Derive an expression for the acceleration of a body of mass m moving with a uniform speed v in circular path of radius r.

Derive an expression for the acceleration of a body of mass m moving w

1.	Derive an expression for the acceleration of a body of mass m moving with a uniform speed v in circular path of radius r.
Answer» Solution :Let A = initial position of the body , B = position of the body after a small time INTERVAL t. So, the angular displacement `theta= angleAOB`, is also small, and the displacement x effectively coincides with the arc AB of the CIRCULAR path. Then, Now at A, the velocity v is horizontal , at B, the velocity makes the same angle `theta` with the horizontal direction, keeping its magnitude v to be the same, as the body moves with a uniform speed v . Again, as `theta` is small, the CHANGE of velocity u, effectively coincides with a circular arc. So, `theta = (u)/(v)` Then we get, `(x)/( r) = (u)/(v)` or,u = `(v)/(r )x` Hence, the acceleration is, a = `("change of velocity")/("time") = (u)/(t) = (vx)/(r t)` = `(v)/(r ) v = (v^(2))/(r )` also SHOWS that, as `theta` is small, u is normal to the velocity v. As v is TANGENTIAL at every point on the circular path, the acceleration a is radial. This is centripetal acceleration of a uniform circular motion.