Chapter 9 Physics Gravity
Give an example of when your weight is greater than mg. Give an example of when your weight is zero.
. In an upward accelerating elevator your weight is greater than mg, in free fall your weight is 0.
A person weighs 150 pounds. He is standing on a scale inside an elevator. If the elevator is free-falling downward (perhaps the cable broke), what is the reading of the scale?
0 pounds Since the person is in free-fall, his acceleration is -9.8 m/s/s and thus the only force acting on him is that of gravity.
A 150 pound person decides to move to the center of the Earth (carving out a little cavity for her house). What is the net force of gravity exerted by the Earth on this person?
0 pounds. Since there is the same amount of material in all directions, at equal distances from the person, the net force is zero.
State Newton's law of universal gravitation in words. Then do the same with one equation.
The force is proportional to the product of two masses and inversely proportional to the square of the distance between their centers. As an equation: F ~ m₁ ×m₂/d²
weight
The force that an object exerts on a supporting surface, or if suspended, a supporting string. Often, but not always, due to the force of gravity.
A person weighs 150 pounds. He steps in an elevator and stands on a scale. If the elevator is moving upwards with a constant speed, what is the reading of the scale?
150 pounds. Since the elevator is not accelerating, the scale provides an upward force equal in magnitude to the force of gravity.
What is a gravitational field, and how can its strength be measured?
A gravitational field is a force field about any mass, and can be measured by the amount of force on a unit of mass located in the field. Typically measured as newtons per kilogram.
What would be the magnitude of the gravitational field anywhere inside a hollow, spherical planet?
Anywhere inside a hollow planet the gravitational field of the planet is zero.
Why are all tides greatest at the time of a full Moon or new Moon?
At full moon and new moon the tides from the Moon and the Sun add because they are in line with Earth.
black hole
Concentration of mass resulting from gravitational collapse, near which gravity is so intense that not even light can escape.
weightlessnes
Condition of free fall toward or around Earth in which an object experiences no support force and exerts no force on a scale.
Inside a planet gravitational fields do what?
Decreases to zero. Because the pull from the mass of Earth below you is partly balanced by what is above you.
perturbation
Deviation of an orbiting object, e.g. a planet, from its path around a center of force, e.g. the sun, caused by the action of an additional center of force, e.g. another planet.
By how much does the gravitational force between two objects decrease when the distance between their centers is doubled? Tripled? Increased tenfold?
The force decreases to 1/4, 1/9, and 1/100 the original value. Because gravitational force = 1/d²
How often do high tides occur?
Every 12 hours
law of universal gravitation formula
F = G(m₁m₂/d²)
What is the magnitude of Earth's gravitational force on a 1-kg body at Earth's surface?
The force is about 10 N, or 9.8 N precisely
law of universal gravitation
For any pair of objects, each particle attracts the other object with a force that is directly proportional to the product of the masses of the objects, and inversely proportional to the square of the distance between them, or their centers of mass if spherical objects, where F is the force, m is the mass, d is distance and G is the gravitation constant.
What is the magnitude of the gravitational force between two 1-kg bodies that are 1 m apart?
G = 6.67 × 10-¹¹Nm²/kg² which = 0.0000000000667 newton. This shows that gravity is a very weak force compared with electrical forces.
Einstein's Theory of Relativity
Gravitational field is a warping of space-time by a planet. The warped space time affects motion of other objects.
If there is an attractive force between all objects, why don't we feel ourselves gravitating toward massive buildings in our vicinity?
Gravity certainly does pull us toward massive buildings and everything else in the universe. The forces between buildings and us are relatively small because their masses are small compared with the mass of Earth. The forces due to the stars are extremely tiny because of their great distances from us. These tiny forces escape our notice when they are overwhelmed by the overpowering attraction to Earth. Physicist Paul A. M. Dirac, 1933 Nobel Prize recipient, put it this way: "Pick a flower on Earth and you move the farthest star!"
For a planet of uniform density, how would the magnitude of the gravitational field halfway to the center compare with the field at the surface?
Half way to the center, the gravitational field is half that of the surface. G/2.
spring tide
High or low tide that occurs when the Sun, Earth, and the Moon are lined up so that the tides due to the sun and Moon coincide, making the high tides higher than average and the low tides lower than average.
wormhole
Hypothetical enormous distortion of space and time, similar to a black hole, but opening out again in some other part of the universe.
ether
Hypothetical invisible medium that was formerly thought to be required for the propagation of electromagnetic waves and thought to fill space through the universe.
