Question

A car travels at 105 km/h on a level road in the positive direction of an x axis. Each tire has a diameter of 63.6 cm. Relative to a woman riding in the car and in unit-vector notation, what are the velocity V at the (a) center, (b) top, and (c) bottom of the tire and the magnitude a of the acceleration at the (d) center, (e) top, and (f) bottom of each tire? Relative to a hitchhiker sitting next to the road and in unit-vector notation, what are the velocity V at the (g) center, (h) top, and (i)bottom of the tire and the magnitude a of the acceleration at the (j) center, (k) top, and (l) bottom of each tire?

Answer 1

Let say be the unit vector in the direction of the car. and let be the velocity of car with respect to ground. So the center of the wheel or tire is moving with velocity V with respect to ground, And the whole tire is moving with speed V in direction with respect to center of the wheel.

(g) Now relative to the hitchhiker or relative to ground frame. the velocity of center of tire is

(h) and velocity of the top of the tire with respect to ground= the velocity of center of tire with respect to ground + velocity of the top of the tire with respect to center (This is a vector sum)

(i) and velocity of the bottom of the tire with respect to ground= the velocity of center of tire with respect to ground + velocity of the bottom of the tire with respect to center

(j) The car has no acceleration so the center of the tire has zero acceleration with respect to ground

(K) Acceleration of the top of the tire with respect to ground= the Acceleration of center of tire with respect to ground + Acceleration of the top of the tire with respect to center.= 0+Acceleration of the top of the tire with respect to center

Acceleration of the top of the tire with respect to center= angular acceleration= in the down word direction that is towards center

(K) similarly acceleration of the bottom with respect to ground is 2679m/s² towards the center or upward direction

(a) Velocity of the center with respect to car or rider is zero

(b) velocity of the top of the tire with respect to car is simply V in the forward direction that is

(C) velocity of the bottom of the tire with respect to car is simply V in the backward direction direction that is

(d) acceleration of center i

(e) /(F) We have already calculated Acceleration of the top and bottom of the tire with respect to center or the rider as it is similar as with respect to ground