension (x):

• Force on M: F - kx = Ma
• Force on m: kx = ma

Here, a is the acceleration of the blocks, and x is the maximum extension of the spring.

Setting Up the Equations

From the first equation, we can express the acceleration a in terms of the force F and the spring extension x:

1. F - kx = Ma

From the second equation, we can express the acceleration a in terms of the mass m:

2. kx = ma

Relating the Two Equations

Since both blocks are connected and will have the same acceleration at maximum extension, we can set the expressions for acceleration equal to each other. From equation 1, we can solve for a:

a = (F - kx) / M

Substituting this into equation 2 gives:

kx = m[(F - kx) / M]

Solving for Maximum Extension

Now, let's rearrange this equation to isolate x:

kxM = m(F - kx)

Expanding this, we get:

kxM = mF - mkx

Now, combine the terms involving x:

kxM + mkx = mF

This can be factored as:

x(kM + mk) = mF

Finally, we can solve for x:

x = (mF) / (k(M + m))

Final Expression for Maximum Extension

The maximum extension of the spring, when a constant force F is applied to the block of mass M, is given by:

x = (mF) / (k(M + m))

This formula shows how the maximum extension depends on the masses of the blocks, the applied force, and the spring constant. The greater the force or the mass of the block m, the larger the extension will be, while a stiffer spring (higher k) will result in a smaller extension.