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Physics Motion in a Plane Mix Matrix Match Questions
Published on: September 12, 2026

A pendulum of length 3.2 m is free to rotate in a vertical circle. Let the velocity of pendulum at its lowest position be v 0 then match the following-

Column – I

Column – II

(i) If v0 = 4m/s,

maximum height attained

by pendulum (in meter)

[A] Zero

(ii) If v0 =  m/s,

minimum velocity of

pendulum (in meter/sec)

[B] 2

(iii) If v0 = 8 m/s,

maximum height attained

by pendulum (in meter)

[C] 3.2

(iv) If v0 = 8 m/s,

minimum velocity of

pendulum (in meter/sec)

[D] 5.4

Correct Matrix Matching

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Text Solution

Verified by Experts
The correct answer is:
A
Step 1: Analyzing the maximum height attained by the pendulum when starting at its lowest point with velocity \( v_0 \). The height attained can be calculated using conservation of energy.
The potential energy at height \( h \) is given by \( PE = mgh \). The kinetic energy at the lowest point is given by \( KE = \frac{1}{2} mv_0^2 \).
On reaching the maximum height, all kinetic energy is converted into potential energy:
\( \frac{1}{2} mv_0^2 = mgh \)
Canceling mass (m), we have: \( \frac{1}{2} v_0^2 = gh \)
Rearranging gives: \( h = \frac{v_0^2}{2g} \)
For \( v_0 = 4 \, m/s \): \( h = \frac{4^2}{2 \times 9.8} = \frac{16}{19.6} \approx 0.816 \, m \)
Thus, for Option (i), the maximum height is not zero.

Step 2: Finding the minimum velocity at the highest point. For a pendulum to just maintain a circular path, the minimum velocity \( v_{min} \) at the top needs to be calculated using the centripetal force required: \( \frac{mv_{min}^2}{L} = mg \) which leads to \( v_{min} = \sqrt{gL} \).
Using \( g \approx 9.8 \, m/s^2 \) and \( L = 3.2 \, m \): \( v_{min} = \sqrt{9.8 \times 3.2} \approx 5.57 \, m/s \). For \( v_0 = 4 \, m/s \), the minimum velocity does not contribute.

Thus the correct matches are:
(i) Zero [A], (ii) 2 [B], (iii) 3.2 [C], (iv) 5.4 [D].
Therefore, the correct option for (i) is [A].

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