.
b) as seen by someone on the plane, the object will fall straight down (since it has the same horizontal velocity as the plane). c) It will land directly below the airplane.
Exercises:
9.16: The acceleration of gravity is les on top of Everest than at sea level. You are 10 km farther from the center of the Earth so that the d in F = GMm/d2 is larger and F is smaller.
9.18: Doubling the mass of the planet doubles the gravitational
force. Doubling the radius quarters the force. The net result is to
halve it. In equations, you have on Earth (ME is the mass
of the Earth, Mp is the mass of the planet, dE
is the radius of the Earth, dp is the radius of the planet,
and m is the mass of the astronaut):
FE = G MEm/dE2
and on the other planet:
Fp = G Mpm/dp2
Fp = G (2ME)m/(2dE2)
Fp = (1/2) G MEm/dE2
Fp = (1/2) FE
9.28: Yes, gravity is acting on the pencil. You and the pencil and the elevator are all accelerating downward at 10 m/s2. Therefore you have the same velocity as each other and the pencil appears to hover.
9.36: If the Moon pulled uniformly on the Earth, so that each kilogram of the Earth experienced the same force, then there would be no tidal forces and no tides. However, the Moon pulls harder on the part of the Earth closest to it, pulling it closer to the Moon. This difference in forces stretches the Earth the same way that a difference in forces stretches the shirt (and tears it).
9.40: Yes, when a high tide is unusually high, the following low tide is unusually low. This is because the water is 'piled up' on the near and far sides of the Earth (relative to the Moon), leaving much less water than usual in between. When your part of the Earth is nearest or farthest the Moon (eg: when there is a full Moon, at noon or midnight) you see the piled up water. At 6:00 (AM or PM) you see the area with much less water than usual. The tides just rearrange the water, they do not create or destroy it.
10.16: Your hang time depends only on the vertical component of velocity. The horizontal and vertical components are completely independent of each other.
10.18: The Moon does not crash into the Earth because it has a large tangential speed. If the Earth did not pull on the Moon, then the Moon would continue in a straight line path and leave us forever. Because the Earth pulls on the Moon, the Moon travels in a circle (ie: it orbits the Earth).
10.22: There are two ways to answer this problem, with and without math. First, without: When we derived the orbital speed (near the surface of the Earth) in class, we used the facts that a) a dropped object falls 5 m in the first second and b) the curvature of the Earth is 5 m for every 8 km. An object with a speed of 8 km/s will travel 8 km in the first second. During that it will fall 5 m and the surface of the Earth will fall away by 5 m. Therefore, the orbit depends only on those two facts. If the mass of the Earth changes, then the acceleration of gravity will change and the satellite will fall a different distance in the first second. If the distance from the satellite to the Earth changes, then the curvature of the orbit will change. If the mass of the satellite changes, neither the gravitational acceleration nor the curvature of the orbit will change. Therefore, the speed of a circling satellite does not depend on the mass of the satellite.
Second, with math: the force of gravity F = GMEm/d2 provides the centripetal force needed to keep the satellite in orbit F = mv2/d where d is the distance from the satellite to the center of the Earth. Setting them equal, we have v2 = GME/d. Thus the speed depends only on the mass of the Earth and the radius of the orbit.
10.46: This is the same as exercise 9.28. The space shuttle, the astronaut and the wrench are all in free fall. (They do not hit the Earth because they have a large tangential speed.) Thus, the wrench will hover in front of the astronaut.
10.extra: a) as seen by an observer on the ground, the object will
follow a parabolic path, like any other object thrown horizontally.
(See figure.) .
b) as seen by someone on the
plane, the object will fall straight down (since it has the same
horizontal velocity as the plane). c) It will land directly below the airplane.
9.4: g = GM/d2 = 6.67*10-11 N m2/kg2 * 3.0*1030 kg / (8*103 m)2 = 3.1*1012 m/s2
9.5 (modified): g = GME/d2 = 6.67*10-11 N m2/kg2 * 6.0*1024 kg / (3*105 m + 6.38*106 m)2 = 9.0 m/s2.
10.6: First we figure out the time it takes the tennis ball to fall 1 m. We get this from d = (1/2)gt2 so that t = sqrt(2d/g) = sqrt(2 * 1 m / 10 m/s2) = sqrt(0.2) s = 0.45 s. Now, the maximum horizontal distance the tennis ball can cover in that time is 12 m (otherwise it will be out of bounds). This gives a maximum speed of v = d/t = 12 m / 0.45 s = 26.8 m/s.