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JEE Main 20208 JanuaryEvening Shift
Exam
JEE Main
Year
2020
Date
8 January
Shift
Evening Shift
Subject
Physics
Chapter
Motion in Two Dimensions

JEE Main 2020 · 8 January · Evening Shift · Physics · Motion in Two Dimensions

A particle moves such that its position vector → r (t) = cos ^ i +...

A particle moves such that its position vector \overset{\to}{r}\left(t\right) = cos \omegat\overset{^}{i} + sin \omegat\overset{^}{j} where ω\omega is a constant and tt is time. Then which of the following statements is true for the velocity v→(t)andacceleration\overset{\to}{v}\left(t\right) and acceleration\overset{\to}{a}\left(t\right) of the particle:

Options

  1. Av→\overset{\to}{v} is perpendicular to r→\overset{\to}{r} and a→\overset{\to}{a} is directed away from the origin
  2. Bv→\overset{\to}{v} and a→\overset{\to}{a} both are perpendicular to r→\overset{\to}{r}
  3. Cv→\overset{\to}{v} and a→\overset{\to}{a} both are parallel to r→\overset{\to}{r}
  4. Dv→\overset{\to}{v} is perpendicular to r→\overset{\to}{r} and a→\overset{\to}{a} is directed towards the origin

Correct answer

D. v→\overset{\to}{v} is perpendicular to r→\overset{\to}{r} and a→\overset{\to}{a} is directed towards the origin

Step-by-step solution

\overset{\to}{r} = cos \omegat\overset{^}{i} + sin \omegat\overset{^}{j}
\overset{\to}{v} = \dfrac{d\overset{\to}{r}}{dt} = \omega\left(- sin \omegat\overset{^}{i} + cos \omegat\overset{^}{j}\right)
\overset{\to}{a} = \dfrac{d\overset{\to}{v}}{dt} = - {\omega}^{2}\left(cos \omegat\overset{^}{i} + sin \omegat\overset{^}{j}\right)
a→=−ω2r→∴a→\overset{\to}{a} = - {\omega}^{2}\overset{\to}{r} ∴ \overset{\to}{a} is antiparallel to r→\overset{\to}{r}
v→.r→=ω(−sinωtcos\omegat+cosωtsin\omegat)=0\overset{\to}{v} . \overset{\to}{r} = \omega\left(- sin \omega tcos \omegat + cos \omega tsin \omegat\right) = 0
So v→⊥r→\overset{\to}{v} ⊥ \overset{\to}{r}
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