0days until NEET 2026Every question counts. โ
๐งช
Did you know?
Hydrogen is the most abundant element in the universe, making up ~75% of all matter.
A particle moves so that its position vector is given by r = x cos(ฯt) iฬ + y sin(ฯt) jฬ, where ฯ is a constant. Which of the following is true?
hard
Motion in a Plane
2016
physics
Velocity is perpendicular to r and acceleration is directed towards the origin.
Velocity is perpendicular to r and acceleration is directed from the origin.
Velocity and acceleration both are perpendicular to r
Velocity and acceleration both are parallel to r.
Explanation
To solve this problem, we need to analyze the motion of the particle given by the position vector r=xcos(ฯt)i^+ysin(ฯt)j^โ.
Our AI powered practice platform can help you achieve your doctor dream.
Practice 2000+ previous year NEET questionsMore practice, more score
AI generated hintsUse hints to get start solving
AI companion chat to clear doubts 24*7Ask any question, get instant answers
AI generated solutionsGet detailed step by step solutions
Check related NCERT contentRead while solving
Track your progressImprove every day
Sign up / Login First, let's find the velocity vector
The velocity
is the time derivative of the position vector
r.v=dtdrโ=dtdโ(xcos(ฯt)i^+ysin(ฯt)j^โ) Using the chain rule, we get:
v=โxฯsin(ฯt)i^+yฯcos(ฯt)j^โ Next, let's find the acceleration vector
The acceleration
is the time derivative of the velocity vector
v.a=dtdvโ=dtdโ(โxฯsin(ฯt)i^+yฯcos(ฯt)j^โ) Again, using the chain rule, we get:
a=โxฯ2cos(ฯt)i^โyฯ2sin(ฯt)j^โ Now, let's analyze the direction of
and
1. Velocity
is given by:
v=โxฯsin(ฯt)i^+yฯcos(ฯt)j^โ The dot product of
and
is:
vโ
r=(โxฯsin(ฯt)i^+yฯcos(ฯt)j^โ)โ
(xcos(ฯt)i^+ysin(ฯt)j^โ)=โx2ฯsin(ฯt)cos(ฯt)+y2ฯcos(ฯt)sin(ฯt)=ฯsin(ฯt)cos(ฯt)(โx2+y2) Since
sin(ฯt)cos(ฯt)๎ =0 and
x2๎ =y2, the dot product is zero, indicating that
is perpendicular to
2. Acceleration
is given by:
a=โxฯ2cos(ฯt)i^โyฯ2sin(ฯt)j^โ This can be rewritten as:
a=โฯ2(xcos(ฯt)i^+ysin(ฯt)j^โ)=โฯ2r This shows that the acceleration
is directed towards the origin, as it is opposite to
Therefore, the correct option is:
Option 1: Velocity is perpendicular to
and acceleration is directed towards the origin.
However, the provided correct option is 2, which is incorrect based on the analysis.