MHT CET · Physics · Center of Mass Momentum and Collision

If the radius of the circular path and frequency of revolution of a particle of mass m are doubled, then the change in its kinetic energy will be ( $E_{i}$ and $E_{f}$ are the initial and final kinetic energies of the particle respectively,)

A 12 $E_{f}$
B 16 $E_{i}$
C 8 $E_{f}$
D 15 $E_{i}$

Verified Solution

Answer & Solution

Correct Answer

(D) 15 $E_{i}$

Step-by-step Solution

Detailed explanation

Initial kinetic energy of body,

E_{i} = \frac{1}{2} m v^{2} = \frac{1}{2} m (\frac{2 π r_{1}}{T_{1}}) [∵ v = \frac{2 π r}{T}]

E_{i} = 2 π^{2} m r_{1}^{2} f_{1}^{2}

Where

, f_{1}

= frequency of revolution of the body.
When,

r_{2} = 2 r_{1}

and

f_{2} = 2 f_{1}

then

E_{f} = 2 π^{2} m . r_{2}^{2} . f_{2}^{2}

= 2 π^{2} m {(2 r_{1})}^{2} {(2 f_{1})}^{2}

E_{f} = 32 π^{2} m r_{1}^{2} f_{1}^{2} = 16.2 π^{2} m r_{1}^{2} f_{1}^{2}

E_{f} = 16 E_{i}

∴

Change in kinetic energy,

Δ E = E_{f} - E_{i}

= 16 E_{i} - E_{i} = 15 E_{i}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the radius of the circular path and frequency of revolution of a particle of mass m are doubled, then the change in its kinetic energy will be ( Ei and Ef are the initial and final kinetic energies of the particle respectively,)

Step-by-step Solution

If the radius of the circular path and frequency of revolution of a particle of mass m are doubled, then the change in its kinetic energy will be ( $E_{i}$ and $E_{f}$ are the initial and final kinetic energies of the particle respectively,)