 A boy on a swing set has a speed of 4.5 m/s and a centripetal acceleration of 8.1 m/s^{2} at the bottom of his swing. How long are the ropes of the swing?
 An amusement park ride features a vertical cylinder 8 m in radius with a horizontal floor. Riders stand on the floor with their backs against the inner surface of the cylinder. The cylinder spins, completing one revolution every 4 s. Once spinning, the axis of the cylinder rotates so it tilts up to 45 degrees. Even as the cylinder tilts sideways, the riders don’t fall off.
 Why don’t they fall off?
 What direction is the force exerted by the cylinder on the riders?
 What is the angular velocity of the riders?
 What is the linear velocity of the riders?
 What centripetal acceleration do the riders experience?
 Compare this to the acceleration of gravity (g).
 What is the angular velocity of Mercury’s orbital motion? (Mercury makes one orbit of the Sun every 88 Earth days.)
 Comets move in highly elliptical orbits. For example, the aphelion, or farthest point of Comet HaleBopp’s orbit is 371 AU from the Sun. The perihelion, or closest approach, is 0.91 AU, within the Earth’s orbital radius!
 Use the orbit equation to calculate the velocity of Comet HaleBopp at aphelion.
 Use the orbit equation to calculate the velocity of Comet HaleBopp at perihelion.
 The Moon is Earth’s only natural satellite. It is much smaller and less massive than Earth, with a radius of 1.74×10^{6} m and a mass of 7.35×10^{22} kg.
 What is the acceleration due to gravity on the surface of the Moon?
 What is your weight on the Moon? (1 lb = 4.45 N, and 1 kg weighs 2.21 lb on Earth.)
   The mass of the Sun is 2×10^{30} kg. Earth’s average orbital radius is 1.52×10^{11} m.
 Use the orbit equation to calculate Earth’s average orbital velocity.
 Suppose Earth increased its velocity by 50%. Calculate the new radius of the planet’s orbit.
 Compare your answer to Part b with Venus’s orbital radius of 1.08×10^{11} m. Given that Venus has a surface temperature hot enough to melt lead, speculate on the possibility for life if Earth had this orbital velocity.
 Suppose that you lived on a planet with Earth’s mass (5.97×10^{24} kg) but only half its radius (that is, a radius of 3,189 km instead of 6,378 km). Suppose, too, that your own mass was 75 kg.
 What would your weight be on the surface of this hypothetical planet? (State your answer in both newtons and in pounds.)
 What would the acceleration due to gravity be on the surface of this madeup planet? (State your answer in newtons per kilogram and compare it to the value on Earth’s surface.)
 Compared to our 1.99×10^{30} kg Sun, a 70 kg person on Earth (1.52×10^{11} m away from the Sun) is very small and light. What is the attractive force due to gravity between the Sun and a person who is on Earth?
 The Sun has a mass of 2.0×10^{30} kg. Jupiter has a mass of 1.9×10^{27} kg. It orbits 7.5×10^{8} km away from the Sun. Assume its orbit is circular.
 What is the gravitational force between Jupiter and the Sun?
 How fast does Jupiter orbit around the Sun?
 What is Jupiter’s angular velocity?
 How may radians does Jupiter travel in one Earth year? How many degrees?
 The orbit of the dwarf planet Haumea is at an orbital semimajor axis of 43 astronomical units (AU), i.e., 43 times larger than the Earth’s orbital radius. Using this information and Kepler’s laws, what do you predict is the orbital period of Haumea?
 On your computer, use the interactive simulation of orbits on page 424 to determine the change in velocity needed to launch a satellite in a transfer orbit from Earth to Venus.
 The planets of our Solar System orbit the Sun. The Sun has a mass of 1.99×10^{30} kg. Mars, a planet 2.37×10^{11} m from the Sun, has a mass of 6.42×10^{23} kg. What is the linear velocity of Mars as it orbits the Sun? Assume Mars has a circular orbit.
