It mass, speed and radius of the circle of a particle moving uniformly in a circular path are all increased by `50%` , the necessary force required to maintain the body moving in the circular path will have to be increased by

Options

A. `225%`
B. `125%`
C. `150%`
D. `100%`

Explanation

To solve the problem, we need to analyze how the centripetal force changes when the mass, speed, and radius of a particle moving in a circular path are all increased by 50%.

### Step-by-Step Solution:

1. **Understanding the Formula for Centripetal Force**:
The centripetal force \( F \) required to keep a particle moving in a circular path is given by the formula:
\[ F = \frac{m v^2}{r} \]
where:
- \( m \) = mass of the particle
- \( v \) = speed of the particle
- \( r \) = radius of the circular path

2. **Calculating the New Values**:
Given that mass, speed, and radius are all increased by 50%, we can express the new values as:
- New mass \( m' = 1.5m \)
- New speed \( v' = 1.5v \)
- New radius \( r' = 1.5r \)

3. **Substituting New Values into the Centripetal Force Formula**:
We substitute the new values into the centripetal force formula:
\[ F' = \frac{m' (v')^2}{r'} \]
Substituting the new values:
\[ F' = \frac{(1.5m) (1.5v)^2}{1.5r} \]

4. **Simplifying the Expression**:
Now we simplify the expression:
\[ F' = \frac{(1.5m) (2.25v^2)}{1.5r} \]
Notice that \( (1.5) \) in the numerator and denominator cancels out:
\[ F' = \frac{2.25mv^2}{r} \]

5. **Relating the New Force to the Original Force**:
We know that the original force \( F \) is:
\[ F = \frac{mv^2}{r} \]
Therefore, we can express the new force \( F' \) in terms of the original force:
\[ F' = 2.25F \]

6. **Calculating the Increase in Force**:
To find the increase in force, we calculate:
\[ \text{Increase in force} = F' - F = 2.25F - F = 1.25F \]
This means the force has increased by \( 1.25F \).

7. **Calculating the Percentage Increase**:
The percentage increase in force is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Original}} \times 100\%\right) = \left(\frac{1.25F}{F} \times 100\%\right) = 125\% \]

### Final Answer:
The necessary force required to maintain the body moving in the circular path will have to be increased by **125%**.