It mass, speed and radius of the circle of a particle moving uniformly in a circular path are all increased by `50%` , the necessary foce 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. **Understand the Formula for Centripetal Force**:
The centripetal force \( F_c \) required to keep a particle moving in a circular path is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is its speed, and \( r \) is the radius of the circular path.

2. **Initial Values**:
Let the initial mass be \( m \), the initial speed be \( v \), and the initial radius be \( r \). The initial centripetal force \( F_{c} \) can be expressed as:
\[ F_{c} = \frac{mv^2}{r} \]

3. **Increase Each Parameter by 50%**:
If we increase the mass, speed, and radius by 50%, the new values will be:
- New mass \( m' = 1.5m \)
- New speed \( v' = 1.5v \)
- New radius \( r' = 1.5r \)

4. **Calculate the New Centripetal Force**:
Substitute the new values into the centripetal force formula:
\[ F_{c}' = \frac{m' (v')^2}{r'} \]
Substituting the new values:
\[ F_{c}' = \frac{(1.5m)(1.5v)^2}{1.5r} \]
Simplifying this:
\[ F_{c}' = \frac{(1.5m)(2.25v^2)}{1.5r} = \frac{(1.5 \times 2.25)mv^2}{1.5r} = \frac{3.375mv^2}{r} \]

5. **Relate New Force to Initial Force**:
Now, we can express the new force in terms of the initial force:
\[ F_{c}' = 3.375 \cdot \frac{mv^2}{r} = 3.375 F_{c} \]

6. **Calculate the Percentage Increase in Force**:
To find the percentage increase in force, we use the formula:
\[ \text{Percentage Increase} = \frac{F_{c}' - F_{c}}{F_{c}} \times 100\% \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{3.375 F_{c} - F_{c}}{F_{c}} \times 100\% = \frac{2.375 F_{c}}{F_{c}} \times 100\% = 237.5\% \]

7. **Final Result**:
The necessary force required to maintain the body moving in the circular path will have to be increased by \( 237.5\% \).