Q: A Ferris wheel of radius $$R$$ rotates at constant speed $$v. $$What is the magnitude of the normal force $$N_{\text{top}}$$ exerted by the seat on a passenger of mass $$m at $$the top of the wheel?

$$N_{\text{top}} = m \left( g - \frac{v^2$}{R} \right)$$

Q: A car of mass $$m$$ rounds a flat curve of radius $$R. If $$the coefficient of static friction between the tires and the road is $$\mu_s$$, what is the maximum speed $$v_{\text{max}} at $$which the car can take the curve without sliding?

$$v_{\text{max}} = \sqrt{\mu_s g R}$$

Q: A car travels around a banked curve of radius $$R at $$speed $$v$$ such that no friction is required. What is the correct expression for the banking angle $$\beta$$?

$$\tan \beta = \frac{v^2$}{g R}$$

Q: A small ball of mass $$m is $$suspended by a cord of length $$\ell$$ and revolves in a circle of radius $$r = \ell \sin \theta. $$What is the correct expression for the speed $$v of $$the ball in terms of $$\ell$$, $$\theta$$, and $$g$$?

$$v = \sqrt{g \ell \sin \theta \tan \theta}$$

Q: At what height $$h$$ above Earth's surface is the value of $$g$$ equal to one-half its value at the surface? (Let $$R_{\text{Earth}} be $$Earth's radius.)

$$h = (\sqrt{2} - 1) R_{\text{Earth}}$$

Question 1

A tether ball of mass $$m$$ swings in a horizontal circle of radius $$R at $$constant speed $$v$$, suspended by a cord of length $$L$$ making an angle $$\beta$$ with the vertical. If the period of revolution is $$T$$, what is the correct expression for the tension $$F_T in $$the cord?

Accepted Answer

$$F_T = \frac{m 4 \pi^2$ L}{T^2$}$$

Question 2

A Ferris wheel of radius $$R$$ rotates at constant speed $$v. $$What is the magnitude of the normal force $$N_{	ext{top}}$$ exerted by the seat on a passenger of mass $$m at $$the top of the wheel?

Accepted Answer

$$N_{	ext{top}} = m \left( g - \frac{v^2$}{R} ight)$$

Question 3

A car of mass $$m$$ rounds a flat curve of radius $$R. If $$the coefficient of static friction between the tires and the road is $$\mu_s$$, what is the maximum speed $$v_{	ext{max}} at $$which the car can take the curve without sliding?

Accepted Answer

$$v_{	ext{max}} = \sqrt{\mu_s g R}$$

Question 4

A car travels around a banked curve of radius $$R at $$speed $$v$$ such that no friction is required. What is the correct expression for the banking angle $$\beta$$?

Accepted Answer

$$	an \beta = \frac{v^2$}{g R}$$

Question 5

A weather satellite is placed in a circular orbit 300 km above Earth's surface. Given Earth's radius $$R_E = 6380 km $$and mass $$M_E = 5.98 	imes 10$$^{24}$$ kg, what is the correct expression for the satellite's orbital speed v$?

Accepted Answer

$$v = \sqrt{\frac{G M_E}{R_E + h}}$$

Question 6

A race car of mass $$m = 600 kg $$travels on a circular arc of radius $$r = 100 m. $$The coefficient of static friction is $$\mu_s = 0.75. If $$the car is on the verge of sliding at $$v = 28.6 m/s$$, and a negative lift $$F_L$$ acts downward, what is the correct equation for $$F_L$$?

Accepted Answer

$$F_L = \frac{m v^2$}{r \mu_s} - m g$$

Question 7

A small ball of mass $$m is $$suspended by a cord of length $$\ell$$ and revolves in a circle of radius $$r = \ell \sin 	heta. $$What is the correct expression for the speed $$v of $$the ball in terms of $$\ell$$, $$	heta$$, and $$g$$?

Accepted Answer

$$v = \sqrt{g \ell \sin 	heta 	an 	heta}$$

Question 8

At what height $$h$$ above Earth's surface is the value of $$g$$ equal to one-half its value at the surface? (Let $$R_{	ext{Earth}} be $$Earth's radius.)

Accepted Answer

$$h = (\sqrt{2} - 1) R_{	ext{Earth}}$$

Question 9

Three moons of Saturn, Tethys, Dione, and Rhea, form a right triangle as shown. If Tethys has mass $$6.2 	imes 10$$^{20}$$ kg, Dione 1.1$$ 	imes $$10^{21} kg$$, and Rhea $$2.3 	imes 10$$^{21}$$ kg, and all sides are 2.3$$ 	imes $$10^5 km$$, what is the direction of the net gravitational force on Tethys due to Dione and Rhea?

Accepted Answer

Along the direction of the diagonal of the right triangle formed by Dione and Rhea

Question 10

Why do astronauts feel weightless while orbiting Earth in a space shuttle 300 km above the surface?

Accepted Answer

Because both the shuttle and astronauts are in free fall, accelerating at the same rate due to gravity

Question 11

A satellite orbits Earth at a height where the acceleration due to gravity is $$0.9g. If$$ $$g at $$the surface is $$9.8 m/s$^2$$, what is the value of $$g at $$this height?

Accepted Answer

$$8.82 m/s$^2$$

Question 12

A ball of mass $$m is $$attached to a string of length $$L$$ and swings in a horizontal circle at constant speed $$v. $$The string makes an angle $$\beta$$ with the vertical. What is the correct expression for $$\beta in $$terms of $$g$$, $$T$$, and $$L$$?

Accepted Answer

$$\beta = \cos$$^{-1}$$ \left( \frac{g T^2$}{4 \pi^2$ L} \right)$$

Question 13

A car of mass $$m is $$moving at speed $$v$$ around a flat curve of radius $$R. If $$the frictional force is just enough to keep the car from sliding, what is the direction of the frictional force?

Accepted Answer

Toward the center of the circle

Question 14

A satellite is in a circular orbit of radius $$r$$ around Earth. What is the period $$T of $$its orbit in terms of $$r$$ and Earth's mass $$M_E$$?

Accepted Answer

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

Question 15

If the gravitational force between two masses $$m_1$$ and $$m_2$$ separated by distance $$r is$$ $$F$$, what happens to the force if the distance is doubled?

Accepted Answer

It becomes one-fourth as large

Question 16

A small mass $$m is $$placed a distance $$x$$ along the perpendicular bisector of two identical point masses $$M$$ separated by $$2R. $$What is the magnitude of the net gravitational force on $$m$$?

Accepted Answer

$$F = \frac{2 G M m x}{(x^2$ + R^2$)^{3/2}$$}$

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