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MCQEasyJEE 2024Circular Motion Dynamics

JEE Physics 2024 Question with Solution

If the radius of curvature of the path of two particles of the same mass are in the ratio 3:43:4, then in order to have constant centripetal force, their velocities will be in the ratio of:

  • A

    3:2\sqrt{3} : 2

  • B

    1:31 : \sqrt{3}

  • C

    3:1\sqrt{3} : 1

  • D

    2:32 : \sqrt{3}

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Two particles have the same mass, and the radii of curvature are in the ratio r1:r2=3:4r_1:r_2 = 3:4.

Find: The ratio of their velocities when the centripetal force is constant.

For centripetal motion,

Fc=mv2rF_c = \frac{mv^2}{r}

Since the masses are the same and the centripetal force is constant for both particles,

mv12r1=mv22r2\frac{mv_1^2}{r_1} = \frac{mv_2^2}{r_2}

Cancelling mm,

v12r1=v22r2\frac{v_1^2}{r_1} = \frac{v_2^2}{r_2}

Rearranging,

v12r2=v22r1v_1^2 r_2 = v_2^2 r_1

Using r1:r2=3:4r_1:r_2 = 3:4,

4v12=3v224v_1^2 = 3v_2^2

So,

v12v22=34\frac{v_1^2}{v_2^2} = \frac{3}{4}

Taking square root,

v1v2=32\frac{v_1}{v_2} = \frac{\sqrt{3}}{2}

Therefore, the velocities are in the ratio 3:2\sqrt{3}:2. The correct option is A.

Direct Ratio Method

Given: Equal masses and constant centripetal force.

Find: Velocity ratio.

From

F=mv2rF = \frac{mv^2}{r}

if FF and mm are constant, then

v2∝rv^2 \propto r

Hence,

v∝rv \propto \sqrt{r}

Therefore,

v1v2=r1r2=34=32\frac{v_1}{v_2} = \sqrt{\frac{r_1}{r_2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}

So the required ratio is 3:2\sqrt{3}:2, which corresponds to A.

Common mistakes

  • Using v∝rv \propto r instead of v∝rv \propto \sqrt{r}. This is wrong because centripetal force depends on v2v^2, not directly on vv. First write mv2r=constant\frac{mv^2}{r} = \text{constant}, then take the square root at the end.

  • Inverting the radius ratio while substituting. This gives the wrong velocity ratio because v12v22=r1r2\frac{v_1^2}{v_2^2} = \frac{r_1}{r_2} for constant centripetal force and equal masses. Substitute r1r2=34\frac{r_1}{r_2} = \frac{3}{4} carefully.

  • Forgetting that the masses are the same. If you do not cancel mm, you may carry unnecessary terms and confuse the relation. Since both particles have equal mass, cancel mm before comparing velocities.

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