A car travels at 90.2 km/h on a level road in the positive direction of an x axi

Question

A car travels at 90.2 km/h on a level road in the positive direction of an x axis. Each tire has a diameter of 61.2 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?

Explanation / Answer

Relative to a woman in the car:

(a) the translational velocity is ZERO since the woman is moving at the same translational velocity as the car relative to the ground.

(b) the magnitude of the translational acceleration of the center of the wheel is also ZERO because the center of the wheel is NOT moving and NOT rotating relative to the woman.

(c) v, velocity of the point at the top of the tire is = 80 km/h because that is how fast the car is moving, thus the wheel must also be rotating at the same speed.

(d) a, acceleration of the point at the top of the tire is given by:

a = v^2/r = (25.055)^2/(0.306) = 2051.48 m/s^2

(e) v, velocity of the point at the bottom of the tire is = -90.2 km/h because at the bottom of the tire, the tire is moving in the opposite direction

(f) a, acceleration of the point at the bottom of the tire is the same as (d) a = v^2/r = 2051.48 m/s^2

Now relative to a hitch hiker on the ground:

(g) v at the center of the wheel is = 90.2 km/h or 25.055 m/s, since the car and the center of the wheel (which is not moving relative to the car) are moving at the same speed relative to the ground.

(h) a at the center of the wheel is = v^2/r = 2051.48m/s^2, since the center of the wheel is rotating about the bottom point of the tire as the tire rolls on the ground.

(i) v at the top of the tire is v(top) = ?d, where ? is the angular velocity of the rotating tire and d is the diameter of the wheel. The reason we use diameter of the wheel is because the top of the tire is rotating abou the bottom of the tire, not the center.

? = v/r = (25.055)/(0.306) = 81.879 rad/sec
v(top) = ?d = (81.879)(0.612) = 50.11 m/s

(j) a at the top of the tire is = v(top)^2/d
= (50.11)^2/(0.612) = 4102.96 m/s^2

(k) v at the bottom of the tire is equal to ZERO because the bottom of the tire is not moving relativ to the hitch hiker or the ground.

(l) a at the bottom of the tire is also equal to ZERO since the velocity is zero.

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