What was the cause of perturbations discovered in the orbit of the planet Uranus? What greater discovery did this lead to?
Neptune caused the perturbations, and later Pluto was discovered.
In what sense does the Moon "fall"?
It falls away from the straight path that it would follow if there were no forces acting on it.
What happens to the strength of the gravitational field at the surface of a star that shrinks?
It increases.
How does the thickness of paint sprayed on a surface change when the sprayer is held twice as far away?
It is 0.25 as thick.
If Earth shrank but there was no change in its mass, what would happen to your weight at the surface?
It would increase.
If Earth shrank, but there was no change in its mass, then what would happen to your weight at the surface?
It would increase.
From Part B, you know that from afar you'll never see the in-falling rocket cross the event horizon, yet it will still eventually disappear from view. Why?
Its light will become so redshifted that it will be undetectable. As the video shows, viewed from afar the light of the in-falling rocket becomes increasingly redshifted. As it approaches the event horizon, the redshift approaches infinity, meaning all its light is stretched to such enormous wavelengths that no detector could see it, even in principle
inverse-square law
Law relating the intensity of an effect to the inverse square of the distance from the cause. Gravity, electric, magnetic, light, sound, and radiation phenomena follow the inverse-square law. Intensity ~ 1/distance²
The force of gravity between two planets depends on their
masses and distances apart. The equation for gravitational force cites only masses and distances as variables.
Newton viewed the curving of the path of a planet as being caused by a force acting on the planet. How did Einstein view the curved path of a planet?
Moving in curved 4-dimensional spacetime
What did Newton discover about gravity?
Newton discovered that gravity is directly proportional to the product of two masses and inversely proportional to the square of the distance separating them.
Gravitational force acts on all bodies in proportion to their masses. Why, then, doesn't a heavy body fall faster than a light body?
Newton's second law (a = F/m) reminds us that greater force acting on greater mass does not result in greater acceleration.
Would the springs inside a bathroom scale be more compressed or less compressed if you weighed yourself in an elevator that was moving upward at constant velocity? Downward at constant velocity?
No changes in compression when moving at constant velocity.
Consider an apple at the top of a tree that is pulled by Earth's gravity with a force of 1 N. If the tree is twice as tall, will the force of gravity be 1/4 as strong? Defend your answer.
No, because an apple at the top of the twice-as-tall apple tree is not twice as far from Earth's center. The taller tree would need a height equal to Earth's radius (6370 km) for the apple's weight at its top to reduce to 1/4 N , we disregard the effects of everyday changes in elevation..
We know that both the Moon and the Sun produce our ocean tides. And we know that the Moon plays the greater role because it is closer. Does the Moon's closeness mean that it pulls on Earth's oceans with a greater gravitational force than the Sun?
No. The Sun's pull is much stronger. But the difference in lunar pulls is greater than the difference in solar pulls. So our tides are due primarily to the Moon
Would a torque on the Moon occur if the Moon were spherical, with both its center of mass and center of gravity in the same location?
No. the torque requires a distance between the center of mass and the center of gravity. There is no lever arm between Earth's gravitational pull and the Moon's axis.
What would happen to Earth if the Sun became a black hole?
Nothing, actually. Letting the equation for gravity guide our thinking, we see that no mass changes, no distance from center to center changes, so there would be NO change in force between the Sun an Earth.
Do tides depend more on the strength of gravitational pull or on the difference in strengths? Explain.
Ocean tides are caused by differences in pulling strengths.
To calculate the gravitational force between two objects we __________, and then multiply by the gravitational constant
multiply the two masses, divide by their distance squared
Which has the higher tides: spring tides or neap tides?
Spring tides are higher, because the tides from the Moon and Sun add together.
Would the springs inside a bathroom scale be more compressed or less compressed if you weighed yourself in an elevator that was accelerating upward? Downward?
Springs are more compressed when accelerating upward, and less compressed when accelerating downward.
neutron star
Star that has undergone a gravitational collapse in which electrons are compressed into protons to form neutrons.
force field
That which exists in the space surrounding a mass, electric charge, or magnet, so that another mass, electric charge, or magnet introduced into this region will experience a force. Examples of force fields are gravitational fields, electric fields, and magnetic field.
Why is a black hole invisible?
The escape velocity of the black hole is greater than the speed of light
Why do both the Sun and the Moon exert a greater gravitational force on one side of Earth than on the other?
The force between the Moon and Earth is stronger on the side facing the Moon. The gravitational force is weaker with increased distance.
