JEE Advanced 2016 · Paper 2 · Physics Q12

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PhysicsMultiple correct+4 / -2Question 12 of 54

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$ . Consider two cases: (i) when the block is at $x_0$ ; and (ii) when the block is at $x=x_0+A$ . In both the cases, a particle with mass $m\,(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass $m$ is placed on the mass $M$ ?

The amplitude of oscillation in the first case changes by a factor of

\sqrt{\dfrac{M}{m+M}}

, whereas in the second case it remains unchanged

The final time period of oscillation in both the cases is same

The total energy decreases in both the cases

The instantaneous speed at

x_0

of the combined masses decreases in both the cases

Answer and worked solution

Official answer: A, B, D

Case (i), at $x_0$ . Momentum is conserved: $v'=\dfrac{Mv}{M+m}$ . With $\omega'=\sqrt{k/(M+m)}$ ,
$A'=\frac{v'}{\omega'}=A\sqrt{\frac M{M+m}}.$

Case (ii), at the extreme. The block is at rest, so the amplitude is unchanged. (A) is true.

(B) Both cases end with $T=2\pi\sqrt{(M+m)/k}$ . True.
(C) Energy is lost in case (i) (a perfectly inelastic "collision") but not in case (ii). False.
(D) In case (i), $v'<v$ . In case (ii), $A\omega'<A\omega$ . Both decrease. True.

(A), (B), (D)

Topics:Spring-mass systems Energy in SHM Conservation of linear momentum

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JEE Advanced 2016 · Paper 2, Physics question 12