How F=mv^2/r? Please explain.

Asked by bhavana147 | 2nd Apr, 2020, 09:16: AM

Expert Answer:

Let us consider a particle of mass 'm' be moving in a circle of radius 'r' with a constant speed 'v'. Consider that the point is at A at any instant and then at B after a short time 't'. 
Thus, AB = vt
The velocity at A is along the tangent AL and at B is along the tangent BM, i.e., the two vectors AL and BM represent the veolcity at A and B respectively and are equal in length (since speed is the same, v)
Draw O'a parallel to and equal to vector AL which represents the velocity 'v' at A. Similarly, draw O'b parallel to and equal to vector BM which represents the velocity 'v' at B, a short time t later.
O'a = O'b = v.
From the law of triangle of velocities, ab represents the change in velocity.
Rate of change of velocity, i.e. acceleration acting on the particle is given as the change in velocity ab taken place in time t, i.e. ab/t.
 
 
 
When the time t is very short, AB is almost a straight line and AOB is a ΔAOB and ΔO'ab are similar.

therefore fraction numerator a b over denominator A B end fraction equals space fraction numerator O apostrophe a over denominator O A end fraction
fraction numerator a b over denominator v t end fraction equals v over r space space left parenthesis because A B thin space equals v t space a n d space O apostrophe a space space equals v space comma space O A space equals r right parenthesis
rightwards double arrow fraction numerator a b over denominator t end fraction equals v squared over r space
rightwards double arrow a c c e l e r a t i o n space equals space v squared over r space
W e space k n o w comma space
F o r c e space equals space m a s s space cross times space a c c e l e r a t i o n space
T h u s comma space
C e n t r i p e t a l space f o r c e comma space F subscript c space equals space m cross times v squared over r space 
 
Hence proved ... 

Answered by Shiwani Sawant | 2nd Apr, 2020, 01:06: PM