The problem consists of multiple parts related to the motion of a car. B. A car moves with constant speed on a straight line. The radius of the tires is 560 mm, and the angular velocity is 60 rad/s. i. Find the angular velocity of the axle and the midpoint between the axle and the edge of the tire. ii. Find the linear velocity of a mud particle stuck to the tire at the midpoint between the axle and the edge. iii. Find the horizontal velocity of the car. C. The car travels to the top of an arched bridge with the same constant speed. The car's wheels remain in good contact with the ground. i. Explain why bridges are arched in their shape, using an expression. D. i. Another car enters a horizontal curve at a speed of 72 km/h. If the coefficient of friction of the curve is $\mu = 0.7$ and the radius of the curve is 50 m, can the car safely navigate the curve? ii. Explain qualitatively how curves should be built to safely navigate them at higher speeds.

1. Problem Description

The problem consists of multiple parts related to the motion of a car.

B. A car moves with constant speed on a straight line. The radius of the tires is 560 mm, and the angular velocity is 60 rad/s.

i. Find the angular velocity of the axle and the midpoint between the axle and the edge of the tire.

ii. Find the linear velocity of a mud particle stuck to the tire at the midpoint between the axle and the edge.

iii. Find the horizontal velocity of the car.

C. The car travels to the top of an arched bridge with the same constant speed. The car's wheels remain in good contact with the ground.

i. Explain why bridges are arched in their shape, using an expression.

D. i. Another car enters a horizontal curve at a speed of 72 km/h. If the coefficient of friction of the curve is $\mu = 0.7$ and the radius of the curve is 50 m, can the car safely navigate the curve?

ii. Explain qualitatively how curves should be built to safely navigate them at higher speeds.

2. Solution Steps

B. i. The angular velocity of the axle and the midpoint between the axle and the edge of the tire is the same as the angular velocity of the tire.

$\omega_{axle} = \omega_{midpoint} = \omega_{tire} = 60 \, rad/s$

B. ii. The radius of the midpoint between the axle and the edge is $r_{midpoint} = \frac{560 \, mm}{2} = 280 \, mm = 0.28 \, m$. The linear velocity of the mud particle is given by:

$v = r \omega = 0.28 \, m \times 60 \, rad/s = 16.8 \, m/s$

B. iii. The horizontal velocity of the car is equal to the linear velocity of the edge of the tire. The radius of the tire is $r = 560 \, mm = 0.56 \, m$.

$v = r \omega = 0.56 \, m \times 60 \, rad/s = 33.6 \, m/s$

C. i. Arched bridges are designed to primarily experience compressive forces rather than bending forces. Compressive forces are much easier to handle in construction materials like stone and concrete. The arch shape ensures that the load is distributed along the curve, transferring the weight of the bridge and any load on it to the supports at either end. This can be visualized by considering the load at the top of the arch. The load is resolved into components along the arch. As the load travels to the supports, this force compresses the arch material.

D. i. First, convert the speed of the car to m/s:

$v = 72 \, km/h = 72 \times \frac{1000 \, m}{3600 \, s} = 20 \, m/s$

The centripetal force required for the car to navigate the curve is:

$F_c = \frac{mv^2}{r}$

The maximum frictional force available is:

$F_f = \mu m g$

For the car to safely navigate the curve, the frictional force must be greater than or equal to the centripetal force:

$\mu m g \ge \frac{m v^2}{r}$

$\mu g \ge \frac{v^2}{r}$

$\mu \ge \frac{v^2}{gr}$

Plugging in the values:

$\frac{v^2}{gr} = \frac{(20 \, m/s)^2}{9.8 \, m/s^2 \times 50 \, m} = \frac{400}{490} \approx 0.816$

Since $\mu = 0.7$ and $\frac{v^2}{gr} \approx 0.816$, then $\mu < \frac{v^2}{gr}$, which means the car cannot safely navigate the curve.

D. ii. To safely navigate curves at higher speeds, the curves should be banked or inclined. Banking provides a component of the normal force that contributes to the centripetal force, reducing the reliance on friction.

3. Final Answer

B. i. 60 rad/s

B. ii. 16.8 m/s

B. iii. 33.6 m/s

C. i. Arched bridges are designed to primarily experience compressive forces. $F_c \approx \frac{F}{\cos(\theta)}$ is the general concept.

D. i. No, the car cannot safely navigate the curve.

D. ii. Curves should be banked to allow for higher speeds.