How does the force of gravity between two bodies change when the distance between them is doubled?
The force decreases to 1/4 of its initial value.
Suppose you weigh 400 pounds and want to weigh only 100 pounds. Instead of going on a diet, you could live in space. How far from the center of the Earth would you have to live?
The distance would be two Earth radii away from the center of the Earth. Since the distance to the center of the Earth doubled, the weight would decrease by a factor of four.
As you found in Part A, your weight will be greater than normal when the elevator is moving upward with increasing speed. For what other motion would your weight also be greater than your normal weight?
The elevator moves downward while slowing in speed.
Halfway to the center of Earth, would the force of gravity on you be less than at the surface of Earth?
The gravitational force on you would be less because there is less mass of Earth below you, which pulls you with less force. If Earth were a uniform sphere of uniform density, the gravitational force halfway to the center would be exactly half that at the surface. But since Earth's core is so dense (about seven times the density of surface rock), the gravitational force halfway down would be somewhat more than half. Exactly how much depends on how Earth's density varies with depth, which is information that is not known today.
How does the gravity in the Space Shuttle compare with the gravity on Earth's surface?
The gravity in the Space Shuttle is approximately equal to the gravity on the surface of the Earth.
The Moon falls around Earth rather than straight into it. If the Moon's tangential velocity were zero, how would it move?
The moon would fall into the Earth
Why are International Space Station occupants weightless when they are firmly in the grip of Earth's gravity?
The occupants are without a support force.
Universal gravitational constant
The proportionality constant G that measures the strength of gravity in the equation for Newton's law of universal gravitation. F = G(m₁m₂/d²)
Why do the astronauts in the Space Shuttle float around?
The space shuttle is in free fall so the shuttle and the astronauts inside it are continuously falling toward the Earth. They thus experience apparent weightlessness.
Tangential velocity angle of a planet or moon
The tangential velocity of a planet or moon moving in a circle is at a right angle to the force of gravity.
What is the Newtonian synthesis?
The union of terrestrial laws and cosmic laws.
Predict how the upward force exerted on the feet by the scale will compare to the man's weight if the elevator is moving downward at a constant speed.
The upward force on the feet will be equal to the man's weight.
Predict how the upward force exerted on the feet by the scale will compare to the weight of the man if the elevator is moving upward at a constant speed.
The upward force on the feet will be equal to the man's weight.
Predict how the upward force exerted on the feet by the scale will compare to the man's weight if the elevator is accelerating downward.
The upward force on the feet will be less than the man's weight.
Predict how the upward force exerted on the feet by the scale will compare to the man's weight if the elevator is accelerating upward.
The upward force on the feet will exceed the man's weight.
neap tide
Tide that occurs when the Moon is halfway between a new Moon and a full Moon, in either direction. The tides due to the Sun and the Moon partly cancel, so that the high tides are lower than average and the low tides are not as low as average.
From the viewpoint of an observer in the orbiting rocket, what happens to time on the other rocket as it falls toward the event horizon of the black hole?
Time runs increasingly slower as the rocket approaches the black hole.
simple harmonic motion
Vibratory or periodic motion, like that of a pendulum, in which the force acting on the vibrating body is proportional to its displacement from its central equilibrium position and acts toward that position.
What would the magnitude of the gravitational field be anywhere inside a hollow, spherical planet?
What would the magnitude of the gravitational field be anywhere inside a hollow, spherical planet?
According to the equation for gravitational force, what happens to the force between two bodies if the mass of one of the bodies is doubled? If both masses are doubled?
When one mass is doubled, the force between it and the other mass doubles. If both masses double, the force is 4 times as great.
inversely
When two values change in opposite directions, so that if one increases and the other decreased by the same factor, they are said to be inversely proportional to each other.
center of gravity
Where the object's weight is considered to act
Why does the gravity in the Space Shuttle compare with the gravity on Earth the way it does?
Why does the gravity in the Space Shuttle compare with the gravity on Earth the way it does?
When G was first measured by Henry Cavendish, newspapers of the time hailed his experiment as the "weighing Earth experiment." Why?
With a known mass, a scale, and the radius of Earth, you can calculate the unknown mass of Earth if you know G
Do tides occur in the molten interior of Earth for the same reason that tides occur in the oceans?
Yes due to the difference in gravitational force across the core
Where do you weigh more: at the bottom of Death Valley or atop one of the peaks of the Sierra Nevada? Why?
