A block B of mass 3 kg lies on a…

A block B of mass 3 kg lies on a smooth horizontal surface and is attached to a horizontal spring of constant 100 N m^-1 whose other end is fixed. A…

One or more options correct

A block

B

of mass

3\ \mathrm{kg}

lies on a smooth horizontal surface and is attached to a horizontal spring of constant

100\ \mathrm{N\,m^{-1}}

whose other end is fixed. A block

A

of mass

1\ \mathrm{kg}

rests on top of

B

, the coefficient of static friction between

A

and

B

being

0.5

. The system oscillates along the spring axis with

A

and

B

moving together. Take

g=10\ \mathrm{m\,s^{-2}}

. Which of the following statements is/are correct?

A. The largest amplitude for which $A$ does not slip on $B$ is $0.20\ \mathrm{m}$ .
B. The time period of the oscillation is $0.2\pi\ \mathrm{s}$ .
C. At the largest non-slipping amplitude the maximum kinetic energy of the system is $4\ \mathrm{J}$ .
D. If the amplitude is $0.10\ \mathrm{m}$ , the maximum friction force on $A$ is $2.5\ \mathrm{N}$ .

Show answer and solution

Answer: A, D

The combined mass is

4\ \mathrm{kg}

, so

\omega^2=k/(M+m)=25\ \mathrm{s^{-2}}

\omega=5\ \mathrm{rad\,s^{-1}}

and

T=2\pi/\omega=0.4\pi\ \mathrm{s}

. Block

A

is accelerated only by friction, so the friction force is

m\omega^2x

and is greatest at the extremes:

m\omega^2A\le\mu mg

, giving

A_{max}=\mu g/\omega^2=5/25=0.20\ \mathrm{m}

. (A) is true. (B) is false because the period is

0.4\pi\ \mathrm{s}

, not

0.2\pi\ \mathrm{s}

. (C) The maximum kinetic energy equals the total energy

\tfrac12kA^2=\tfrac12\times100\times0.04=2\ \mathrm{J}

, so

4\ \mathrm{J}

is false. (D) For

A=0.10\ \mathrm{m}

the friction force is

1\times25\times0.10=2.5\ \mathrm{N}

, which is below the limit

5\ \mathrm{N}

, so it is true. Correct options: (A) and (D).

Original question written in the JEE Advanced 2019 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake