A wheel has radius of 4 inches Point P on the rim of the wh

A wheel has radius of 4 inches . Point P on the rim of the wheel in its initial position has coordinates of (3,root7) relative to the center of the wheel. The wheel then rolls 19pi inches to the left. Find the final coordniates of the point P after this rotation, relative to the center of the wheel.

Solution

( 3 , sqrt7)

Assume wheel is a circle centred at origin.

r = sqrt[ 3^2 +7] = 4

theta = tan^-1(sqrt(7)/3) = 0.72 rad

After rotation = 19pi + 0.72 = 60.41 rad

x = 4cos60.41 = -3

y = 4sin60.41 = -2.63

( x, y) = ( -3 , -2.63)