Question

A car travels at 80 km/h on a level road in the positive direction of an x axis. Each tire has a diameter of 66 cm. Relative to a woman riding in the car and in unit-vector notation, what are the velocity 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 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

The velocity and acceleration of the car and tire relative to a woman in the car and a hitchhiker next to the road are given. The velocity at the center, top, and bottom of the tire can be determined by considering the linear and angular velocities. The acceleration at the center, top, and bottom of the tire can be found by considering the centripetal acceleration due to the tire's rotation.

Step-by-step explanation:

In order to determine the velocity and acceleration of the car and the tire, we need to use the relationship between linear and angular velocities. Let's consider the velocity and acceleration relative to the woman in the car first.

(a) The velocity at the center of the tire is the same as the velocity of the car, which is 80 km/h in the positive x direction.

(b) The top of the tire is moving with a velocity equal to the linear velocity plus the tangential velocity due to rotation. Since the tire is rotating clockwise, the velocity at the top will be 80 km/h + (angular velocity x radius).

(c) Similarly, the bottom of the tire is moving with a velocity equal to the linear velocity minus the tangential velocity due to rotation. The velocity at the bottom will be 80 km/h - (angular velocity x radius).

(d) The acceleration at the center of the tire is zero since the car is moving at a constant velocity.

(e) The acceleration at the top of the tire is the centripetal acceleration due to the rotation of the tire, which is equal to (angular velocity)^2 x radius.

(f) The acceleration at the bottom of the tire is also the centripetal acceleration due to the rotation, but in the opposite direction.

To determine the velocity and acceleration relative to the hitchhiker, we need to consider their relative motion. If the hitchhiker is sitting next to the road, they are stationary relative to the road.

(g) The velocity at the center of the tire relative to the hitchhiker is the same as the velocity of the car - 80 km/h in the positive x direction.

(h) The velocity at the top of the tire relative to the hitchhiker will be the velocity of the car plus the tangential velocity due to the rotation of the tire.

(i) The velocity at the bottom of the tire relative to the hitchhiker will be the velocity of the car minus the tangential velocity due to the rotation.

(j) The acceleration at the center of the tire relative to the hitchhiker is zero.

(k) The acceleration at the top of the tire relative to the hitchhiker is the same as the acceleration at the top relative to the woman.

(l) The acceleration at the bottom of the tire relative to the hitchhiker is the same as the acceleration at the bottom relative to the woman.

Answer 2

80 km/h = 22.2 m/s
66 cm = 0.66 m

Relative to the woman riding in the car:

(a) Since the center of the tire is not moving, and the woman goes along with the movement of the car, v=0
(b) v = +22.2 km/h, since the wheel turns at this velocity from left to right
(c) v = -22.2 km/h, since the wheel turns at this velocity from right to left
(d) a = 0, since v=0
(e) a = v2/r = (+22.2)^2/(0.66 ÷2) = 1493.45 m2/s
(f) a = v2/r = (-22.2)^2/(0.66 ÷2) = -1493.45 m2/s

Relative to hitchhiker:

(g) v = +22.2 m/s, since from an external body, the velocity of the car is still observed
(h) v = +22.2 x 2 = 44.4 m/s, since the top runs twice faster to be able to move the car
(i) v = 0, since from an external body, the bottom part of the tire is always in contact with the ground
(j) a = 0, because velocity is constant
(k) a = 1493.45 m2/s, same with that of the woman because both are inertial reference
(l) a = -1493.45 m2/s, same with that of the woman because both are inertial reference