96 PART 1^ |^ EXPLORING THE SKY
Review Questions
- Why wouldn’t Aristotle’s explanation of gravity work if Earth is not the
center of the universe? - According to the principles of Aristotle, what part of the motion of a
baseball pitched across the home plate is natural motion? What part is
violent motion? - If you drop a feather and a steel hammer at the same moment, they
should hit the ground at the same instant. Why doesn’t this work on
Earth, and why does it work on the moon? - What is the difference between mass and weight? Between speed and
velocity? - Why did Newton conclude that some force had to pull the moon toward
Earth? - Why did Newton conclude that gravity has to be mutual and universal?
- How does the concept of a fi eld explain action at a distance? Name
another kind of fi eld also associated with action at a distance. - Why can’t a spacecraft go “beyond Earth’s gravity?”
- What is the center of mass of the Earth–moon system? Where is it?
- How do planets orbiting the sun, and skaters doing a spin, both
conserve angular momentum? - Why is the period of an open orbit undefi ned?
- How does the fi rst postulate of special relativity imply the second?
- When you ride a fast elevator upward, you feel slightly heavier as
the trip begins and slightly lighter as the trip ends. How is this
phenomenon related to the equivalence principle? - From your knowledge of general relativity, would you expect radio
waves from distant galaxies to be defl ected as they pass near the sun?
Why or why not? - How Do We Know? Why would science be impossible if some natural
events happened without causes? - How Do We Know? Why is it important that a theory make testable
predictions?
Discussion Questions
- How did Galileo idealize his inclines to conclude that an object in
motion stays in motion until it is acted on by some force? - Give an example from everyday life to illustrate each of Newton’s laws.
- People who lived before Newton may not have believed in cause and
effect as strongly as you do. How do you suppose that affected how
they saw their daily lives?
Problems
- Compared with the strength of Earth’s gravity at its surface, how much
weaker is gravity at a distance of 10 Earth radii from Earth’s center? At
20 Earth radii? - Compare the force of lunar gravity on the surface of the moon with the
force of Earth’s gravity at Earth’s surface. - If a small lead ball falls from a high tower on Earth, what will be its
velocity after 2 seconds? After 4 seconds? - What is the circular velocity of an Earth satellite 1000 km above
Earth’s surface? (Hint: Earth’s radius is 6380 km.) - What is the circular velocity of an Earth satellite 36,000 km above
Earth’s surface? What is its orbital period? (Hint: Earth’s radius is
6380 km.) - What is the orbital period of an imaginary satellite orbiting just above
Earth’s surface? Ignore friction with the atmosphere. - Repeat the previous problem for Mercury, Venus, the moon, and Mars. (Hint:
Find the masses and radii of each of these objects in the Appendix A tables.)
▶ (^) From his mathematical analysis, Newton was able to show that the
force of gravity between two masses is proportional to the product of
their masses and obeys the inverse square law (p. 81). That is, the
force of gravity is inversely proportional to the square of the distance
between the two objects.
▶ (^) To explain how gravity can act at a distance, scientists describe it as a
fi eld (p. 81).
▶ (^) An object in space near Earth would move along a straight line and
quickly leave Earth were it not for Earth’s gravity accelerating the
object toward Earth’s center and forcing it to follow a curved path,
an orbit. Objects in orbit around Earth are falling (being accelerated)
toward Earth’s center. If there is no friction, the object will fall around
its orbit forever.
▶ (^) An object in a closed orbit (p. 85) follows an elliptical path. A circle
is just a special case of an ellipse with zero eccentricity. To follow a
circular orbit, an object must orbit with circular velocity (p. 84). At a
certain distance from Earth, a geosynchronous satellite (p. 84) stays
above a spot on Earth’s equator as Earth rotates.
▶ (^) If a body’s velocity equals or exceeds the escape velocity, Ve (p. 85),
it will follow a parabola or hyperbola. These orbits are termed open
orbits (p. 85) because the object never returns to its starting place.
▶ (^) Two objects in orbit around each other actually orbit their common
center of mass (p. 85).
▶ (^) Newton’s laws explain Kepler’s three laws of planetary motion. The
planets follow elliptical orbits because gravity obeys the inverse square
law. The planets move faster when closer to the sun and slower when
farther away because they conserve angular momentum (p. 83).
A planet’s orbital period squared is proportional to its orbital radius
cubed because the moving planet conserves energy.
▶ (^) Energy (p. 86) refers to the ability to produce a change. Kinetic
energy (p. 86) is an object’s energy of motion, and potential energy
(p. 86) is the energy an object has because of its position in a
gravitational fi eld. The unit of energy is the joule (J) (p. 86).
▶ (^) Tides are caused by differences in the force of gravity acting on differ-
ent parts of a body. Tides on Earth occur because the moon’s gravity
pulls more strongly on the near side of Earth than on the center of
Earth, and more strongly on the center of Earth than on the far side
of Earth. As a result, there are two tidal bulges on Earth caused by the
moon’s gravity, one toward the moon on Earth’s near side and one away
from the moon on Earth’s far side.
▶ (^) Tides produced by the moon combine with tides produced by the sun
to cause extreme tides (called spring tides) (p. 87) at new and full
moons. The moon and sun work against each other to produce smallest
tides (neap tides) (p. 87) at quarter moons.
▶ (^) Friction from tides can slow the rotation of a rotating world, and the
gravitational pull of tidal bulges can make orbits change slowly.
▶ (^) Einstein published two theories that extended Newton’s laws of motion
and gravity, the special theory of relativity and the general theory of
relativity.
▶ (^) Special theory of relativity (p. 91) says that uniform (unacceler-
ated) motion is relative. Observers cannot detect their uniform motion
through space except relative to outside objects. This is known as the
fi rst postulate. This leads to the second postulate: The speed of light is
a constant for all observers.
▶ (^) A consequence of special relativity is that mass and energy are related.
▶ (^) The general theory of relativity (p. 92) says that a gravitational
fi eld is a curvature of space-time caused by the presence of a mass. For
example, Earth’s mass curves space-time, and the mass of your body
responds to that curvature by accelerating toward Earth’s center.
▶ (^) The curvature of space-time was confi rmed by the slow advance in
perihelion (precession) of the orbit of Mercury, and by the defl ection
of starlight passing near the sun observed during a 1919 total solar
eclipse.