A block of mass m is tied with an inextensible light string of length ℓ . One end of the string is fixed at point O. Block is released (from rest) at A. Find acceleration of particle during its motion

Question

A block of mass m is tied with an inextensible light string of length ℓ . One end of the string is fixed at point O. Block is released (from rest) at A. Find acceleration of particle during its motion in a vertical plane at positions specified in List-I and match them with List-II. Given that A and O are at the same horizontal level. List-I List-II (P) Highest point (1) Acceleration is horizontal. (Q) At lowest point (2) Acceleration is vertically upwards. (R) At θ = tan − 1 ( 3 ​ ) with vertical (3) Acceleration is vertically downwards. (S) At θ = tan − 1 ( 1/ 3 ​ ) with vertical (4) Acceleration has both horizontal and vertical components. (5) Acceleration is zero.

TestHub · Accepted Answer

Take the angle from the downward vertical. Energy conservation gives

$v^{2} = 2 g ℓ cos θ$

The radial and tangential accelerations are

$a_{r} = 2 g cos θ, a_{t} = g sin θ$

P: speed is zero, so gravity acts downward. Q: tangential acceleration is zero; radial acceleration acts upward.

On the right-hand side:

$a_{x} = - 3 g sin θ cos θ$

$a_{y} = g (3 cos^{2} θ - 1)$

$θ = 6 0^{\circ} : a_{x} = - \frac{3 3}{4} g, a_{y} = - \frac{1}{4} g$

$θ = 3 0^{\circ} : a_{x} = - \frac{3 3}{4} g, a_{y} = \frac{5}{4} g$

Both components are nonzero for R and S.

P to 3, Q to 2, R to 4, S to 4. Answer: B.

Physics - W.E.P. Question with Solution | TestHub

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Doubts & Discussion

JEE Main 2028 (Class 11)

List-I	List-II
(P) Highest point	(1) Acceleration is horizontal.
(Q) At lowest point	(2) Acceleration is vertically upwards.
(R) At $θ = tan^{- 1} (3)$ with vertical	(3) Acceleration is vertically downwards.
(S) At $θ = tan^{- 1} (1/ 3)$ with vertical	(4) Acceleration has both horizontal and vertical components.
	(5) Acceleration is zero.