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A particle moves so that its position vector is given by r = x cos(Ļt) iĢ + y sin(Ļt) jĢ, where Ļ 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(Ļt)i^+ysin(Ļt)j^ā.
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Sign up / Login First, let's find the velocity vector
The velocity
is the time derivative of the position vector
r.v=dtdrā=dtdā(xcos(Ļt)i^+ysin(Ļt)j^ā) Using the chain rule, we get:
v=āxĻsin(Ļt)i^+yĻcos(Ļt)j^ā Next, let's find the acceleration vector
The acceleration
is the time derivative of the velocity vector
v.a=dtdvā=dtdā(āxĻsin(Ļt)i^+yĻcos(Ļt)j^ā) Again, using the chain rule, we get:
a=āxĻ2cos(Ļt)i^āyĻ2sin(Ļt)j^ā Now, let's analyze the direction of
and
1. Velocity
is given by:
v=āxĻsin(Ļt)i^+yĻcos(Ļt)j^ā The dot product of
and
is:
vā
r=(āxĻsin(Ļt)i^+yĻcos(Ļt)j^ā)ā
(xcos(Ļt)i^+ysin(Ļt)j^ā)=āx2Ļsin(Ļt)cos(Ļt)+y2Ļcos(Ļt)sin(Ļt)=Ļsin(Ļt)cos(Ļt)(āx2+y2) Since
sin(Ļt)cos(Ļt)ī =0 and
x2ī =y2, the dot product is zero, indicating that
is perpendicular to
2. Acceleration
is given by:
a=āxĻ2cos(Ļt)i^āyĻ2sin(Ļt)j^ā This can be rewritten as:
a=āĻ2(xcos(Ļt)i^+ysin(Ļt)j^ā)=āĻ2r 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.