Physics 110
Black Holes
-Stars can collapse into black holes -Center of galaxies have black holes
Escape Speed
11.2 km/sec
An asteroid exerts a 360-N gravitational force on a nearby spacecraft. If the spacecraft moves to a point 3 times as far from the center of the asteroid, what is the gravitational force?
40 N
Newton's law of Universal Gravitation
F = Gm1m2 / d^2
If all of Earth's inhabitants moved to the equator, how would this affect Earth's rotational intertia? How would it affect the length of a day?
Increase rotational inertia, by angular momentum conservation, the rotation of the earth would decrease (just as a skater spins slower with arms outstreched), making days longer
Rotational speed
Number of rotations in a given time interval
When a satellite in circular orbit slows, perhaps due to the firing of a "retro rocket" it ends up gaining more speed than it had initially, Why?
Upon slowing, it spirals in toward the earth and in doing so has a component of gravitational force in its direction of motion which causes it to gain speed.
An object is placed exactly halfway between the Earth and moon. What will the object will move towards?
earth
Universal Gravitational Constant
6.67 x 10^-11 Nm^2/kg^2
A projectile is launched at 20 degrees. At what other angle can the projectile be launched to land in the same placee?
70 degree
Is gravitational force acting on a person who falls off a cliff? On an astronaut inside an orbiting space shuttle?
Gravitational force is indeed acting on a person who falls off a cliff, and on a person in a space shuttle. Both are falling under the influence of gravity.
Halfway to the center of earth, would the force of gravity on you be less than at the surface of earth?
Gravitational force, on you would be less, because there is less mass of earth below you, which pulls you with less force.
An astronaut lands on a planet that has the same mass as earth but twice the diameter. How does the austrnaut's wight differ from that on earth?
Letting the equation for gravitation guide your thinking, twice the diameter is twice the radius, which corresponds to 1/4 the astronaut's weight at the planet's surface.
At what point in its trajectory does a batted baseball have its minimum speed? If air drag can be neglected, how does this compare with the horizontal component of its velocity at other points?
Minimum speed occurs at the top, which is the same as the horizontal component of velocity anywhere along the path.
A friend says that up above the atmosphere, in space shuttle territory, Earth's gravitational field is zero. Explain how your friend has a misconception, and use the equation for gravitational force in your explanation.
Misconception is popular - investigation of the gravitational equation shows that no matter how big the distance, the force never gets to zero.
Which requires more fuel - a rocket going from Earth to the Moon or a rocket coming from the Moon to Earth? Why?
More fuel is required for a rocket that leaves Earth to go to the moon than the other way around. This is because a rocket must move against the greater gravitational field of the earth most of the way. ( If launched from the moon to earth, then it would be traveling with the earth's field most of the way )
Somewhere between Earth and the Moon, gravity from these two bodies on a space pod would cancel. Is this location nearer Earth or the Moon?
Nearer the Moon, because of its smaller mass and lesser pull at equal distances
Rotational inertia
Resistance of an object to change its rotational state of motion *The further the mass is from the axis of rotation, the greater the roational inertia
A softball and a basketball start from rest and roll down an incline. Which ball reaches the bottom first?
The softball because it has the least rotational inertia compared with its mass.
A large wheel is coupled to a wheel with half the diameter. How does the rotational speed of the small wheel compare with that of the larger wheel? How do the tangential speeds at the rims compare (assuming the belt doesn't slip)?
The tangential speeds are equal - same speed as the belt. The smaller wheel rotates twice as fast because for the same tangential speed, and r half, w must be twice. v(big wheel) = rw(small wheel) = (r/2 x 2w)
What two forces act on you while you are in a moving elevator? When are these forces equal magnitude and when are they not?
The two forces are the normal force and mg, which are equal when the elevator doesn't accelerate, and unequal when the elevator accelerates.
