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 + n)`
C. (c)`(PM)/(M - n)`
D. (d)p

Explanation

To solve the problem of finding the force exerted by the rope on the block, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the System**:
We have a block of mass \( M \) and a rope of mass \( m \) on a frictionless surface. A force \( P \) is applied at the free end of the rope.

2. **Determine the Total Mass**:
The total mass of the system (block + rope) is \( M + m \).

3. **Calculate the Acceleration of the System**:
Since the entire system is being pulled by the force \( P \), we can use Newton's second law to find the acceleration \( A \) of the system:
\[ A = \frac{P}{M + m} \]

4. **Analyze the Forces on the Block**:
The block experiences a force from the rope. Let’s denote this force as \( F \). According to Newton's second law, the force exerted by the rope on the block can be expressed as:
\[ F = M \cdot A \]

5. **Substitute the Acceleration**:
Now, substitute the expression for acceleration \( A \) into the equation for \( F \):
\[ F = M \cdot \left(\frac{P}{M + m}\right) \]

6. **Final Expression for the Force**:
Simplifying this gives:
\[ F = \frac{M \cdot P}{M + m} \]

Thus, the force exerted by the rope on the block is:
\[ F = \frac{M \cdot P}{M + m} \]