Q: A 60 kg passenger rides a roller coaster through a vertical loop of radius 10 m at a speed of 14 m/s. What is her apparent weight at the bottom of the loop?

$$w_{app} = 60 \times 9.8 + \frac{60 \times 14^2$}{10} = 588 + 1176 = 1764\ \mathrm{N}$$

Question 1

A car of mass 1200 kg is moving at a constant speed around a flat, unbanked curve of radius 25 m. If the coefficient of static friction between the tires and the road is 0.8, what is the maximum speed the car can have without skidding?

Accepted Answer

$$v_{max} = \sqrt{\mu_s g r} = \sqrt{(0.8)(9.8\ \mathrm{m/s^2$})(25\ \mathrm{m})} \approx 14\ \mathrm{m/s}$$

Question 2

Which of the following statements about uniform circular motion is correct?

Accepted Answer

The speed is constant, but the velocity is not because its direction changes.

Question 3

A roller coaster passenger of mass $$m is at $$the bottom of a vertical loop moving at speed $$v. $$What is her apparent weight at that point?

Accepted Answer

$$w_{app} = n = w + \frac{mv^2$}{r}$$

Question 4

A satellite orbits Earth in a circular path of radius $$r. $$What is the correct expression for its orbital speed?

Accepted Answer

$$v_{orbit} = \sqrt{\frac{GM}{r}}$$

Question 5

If the gravitational force between two spheres is $$0.010\ \mathrm{N}$$ when they are $$20\ \mathrm{m}$$ apart, what will the force be if the distance is reduced to $$5.0\ \mathrm{m}$$?

Accepted Answer

$$0.160\ \mathrm{N}$$

Question 6

A 70 kg astronaut feels weightless while orbiting Earth. What is the best explanation for this phenomenon?

Accepted Answer

The astronaut and the spacecraft are both in free fall, so there is no normal force acting on her.

Question 7

Which force provides the centripetal acceleration for a car turning a corner on a flat road?

Accepted Answer

Static friction

Question 8

The period $$T of a $$satellite in a circular orbit of radius $$r$$ around a planet of mass $$M is $$given by:

Accepted Answer

$$T = 2\pi \sqrt{\frac{r^3$}{GM}}$$

Question 9

If the distance between two masses is tripled, by what factor does the gravitational force between them change?

Accepted Answer

It decreases by a factor of 9.

Question 10

A 60 kg passenger rides a roller coaster through a vertical loop of radius 10 m at a speed of 14 m/s. What is her apparent weight at the bottom of the loop?

Accepted Answer

$$w_{app} = 60 	imes 9.8 + \frac{60 	imes 14^2$}{10} = 588 + 1176 = 1764\ \mathrm{N}$$

Question 11

Which of the following best describes Newton's law of universal gravitation?

Accepted Answer

The force is directly proportional to the product of the masses and inversely proportional to the square of the distance.

Question 12

A planet has twice the mass of Earth and half its radius. What is the acceleration due to gravity on its surface?

Accepted Answer

$$g_{planet} = \frac{GM_{planet}}{R_{planet}^2} = \frac{G(2M_{Earth})}{(0.5R_{Earth})^2$} = \frac{2G M_{Earth}}{0.25 R_{Earth}^2} = 8g_{Earth}$$

Question 13

A satellite is in a circular orbit around Earth at a radius $$r. If $$the mass of Earth were doubled, but the radius of the orbit remained the same, how would the satellite's orbital speed change?

Accepted Answer

It would increase by a factor of $$\sqrt{2}.$$

Question 14

A car goes into a dip in the road at constant speed. At the very bottom of the dip, how does the normal force of the road on the car compare to the car's weight?

Accepted Answer

The normal force is greater than the car's weight.

Question 15

The centripetal acceleration for an object moving in a circle of radius $$r at $$speed $$v is $$given by:

Accepted Answer

$$a = \frac{v^2$}{r}$$

Question 16

A satellite orbits a planet of mass $$M at $$radius $$r. $$What is the correct formula for the gravitational force acting on the satellite?

Accepted Answer

$$F = \frac{GMm}{r^2$}$$

Question 17

A projectile is launched horizontally from the surface of a smooth, airless Earth. What speed must it have to maintain a circular orbit just above the surface?

Accepted Answer

$$v_{orbit} = \sqrt{g R_{Earth}}$$

Question 18

The period $$T of a $$satellite orbiting at radius $$r$$ around a planet of mass $$M is $$found by squaring the orbital speed equation and rearranging. What is the resulting formula?

Accepted Answer

$$T^2$$ = \frac{4\$$pi^2$$ $$r^3$$}{GM}$$

Chapter 6: Circular Motion, Orbits, and Gravity – Study Notes

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