A child of mass m is standing at the edge of a carousel. Both the carousel and the child are initially stationary. The carousel is a large flat, uniform disc of mass M, and radius R, which turns on a frictionless pivot at its center. It has moment of inertia I = MR2 /2. The child is very small compared to the carousel. The child starts to run around the edge of the carousel at speed v. The direction of child is counter-clockwise when the carousel is viewed from above. Taking the z-axis as upwards, what is the angular velocity of the carousel after the child has started running?

the angular velocity of the carousel after the child has started running =

Explanation:

Given that

the mass of the child = m

The radius of the disc = R

moment of inertia I =

change in time =

By using the torque around the inertia ; we have:

T = I×∝

where

R×F = I × ∝

R×F = ∝

F = ∝

∝ =           ( expression for angular  angular acceleration)

The first equation of motion of rotating wheel can be expressed as :

where ;

∝ =    

Then;

∴ the angular velocity of the carousel after the child has started running =

the old price per ounce is c/4.

the new price per ounce is c/3.6.

the ratio of the old price to the new one is

(c/4)/(c/3.6) = (c/c)×(3.6/4) = 0.9 . . = 9/10 . . "nine tenths" in either form