, velocity, and radius as $ m $, $ v $, and $ r $, respectively. The initial force is:

$$ F_{\text{initial}} = \frac{m v^2}{r} $$

Now, if mass, velocity, and radius each increase by 20%, the new values will be:

New mass = $ m' = 1.2m $

New velocity = $ v' = 1.2v $

New radius = $ r' = 1.2r $

The new force required, $ F_{\text{new}} $, is:

$$ F_{\text{new}} = \frac{m' (v')^2}{r'} = \frac{(1.2m)(1.2v)^2}{1.2r} $$

Calculating the new force, we have:

$$ F_{\text{new}} = \frac{1.2m \times (1.44v^2)}{1.2r} $$

Simplifying the expression:

$$ F_{\text{new}} = \frac{1.2 \times 1.44 \times m \times v^2}{1.2 \times r} $$

$$ F_{\text{new}} = \frac{1.44m v^2}{r} $$

The change in force can be calculated as:

$$ \text{Change in force} = F_{\text{new}} - F_{\text{initial}} $$

Substituting the expressions for $ F_{\text{new}} $ and $ F_{\text{initial}} $:

$$ \text{Change in force} = \frac{1.44m v^2}{r} - \frac{mv^2}{r} $$

Simplifying the change in force expression:

$$ \text{Change in force} = \left(1.44 - 1\right) \frac{mv^2}{r} $$

$$ \text{Change in force} = 0.44 \frac{mv^2}{r} $$

The percentage change in force is:

$$ \text{Percentage change} = \left(\frac{\text{Change in force}}{F_{\text{initial}}}\right) \times 100\% $$

$$ \text{Percentage change} = \left(\frac{0.44 \frac{mv^2}{r}}{\frac{mv^2}{r}}\right) \times 100\% $$

$$ \text{Percentage change} = 0.44 \times 100\% $$

$$ \text{Percentage change} = 44\% $$

Therefore, the necessary change in force required is 44%. The correct answer is Option B: $ 44\% $.

Circular Motion

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