Problem Statement:
A train of mass M is moving on a circular track of radius R with constant speed v. The length of the train is half of the perimeter of the track. We need to determine the linear momentum of the train.

Solution:
To find the linear momentum of the train, we need to determine the product of its mass and velocity.

Understanding the Problem:
Before we start solving the problem, let's first understand the given information:
- The train is moving on a circular track with a constant speed v.
- The radius of the track is R.
- The length of the train is half of the perimeter of the track.

Key Concepts:
To solve this problem, we need to apply the concept of linear momentum, which is defined as the product of an object's mass and its velocity. The formula for linear momentum is:

Linear Momentum (p) = Mass (M) * Velocity (v)

Analysis:
Let's break down the problem into smaller parts and analyze the given information.

1. Perimeter of the Track:
The length of the train is half of the perimeter of the track. The perimeter of the track is given by the formula:

Perimeter = 2 * π * R

Since the length of the train is half of the perimeter, the length of the train can be calculated as:

Length of Train = (1/2) * 2 * π * R = π * R

2. Linear Momentum:
The linear momentum of the train can be calculated using the formula:

Linear Momentum (p) = Mass (M) * Velocity (v)

Solution:
Now that we have analyzed the given information, let's calculate the linear momentum of the train.

The length of the train is given as π * R, and the mass of the train is given as M. The velocity of the train is given as v.

Using the formula for linear momentum, we can calculate the linear momentum of the train as:

Linear Momentum (p) = Mass (M) * Velocity (v)
= M * v

Final Answer:
Therefore, the linear momentum of the train is Mv (Option d).

Summary:
- A train of mass M is moving on a circular track of radius R with constant speed v.
- The length of the train is half of the perimeter of the track, which is given by π * R.
- The linear momentum of the train is given by the formula p = Mv.
- Therefore, the linear momentum of the train is Mv (Option d).

D option