Question 282: Work, Energy and Power & Circular Motion

Question

A particle is moving along the circular path with constant speed and centripetal acceleration 'a'. If the speed is doubled, the ratio of its acceleration after and before the change is

Accepted Answer

To determine the ratio of the particle's acceleration after and before the change in speed, let's start with the formula for centripetal acceleration.
The centripetal acceleration of a particle moving with a constant speed $ v $ along a circular path of radius $ r $ is given by:
$$ a = \frac{v^2}{r} $$
If the speed of the particle is doubled, the new speed $ v' $ will be:
$$ v' = 2v $$
The new centripetal acceleration $ a' $ for this doubled speed is:
$$ a' = \frac{{v'}^2}{r} $$
Substituting $ v' = 2v $ into the equation for $ a' $:
$$ a' = \frac{{(2v)}^2}{r} = \frac{4v^2}{r} $$
The initial centripetal acceleration $ a $ is:
$$ a = \frac{v^2}{r} $$
Now, the ratio of the new centripetal acceleration $ a' $ to the initial centripetal acceleration $ a $ is:
$$ \frac{a'}{a} = \frac{\frac{4v^2}{r}}{\frac{v^2}{r}} = \frac{4v^2}{v^2} = 4 $$
Thus, the ratio of the particle's acceleration after doubling the speed to the acceleration before the speed change is:
$$ 4: 1 $$
Therefore, the correct option is:
Option D: 4: 1

Answer

$$3: 1$$

Answer

$$1: 4$$

Answer

$$2: 1$$

Answer

$$4: 1$$

Work, Energy and Power & Circular Motion

A particle is moving along the circular path with constant speed and centripetal acceleration 'a'. If the speed is doubled, the ratio of its acceleration after and before the change is

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