What mistake did I make in finding the net force on a rotating disc?

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    Irodov Mechancis
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SUMMARY

The discussion focuses on calculating the net force exerted by a rotating disc on a mass moving along its diameter. The key mistake identified is the confusion between speed and velocity; the expression for net velocity at radius r is correctly given as (v² + (wr)²)^(1/2), but this represents speed, not velocity. The distinction is crucial as acceleration is defined as the rate of change of velocity, which includes both speed and direction. For accurate calculations, one must consider the vector nature of velocity and apply the appropriate equations of motion in polar coordinates.

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Homework Statement


A horizontal disc is rotating with an angular velocity w about its axis perpendicular to its plane. A small body of mass m moves with velocity v relative to the disc alog the diameter. Find the net force exerted by the disk on the body as function of radius from centre.

Homework Equations

The Attempt at a Solution


Attepmt-the velocity component perpendicular to radius is wr and its normal component is v. Thus net velocity at radius r is (v^2+(wr)^2)^(1/2).
I differentiate it wrt time but the answer comes to be wrong. Could anyone point out my mistake?
 
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Brilli said:
Thus net velocity at radius r is (v^2+(wr)^2)^(1/2).
I differentiate it wrt time but the answer comes to be wrong. Could anyone point out my mistake?
The expression (v2+(wr)2)1/2 represents the speed of the particle, not the velocity. Velocity is a vector quantity.
Acceleration is defined as rate of change of velocity, not rate of change of speed. So, acceleration takes into account change in direction of velocity as well as change in speed.

For a review of velocity and acceleration in polar coordinates, see
https://ocw.mit.edu/courses/aeronau...fall-2009/lecture-notes/MIT16_07F09_Lec05.pdf

Note equation (4), which gives the acceleration.
 
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