JEE Advanced 2019 · Paper 2 · Physics Q2

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PhysicsMultiple correct+4 / -1Question 2 of 54

A block of mass $2M$ is attached to a massless spring with spring-constant $k$ . This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1$ , $a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$ . Which of the following option(s) is/are correct?

[ $g$ is the acceleration due to gravity. Neglect friction]

A block 2M on a smooth horizontal table, tied to a wall by a spring, accelerates right (a₁). Its string passes over a fixed pulley at the table edge and holds a movable pulley; over the movable pulley hang a block M (acceleration a₂, down) and a block 2M (acceleration a₃, down).

x_0 = \dfrac{4Mg}k

When spring achieves an extension of

\dfrac{x_0}2

for the first time, the speed of the block connected to the spring is

3g\sqrt{\dfrac M{5k}}

At an extension of

\dfrac{x_0}4

of the spring, the magnitude of acceleration of the block connected to the spring is

\dfrac{3g}{10}

a_2 - a_1 = a_1 - a_3

Answer and worked solution

Official answer: D

Constraint. The movable pulley moves down as fast as the table block moves right, and it is the average of the two hanging blocks:
$a_1=\frac{a_2+a_3}2\implies a_2-a_1=a_1-a_3.$
So (D) is true.

Dynamics. Let $T$ be the tension in the lower string; the upper string carries $2T$ , and $x$ is the spring's extension.
$2Ma_1=2T-kx,\qquad Ma_2=Mg-T,\qquad 2Ma_3=2Mg-T.$
Eliminating $T$ gives
$a_1=\frac{4g}7-\frac{3k}{14M}x.$
This is SHM with $\omega^2=\dfrac{3k}{14M}$ about the equilibrium extension $x_e=\dfrac{8Mg}{3k}$ .

(A) The block starts from rest at $x=0$ , so $x_0=2x_e=\dfrac{16Mg}{3k}$ , not $\dfrac{4Mg}k$ . False.

(B) At $x_0/2=x_e$ the speed is the maximum,
$\omega x_e=\sqrt{\frac{3k}{14M}}\cdot\frac{8Mg}{3k}=8g\sqrt{\frac M{42k}}\approx1.23\,g\sqrt{\frac Mk}.$
The option gives $3g\sqrt{\frac M{5k}}\approx1.34\,g\sqrt{\frac Mk}$ . False.

(C) At $x_0/4$ the displacement from equilibrium is $x_e/2$ , so
$|a_1|=\omega^2\frac{x_e}2=\frac{3k}{14M}\cdot\frac{4Mg}{3k}=\frac{2g}7,$
not $\frac{3g}{10}$ . False.

(D)

Topics:Constraint motion and pulleys Spring potential energy Spring-mass systems

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JEE Advanced 2019 · Paper 2, Physics question 2