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

[Brilli]

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?

 

[TSny]

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.