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Question

A particle moves so that its position vector is given by r=cosωt^x+sinωt^y. Where ω is a constant. Which of the following is true?
  1. Velocity an acceleration both are perpendicular to r
  2. Velocity and acceleration both are parallel to r
  3. Velocity is perpendicular to r and acceleration is directed towards the origin
  4. Velocity is perpendicular to r and acceleration is directed away from the origin.

A
Velocity and acceleration both are parallel to r
B
Velocity is perpendicular to r and acceleration is directed towards the origin
C
Velocity an acceleration both are perpendicular to r
D
Velocity is perpendicular to r and acceleration is directed away from the origin.
Solution
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Step 1: Velocity and acceleration calculation
r=cosωt^x+sinωt^y ....(1)
Velocity is given by v=drdt

v=ωsinωt^x+ωcosωt^y m/s ....(2)

Acceleration is given by a=dvdt

a=ω2cosωt^xω2sinωt^y m/s2 ....(3)

Step 2: Analysing r,v and a
If the dot product between two vectors is zero that means the vectors are perpendicular to each other.
vr=ωsinωtcosωt+ωsinωtcosωt=0

v is perpendicular to r

ar=ω2cos2ωtω2sin2ωt0

a is not perpendicular to r

Negative sign in eqn(3) indicates acceleration is towards the origin (Opposite to r)
Hence , v is perpendicular to r and the acceleration is directed towards the origin
Option C is the correct answer.

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