You weight more in Death Valley because you are closer to the center of the Earth.
When is your weight measured as mg
Your weight is measured as mg when you are strongly supported in a gravitational field of g and in equilibrium.
What is the magnitude of the gravitational field at Earth's center?
Zero.
gravitational field
a force field that exists in the space around every mass or group of masses; measured in newtons per kilogram.
If you were inside the rocket that falls toward the event horizon, from your own viewpoint you would __________.
accelerate as you fall and cross the event horizon completely unhindered. From your point of view, time runs normally in your spaceship and gravity must accelerate you as you fall toward the black hole. There is no physical barrier at the event horizon, so you cross it unhindered.
If you were inside the rocket that falls toward the event horizon, you would notice your own clock to be running __________.
at a constant, normal rate as you approach the event horizon. All motion is relative, and you will always consider your own clock to be running at a normal rate
gravitation
attraction between objects due to mass.
If the mass of one planet is somehow doubled, the force of gravity between it and a neighboring planet would
double. From F = G(m₁m₂)/ d² If one doubles then the force will double
Newton's most celebrated synthesis is of
earthly and heavenly laws
When an elevator accelerates upward, your weight reading on a scale is
greater. The support force pressing on you is greater.
A person weighs 150 pounds. He is standing on a scale inside an elevator. The elevator is moving downward and slowing down (coming to a stop). What is the reading of the scale?
higher than 150 pounds Since the elevator is slowing down while moving downwards, the person is accelerating upwards. This means that the normal force must be greater than the magnitude of the force of gravity.
A 150 pound astronaut is inside the space shuttle, orbiting around the Earth just above the Earth's atmosphere (the atmosphere's thickness is very small compared to the radius of the Earth). The force of gravity acting on the astronaut
is roughly 150 pounds. Even though the astronaut is in a state of free fall and feels weightless, the force of gravity is still the same as if he were standing on Earth's surface.
When an elevator accelerates downward, your weight reading is
less. The support force is less.
The constant G in Newton's equation _______.
makes the units of measurement consistent
An object is lifted from the surface of a spherical planet to an altitude equal to the radius of the planet. As a result, which of the following changes in the properties of the object take place?
mass remains the same; weight decreases
Suppose you stepped into a hole bored clear through the center of Earth and made no attempt to grab the edges at either end. Neglecting air drag, what kind of motion would you experience?
ou would oscillate back and forth. If Earth were an ideal sphere of uniform density and there were no air drag, your oscillation would be what is called simple harmonic motion. Each round-trip would take nearly 90 minutes. We will see in Chapter 10 that an Earth satellite in close orbit about Earth also takes 90 minutes to make a complete round-trip. (This is no coincidence: If you study physics further, you'll learn that "backand- forth" simple harmonic motion is simply the vertical component of uniform circular motion—interesting stuff.)
If the masses of two planets are somehow doubled, the force of gravity____
quadruples
If the masses of two planets are each somehow doubled, the force of gravity between them
quadruples. From F = G(m₁m₂)/ d² If both masses are doubled then the force would four times greater.
Suppose a rocket ship, while sitting on Earth, has a weight of 10,000 N. The rocket ship takes off and reaches a distance from Earth's center of 3REarth. What is the force due to gravity acting on the spaceship?
roughly 1,100 N The force of gravity here is 1/9th that when on the surface.
A person weighs 150 pounds. He is standing on a scale in an elevator and stands on a scale. The elevator is accelerating upwards at a rate of 10 m/s/s, what is the reading of the scale?
roughly 300 pounds The upward force of the scale minus the force of gravity is equal to the mass times acceleration. Since the acceleration is roughly g (9.8 m/s/s), this means that the upward force needs be 2mg.
As the falling rocket plunges toward the event horizon, an observer in the orbiting rocket would see that the falling rocket __________.
slows down as it approaches the event horizon, and never actually crosses the event horizon
Consider Earth and the Moon. As you should now realize, the gravitational force that Earth exerts on the Moon is equal and opposite to that which the Moon exerts on Earth. Therefore, according to Newton's second law of motion __________.
the Moon has a larger acceleration than Earth, because it has a smaller mass. Newton's second law of motion, F=ma, means that for a particular force F, the product mass x acceleration must always be the same. Therefore if mass is larger, acceleration must be smaller, and vice versa.
If you are standing on a scale in an elevator, what exactly does the scale measure?
the force exerted on the scale
When the elevator cable breaks, the elevator falls freely so your weight reading is
zero. There is zero support force on you.