Projectile motion
a.) horizontal velocity is constant b.) vertical acceleration is constant (gravity) c.) the horizontal direction and the vertical direction are independent of one another d.) maximum range - 45 degree w/o air - 39-42 degrees w/air
Fragments of fireworks illuminate the sky. a.) what specific path is ideally traced by each fragment? b.) What paths would the fragments trace in a gravity-free region?
a.) the paths are parabolas b.) the paths would be straight lines
If the Earth's mass decreased to one-half its original mass with no change in radius, how will your weight change?
decrease to one half your original weight
The moon falls around the earth rather than straight into it, if the moon 's tangential velocity were zero, how would it more?
if the moons tangential velocity were zero, it would fall straight down and crash into earth
If the mass of the Earth somehow increased with no change in radius, how would your weight change?
increase also
How is the FORCE between two planets altered when they are moved twice as far apart?
it's 1/4 as much 2 x = 1/4 3x = 1/9 4x = 1/16 etc
At what point in its elliptical orbit about the Sun is the acceleration of Earth at a maximum? At what point is it at a minimum?
maximum where gravitational force is maximum, when earth is closest to the sun, at the perigee. At the apogee, force and accelerate are minimum.
Fast moving projectiles - satelittes
satellites move at high tangential velocity - earth satellite has a speed of 8 km/sec (85-90 min)
Suppose you stood atop a ladder so tall that you were 3 times as far from earths center as you presently are. Show that your weight would be one ninth of its present value.
squared is 9d2, which means the force is one ninth of surface weight
A weight watcher who normally weighs 400 N stands on top of a very tall ladder so she is one Earth radius above the Earth's surface. How much would she weigh there?
100 N
Why does a person in free fall experience weightlessness, while a person falling at terminal velocity does not?
A person is weightless when the only force acting is gravity, and there is no support force. Hence the person in free fall is weightless. But more than gravity acts on the person falling at terminal velocity. In addition to gravity, the falling person is "supported" by air drag.
Centripital Force
Any force that causes a body to move in a circular path *Stands for "center seeking"
Sitting at the middle of a turntable as it is set spinning and then allowed to spin freely. When you crawl toward the edge of the turntable, does the rate of rotation increase, decrease, or remain unchanged? What physics principle supports your answer?
As you crawl outward, the rotational inertia increases. In accord with the conservation of angular momentum, crawling toward the outer rim increases the rotational inertia of the spinning system and decreases angular speed.
A friend says that the international space station shown on the cover of this book is beyond the pull of earths gravity. Correct your friends statement.
Astronauts are weightless because they lack a support force, but they are well in the grips of Earth gravity, which accounts for them circling the Earth rather than going off in a straight line in outer space.
Two golfers each hit a ball at the same speed, but one at 60 degrees with the horizontal and the other at 30 degrees. Which ball goes farther? Which hits the ground first? (Ignore air resistance)
Both balls have the same range. Te ball with the initial projection angle of 30 degrees, however, is in the air for a shorter time and hits the ground first.
If you stopped an Earth satellite dead in its tracks, it would simply crash into Earth. why, then, don't the communications satellites that "hover motionless" abolve the same spot on Earth crash into Earth?
Communication satellites only appear motionless because their orbital period concides with the daily rotation of the Earth
Your friend says that the primary reason astronauts in orbit feel weightless is that they are beyond the main pull of Earth's gravity. Why do you agree or disagree?
Disagree, for the force of gravity on oribiting astronauts is almost as strong as at Earth's surface. They feel weightless because of the absence of a support force.
Of all the United States, why is hawaii the most efficient launching site for nonpolar satellites? (look at the spinning earth from above either pole and compare it to a spinning turntable)
Hawaii is closer to the equator, and therefor has a greater tangential speed about the polar axis. This speed could be added to the launch speed of a satellite and thereby save fuel
Another friend says that the Moon's gravity would prevent rocks dropping from the Moon to Earth, but that if the Moon's gravity somehow no longer pulled on its own rocks, then rocks on the moon would fall to earth. What is wrong with this assumption?
