A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m. If a force P is applied at the free end of the rope, the force exerted by the rope on the block is

Options

A. (a) `(Pm)/(M+m)`
B. (b) `(Pm)/(M-m)`
C. (c) P
D. (d) `(PM)/(M+m)`

Explanation

To solve the problem of finding the force exerted by the rope on the block when a force \( P \) is applied at the free end of the rope, we can follow these steps:

### Step 1: Understand the System
We have a block of mass \( M \) on a frictionless surface, and it is being pulled by a rope of mass \( m \). A force \( P \) is applied at the free end of the rope.

### Step 2: Identify the Total Mass Being Accelerated
When the force \( P \) is applied, both the rope and the block will accelerate. The total mass that is being accelerated is the sum of the mass of the block and the mass of the rope:
\[ \text{Total Mass} = M + m \]

### Step 3: Apply Newton's Second Law
According to Newton's second law, the net force acting on the system is equal to the total mass times the acceleration (\( A \)):
\[ P = (M + m) \cdot A \]
From this, we can express the acceleration \( A \):
\[ A = \frac{P}{M + m} \]

### Step 4: Calculate the Tension in the Rope
The tension \( T \) in the rope can be found by considering the block of mass \( M \) alone. The only force acting on the block is the tension \( T \), which causes it to accelerate:
\[ T = M \cdot A \]
Substituting the expression for \( A \) from Step 3:
\[ T = M \cdot \left(\frac{P}{M + m}\right) \]
Thus, the tension \( T \) (which is the force exerted by the rope on the block) is:
\[ T = \frac{M \cdot P}{M + m} \]

### Final Answer
The force exerted by the rope on the block is:
\[ \boxed{\frac{M \cdot P}{M + m}} \]