Q: The image shows a car driving on a sharply banked curve. If the radius of the curve is $$100\ \mathrm{m}$$ and the banking angle is $$30^\circ$$, what is the maximum speed at which the car can travel around the curve without relying on friction?

$$v = \sqrt{r g \tan \theta} = \sqrt{100 \times 9.8 \times \tan 30^\circ} = 23.8\ \mathrm{m/s}$$

Q: A car is moving at $$36.0\ \mathrm{m/s}$$ around a flat, unbanked curve of radius $$300\ \mathrm{m}. $$What is the minimum coefficient of static friction required to prevent the car from sliding?

$$\mu_s = \frac{v^2$}{r g} = \frac{36.0^2$}{300 \times 9.8} = 0.441$$

Q: A roller coaster car of mass $$m$$ starts at the bottom of a $$12.0\ \mathrm{m}$$ diameter vertical loop. What is the minimum speed at the top of the loop to avoid falling out?

$$v_{min} = \sqrt{g R} = \sqrt{9.8 \times 6.0} = 7.67\ \mathrm{m/s}$$

Q: A car is designed to safely navigate a banked turn of radius $$75.0\ \mathrm{m} at$$ $$24.59\ \mathrm{m/s}$$ even in icy conditions. What should the banking angle be?

$$\theta = \tan$$^{-1}$$\left(\frac{v^2$}{r g}\right) = \tan$$^{-1}$$\left(\frac{24.59^2$}{75.0 \times 9.8}\right) = 39.4^\circ$$

Question 1

The image shows a car driving on a sharply banked curve. If the radius of the curve is $$100\ \mathrm{m}$$ and the banking angle is $$30^\circ$$, what is the maximum speed at which the car can travel around the curve without relying on friction?

Accepted Answer

$$v = \sqrt{r g 	an 	heta} = \sqrt{100 	imes 9.8 	imes 	an 30^\circ} = 23.8\ \mathrm{m/s}$$

Question 2

Which component of the normal force provides the centripetal force required for a car to navigate a banked turn without friction?

Accepted Answer

The x-component of the normal force, $$F_{N,x}$$

Question 3

A car is moving at $$36.0\ \mathrm{m/s}$$ around a flat, unbanked curve of radius $$300\ \mathrm{m}. $$What is the minimum coefficient of static friction required to prevent the car from sliding?

Accepted Answer

$$\mu_s = \frac{v^2$}{r g} = \frac{36.0^2$}{300 	imes 9.8} = 0.441$$

Question 4

In a vertical circle, where does a rider feel the greatest apparent weight?

Accepted Answer

At the bottom of the circle

Question 5

If a motorcycle is at the top of a vertical circle and the normal force is zero, what is true about the forces acting on the rider?

Accepted Answer

The gravitational force provides all the required centripetal force

Question 6

A roller coaster car of mass $$m$$ starts at the bottom of a $$12.0\ \mathrm{m}$$ diameter vertical loop. What is the minimum speed at the top of the loop to avoid falling out?

Accepted Answer

$$v_{min} = \sqrt{g R} = \sqrt{9.8 	imes 6.0} = 7.67\ \mathrm{m/s}$$

Question 7

Why is kinetic energy not conserved as an object moves around a vertical circle (assuming negligible friction)?

Accepted Answer

Because gravitational potential energy changes as the object moves up and down

Question 8

A car is designed to safely navigate a banked turn of radius $$75.0\ \mathrm{m} at$$ $$24.59\ \mathrm{m/s}$$ even in icy conditions. What should the banking angle be?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{v^2$}{r g}ight) = 	an$$^{-1}$$\left(\frac{24.59^2$}{75.0 	imes 9.8}ight) = 39.4^\circ$$

Question 9

In a banked turn with friction, what is the effect of friction on the maximum possible speed through the turn?

Accepted Answer

Friction increases the maximum possible speed

Question 10

At the sides of a vertical circle, what force provides the normal force that the rider feels?

Accepted Answer

Only inertia (centripetal force) pushing against the wall

Question 11

If a car is moving too slowly at the top of a vertical loop, what will happen?

Accepted Answer

The normal force becomes negative and the car will fall

Question 12

Which equation correctly relates the normal force and gravitational force at the bottom of a vertical circle?

Accepted Answer

$$F_N = F_c + F_g$$

Question 13

For a banked turn, which equation gives the required banking angle $$	heta$$ for a given speed $$v$$, radius $$r$$, and gravitational acceleration $$g ($$assuming no friction)?

Accepted Answer

$$	an 	heta = \frac{v^2$}{r g}$$

Question 14

A car is moving in a circle of radius $$R at $$constant speed $$v. $$What is the direction of the centripetal force acting on the car?

Accepted Answer

Toward the center of the circle

Question 15

Which of the following best describes 'apparent weight' in the context of circular motion?

Accepted Answer

It is the normal force exerted on an object in an accelerating frame of reference

Circular Motion: Vertical Circles and Banked Turns

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