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Physics · Rotational Dynamics

JEE Advanced 2021 — Paper 1 — Question 5

A particle of mass M=0.2 kg\mathrm{M}=0.2 \mathrm{~kg} is initially at rest in the xy -plane at a point (x=−ℓ,y=−h)(\mathrm{x}=-\ell, \mathrm{y}=-\mathrm{h}), where ℓ=10 m\ell=10 \mathrm{~m} and h=1 m\mathrm{h}=1 \mathrm{~m}. The particle is accelerated at time t=0\mathrm{t}=0 with a constant acceleration a=10 m/s2\mathrm{a}=10 \mathrm{~m} / \mathrm{s}^{2} along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by L→\overrightarrow{\mathrm{L}} and τ⃗\vec{\tau} respectively. i^,j^\hat{\mathrm{i}}, \hat{\mathrm{j}} and k^\hat{\mathrm{k}} are unit vectors along the positive x , y and z-directions, respectively. If k^=i^×j^\hat{k}=\hat{i} \times \hat{j} then which of the following statement(s) is(are) correct?

  1. Option A:

    The particle arrives at the point (x=ℓ,y=−h)(x=\ell, \mathrm{y}=-\mathrm{h}) at time t=2 s\mathrm{t}=2 \mathrm{~s}.

    Correct
  2. Option B:

    τ⃗=2k^\vec{\tau}=2 \hat{k} when the particle passes through the point (x=ℓ,y=−h)(x=\ell, y=-h)

    Correct
  3. Option C:

    L→=4k^\overrightarrow{\mathrm{L}}=4 \hat{\mathrm{k}} when the particle passes through the point (x=ℓ,y=−h)(\mathrm{x}=\ell, \mathrm{y}=-\mathrm{h})

    Correct
  4. Option D:

    τ⃗=k^\vec{\tau}=\hat{\mathrm{k}} when the particle passes through the point (x=0,y=−h)(\mathrm{x}=0, \mathrm{y}=-\mathrm{h})

Answer: A, B, C

Step-by-step solution

Time taken to reach from (−ℓ,−h)(-\ell,-\mathrm{h}) to (ℓ,−h)(\ell,-\mathrm{h}) is given by 2ℓ=12at22 \ell=\frac{1}{2} \mathrm{at}^{2}

20=12(10)t2t=2sec20=\frac{1}{2}(10) \mathrm{t}^{2} \quad \mathrm{t}=2 \mathrm{sec} L→=mvh⁡k^=math⁡k^\overrightarrow{\mathrm{L}}=\operatorname{mvh} \hat{\mathrm{k}}=\operatorname{math} \hat{\mathrm{k}} L→=(0.2)(10)2(1)k^=4k^\overrightarrow{\mathrm{L}}=(0.2)(10) 2(1) \hat{\mathrm{k}}=4 \hat{\mathrm{k}}

[when particle passes through (ℓ,−h)(\ell,-\mathrm{h})

τ⃗=dL→dt=mah⁡k^=(0.2)(10)(1)k^=2k^\vec{\tau}=\frac{\mathrm{d} \overrightarrow{\mathrm{L}}}{\mathrm{dt}}=\operatorname{mah} \hat{\mathrm{k}}=(0.2)(10)(1) \hat{\mathrm{k}}=2 \hat{\mathrm{k}} (always)

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2021
Paper
Paper 1
Subject
Physics
Chapter
Rotational Dynamics
Topic
Angular Momentum and its Conservation

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