The expression from centripetal force depends upon mass of body, speed of the body and the radius of circular path. Find the expression for centripetal force.

Options

A. `(Mv^(2))/(R)`
B. `(M^(2)v)/(R)`
C. `(Mv)/(R^(2))`
D. `(Mv)/(R)`

Explanation

To derive the expression for centripetal force, we can follow these steps:

### Step 1: Understand Circular Motion
When an object moves in a circular path, it continuously changes direction. This change in direction requires a force directed towards the center of the circle, known as centripetal force.

### Step 2: Identify the Variables
- Let \( m \) be the mass of the object.
- Let \( v \) be the speed of the object.
- Let \( r \) be the radius of the circular path.

### Step 3: Use the Formula for Centripetal Acceleration
The centripetal acceleration \( a_c \) of an object moving in a circle is given by the formula:
\[ a_c = \frac{v^2}{r} \]
This formula shows that the acceleration depends on the square of the speed and inversely on the radius of the circular path.

### Step 4: Relate Centripetal Force to Centripetal Acceleration
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = m \cdot a \]
In the case of centripetal force, we substitute the centripetal acceleration:
\[ F_c = m \cdot a_c \]
Substituting the expression for centripetal acceleration:
\[ F_c = m \cdot \frac{v^2}{r} \]

### Step 5: Write the Final Expression for Centripetal Force
Thus, the expression for centripetal force \( F_c \) is:
\[ F_c = \frac{m v^2}{r} \]

### Summary
The centripetal force acting on an object moving in a circular path is given by the formula:
\[ F_c = \frac{m v^2}{r} \]