Question 221: Work, Energy and Power & Circular Motion

Question

A particle of mass ' $m$ ' performs uniform circular motion of radius ' $r$ ' with linear speed ' $v$ ' under the application of force ' $F$ '. If ' $m$ ', ' $v$ ' and $' \mathrm{r}$ ' are all increased by $20 \%$ the necessary change in force required to maintain the particle in uniform circular motion, is

Accepted Answer

In uniform circular motion, the centripetal force required to maintain the motion is given by the formula:
$$ F = \frac{m v^2}{r} $$
Let's denote the initial values of mass, velocity, and radius as $ m $, $ v $, and $ r $, respectively. The initial force is:
$$ F_{	ext{initial}} = \frac{m v^2}{r} $$
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_{	ext{new}} $, is:
$$ F_{	ext{new}} = \frac{m' (v')^2}{r'} = \frac{(1.2m)(1.2v)^2}{1.2r} $$
Calculating the new force, we have:
$$ F_{	ext{new}} = \frac{1.2m 	imes (1.44v^2)}{1.2r} $$
Simplifying the expression:
$$ F_{	ext{new}} = \frac{1.2 	imes 1.44 	imes m 	imes v^2}{1.2 	imes r} $$
$$ F_{	ext{new}} = \frac{1.44m v^2}{r} $$
The change in force can be calculated as:
$$ 	ext{Change in force} = F_{	ext{new}} - F_{	ext{initial}} $$
Substituting the expressions for $ F_{	ext{new}} $ and $ F_{	ext{initial}} $:
$$ 	ext{Change in force} = \frac{1.44m v^2}{r} - \frac{mv^2}{r} $$
Simplifying the change in force expression:
$$ 	ext{Change in force} = \left(1.44 - 1ight) \frac{mv^2}{r} $$
$$ 	ext{Change in force} = 0.44 \frac{mv^2}{r} $$
The percentage change in force is:
$$ 	ext{Percentage change} = \left(\frac{	ext{Change in force}}{F_{	ext{initial}}}ight) 	imes 100\% $$
$$ 	ext{Percentage change} = \left(\frac{0.44 \frac{mv^2}{r}}{\frac{mv^2}{r}}ight) 	imes 100\% $$
$$ 	ext{Percentage change} = 0.44 	imes 100\% $$
$$ 	ext{Percentage change} = 44\% $$
Therefore, the necessary change in force required is
44%
. The correct answer is
Option B: $ 44\% $
.

Answer

$12 \%$

Answer

$44 \%$

Answer

$14 \%$

Answer

$144 \%$

Work, Energy and Power & Circular Motion

A particle of mass ' m ' performs uniform circular motion of radius ' r ' with linear speed ' v ' under the application of force ' F '. If ' m ', ' v ' and ' \mathrm{r} ' are all increased by 20 \% the necessary change in force required to maintain the particle in uniform circular motion, is

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