Q: A car is navigating a flat circular intersection with a radius of $$12\,\mathrm{m}. If $$the coefficient of static friction between the tires and the road is $$\mu_s = 1.1$$, what is the maximum safe speed at which the car can travel without slipping? Use $$g = 9.8\,\mathrm{m/s^2$}.$$

$$v_{\text{max}} = \sqrt{\mu_s g r} = \sqrt{1.1 \times 9.8 \times 12} \approx 11.3\,\mathrm{m/s}$$

Q: On a rainy day, the coefficient of static friction drops to $$\mu_s = 0.7. $$What is the new maximum safe speed for the same $$12\,\mathrm{m}$$ radius intersection?

$$v_{\text{max}} = \sqrt{0.7 \times 9.8 \times 12} \approx 9.1\,\mathrm{m/s}$$

Q: If the coefficient of static friction drops further to $$\mu_s = 0.3 on a $$snowy day, what is the maximum safe speed for the same intersection?

$$v_{\text{max}} = \sqrt{0.3 \times 9.8 \times 12} \approx 5.9\,\mathrm{m/s}$$

Q: An exit ramp is designed for a car to safely travel at $$25\,\mathrm{m/s} on a $$curve with $$\mu_s = 0.7. $$What is the minimum radius required for the curve?

$$r = \frac{v^2$}{\mu_s g} = \frac{25^2$}{0.7 \times 9.8} \approx 91\,\mathrm{m}$$

Q: A car is on a banked curve with angle $$\theta$$ and coefficient of static friction $$\mu_s. $$What is the general formula for the maximum speed $$v_{\text{max}}$$ the car can have without sliding up the bank?

$$v_{\text{max}} = \sqrt{\frac{g r (\sin \theta + \mu_s \cos \theta)}{\cos \theta - \mu_s \sin \theta}}$$

Q: On a banked curve with $$\theta = 20^\circ$$, $$r = 50\,\mathrm{m}$$, and $$\mu_s = 0.3$$, what is the minimum speed required to avoid sliding down the bank?

$$v_{\text{min}} = \sqrt{\frac{g r (\sin \theta - \mu_s \cos \theta)}{\cos \theta + \mu_s \sin \theta}}$$

Question 1

A car is navigating a flat circular intersection with a radius of $$12\,\mathrm{m}. If $$the coefficient of static friction between the tires and the road is $$\mu_s = 1.1$$, what is the maximum safe speed at which the car can travel without slipping? Use $$g = 9.8\,\mathrm{m/s^2$}.$$

Accepted Answer

$$v_{	ext{max}} = \sqrt{\mu_s g r} = \sqrt{1.1 	imes 9.8 	imes 12} \approx 11.3\,\mathrm{m/s}$$

Question 2

On a rainy day, the coefficient of static friction drops to $$\mu_s = 0.7. $$What is the new maximum safe speed for the same $$12\,\mathrm{m}$$ radius intersection?

Accepted Answer

$$v_{	ext{max}} = \sqrt{0.7 	imes 9.8 	imes 12} \approx 9.1\,\mathrm{m/s}$$

Question 3

If the coefficient of static friction drops further to $$\mu_s = 0.3 on a $$snowy day, what is the maximum safe speed for the same intersection?

Accepted Answer

$$v_{	ext{max}} = \sqrt{0.3 	imes 9.8 	imes 12} \approx 5.9\,\mathrm{m/s}$$

Question 4

An exit ramp is designed for a car to safely travel at $$25\,\mathrm{m/s} on a $$curve with $$\mu_s = 0.7. $$What is the minimum radius required for the curve?

Accepted Answer

$$r = \frac{v^2$}{\mu_s g} = \frac{25^2$}{0.7 	imes 9.8} \approx 91\,\mathrm{m}$$

Question 5

Why does the frictional force $$F_{fs}$$ point up the incline when a car is moving too slowly on a banked curve?

Accepted Answer

Because $$F_{fs}$$ must act up the incline to provide the necessary centripetal force to keep the car from sliding down.

Question 6

A car is on a banked curve with angle $$	heta$$ and coefficient of static friction $$\mu_s. $$What is the general formula for the maximum speed $$v_{	ext{max}}$$ the car can have without sliding up the bank?

Accepted Answer

$$v_{	ext{max}} = \sqrt{\frac{g r (\sin 	heta + \mu_s \cos 	heta)}{\cos 	heta - \mu_s \sin 	heta}}$$

Question 7

On a banked curve with $$	heta = 20^\circ$$, $$r = 50\,\mathrm{m}$$, and $$\mu_s = 0.3$$, what is the minimum speed required to avoid sliding down the bank?

Accepted Answer

$$v_{	ext{min}} = \sqrt{\frac{g r (\sin 	heta - \mu_s \cos 	heta)}{\cos 	heta + \mu_s \sin 	heta}}$$

Question 8

If a car is moving too slowly on a banked curve in snowy conditions ($$\mu_s = 0.3$$), what could happen?

Accepted Answer

The car could slide down the bank because friction is not enough to keep it on the curve.

Question 9

A car is driving on a flat curve with radius $$r$$ and coefficient of static friction $$\mu_s. $$Which force provides the centripetal acceleration?

Accepted Answer

The frictional force between the tires and the road.

Question 10

Why is it necessary for a road to be banked at a certain angle for cars to safely complete a turn at higher speeds without relying on friction?

Accepted Answer

Banking the road allows the normal force to provide the required centripetal force for the turn.

Question 11

For a banked curve with no friction, what is the formula for the speed at which a car can safely navigate the turn?

Accepted Answer

$$v = \sqrt{g r 	an 	heta}$$

Question 12

A car is driving on a banked curve with $$	heta = 30^\circ$$ and $$r = 40\,\mathrm{m}. $$What is the speed at which no friction is required to make the turn?

Accepted Answer

$$v = \sqrt{g r 	an 30^\circ} \approx \sqrt{9.8 	imes 40 	imes 0.577} \approx 15.0\,\mathrm{m/s}$$

Question 13

If a car is driving on a banked curve and the frictional force is acting down the incline, what does this indicate about the car's speed?

Accepted Answer

The car is moving faster than the speed for which the bank was designed.

Question 14

A car is driving on a flat curve with $$r = 20\,\mathrm{m}$$ and $$\mu_s = 0.5. $$What is the maximum speed before slipping occurs?

Accepted Answer

$$v_{	ext{max}} = \sqrt{0.5 	imes 9.8 	imes 20} \approx 9.9\,\mathrm{m/s}$$

Question 15

If the coefficient of static friction is reduced due to ice on the road, what happens to the maximum safe speed for a car on a curve?

Accepted Answer

The maximum safe speed decreases because friction is reduced.

Question 16

A car is driving on a banked curve with $$	heta = 25^\circ$$, $$r = 60\,\mathrm{m}$$, and $$\mu_s = 0.4. $$Calculate the maximum speed before sliding up the bank.

Accepted Answer

$$v_{	ext{max}} = \sqrt{\frac{g r (\sin 25^\circ + 0.4 \cos 25^\circ)}{\cos 25^\circ - 0.4 \sin 25^\circ}}$$

Question 17

Why is it dangerous to drive too slowly on a banked curve in snowy conditions?

Accepted Answer

Because friction may not be enough to prevent the car from sliding down the bank.

Circular Motion and Friction in Vehicle Dynamics: Study Notes

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