If gravity between the Moon and its rocks vanished, the rocks, like the moon, would continue in their orbital path around the Earth. The assumption ignores the law of inertia.
Conservation of Angular Momentum
If there is no external torque, the angular momentum of a system stays the same
Why is it important to secure fil cabinets to the floor, especially cabinets with heavy loads in top drawers?
If you pull open the upper drawers of a fully-loaded file cabinet that isn't secured to the floor, the CG of the cabinet can easily be shifted beyond the support base of the cabinet, causing it to topple over.
Larry weighs 300 N at the surface of earth. What is the weight of earth in the gravitational field of Larry?
In accord with Newton's 3rd law, the weight of the Earth in the gravitational field of the apple is 1 N; the same as the weight of the apple in the Earth's gravitational field.
If the world's population moved to the North Pole and the South pole, would the 24-hour day become longer, shorter, or stay the same?
In accord with the conservation of angular momentum, if mass moves closer to the axis of rotation, rotational speed increases. So the day would be slightly shorter.
Show that the gravitational force between two planets is quadrupled if the masses of both planets are doubled but the distance between them stays the same.
In accord with the inverse-square law, four times as far from the Earth's center diminishes the value of g to g/42, or g/16, or 0.6 m/s2.
How could an astronaut in a space shuttle "drop" an object vertically to Earth?
It could be dropped by firing it straight backward at the sme speed of the satellite
By how much does the gravitational force between two objects decrease when the distance between their centers is doubled? Tripled? Increased tenfold?
It decreases to 1/4, 1/9, and 1/100 to the original value.
Find the change in the force of gravity between two planets when distance between them is decreased by 10.
It is g = GM/r2 = (6.67 x 10-11) (3.0 x 1030)/(8.0 x 103)2 = 3.1 x 10^12 m/s2, 300 billion times g on Earth.
Consider an apple at the top of a tree that is pulled by earths gravity with a force 1 N. If the tree were twice as tall, would the force of gravity be 1/4 as strong?
No, because an apple at the top of the twice-as tall apple tree is not twice as far from earths center. The taller tree would need a height equal to the earths radius for the apple's weight at its top to reduce 1/4 N. Before its weight decreases by 1%, an apple or any object must be raised 32 km--nearly 4 times the height of Mt. Everest. So, as a practical matter we disregard the effects of everyday changes in elevation.
Comment on whether or not the following label on a consumer product should be cause for concern: CAUTION: The mass of this product pulls on every other mass in the universe, with an attracting orce that is proportional of the masses and inversely proportional to the square of the distance between them.
Nothing to be concerned about on this consumer label. It simply states the universal law of gravitation, which applies to ALL products. It looks like the manufacturer knows some physics and has a sense of humor.
When an Earth satellite is placed into a higher orbit, what happens to its period?
Period is greater for satellites farther from earth
Center of mass
The average position of all the mass that makes up the object *Things balance at the center mass For something to be stable, the center of mass has to be over the area of support
Tangential Speed
Speed of an object
Does the fact that one side of the Moon always faces Earth mean that the Moon rotates about its axis (like a top) or that it doesn't rotate about its axis?
The Moon does rotate like a top as it circles Earth. It rotates once per revolution, which is why we only see the same face. If it didn't rotate, we'd see the back side every half month.
Center of Gravity
The average position of weight distribution
A heavy crate falls from a high-flying airplace just as it flies directly above a shiny red Porsche smartly parked in a car lot. Relative to the Porsche, where will the crate crash? (pic pg 189)
The crate will not hit the Porsche, but will crash a distance beyond it determined by the height and speed of the plane.
Centrifugal Force
The feeling ou get being pulled outwad when going around in a circle *Result of rotation
A friend says that, since earth's gravity is so much stronger than the moons gravity, rocks on the moon could be dropped to earth. What is wrong with this assumption?
The force of gravity on moon rocks at the Moon's surface is considerably stronger than the force of gravity of the distant Earth. Rocks dropped on the Moon fall onto the Moon surface. (The force of the Moon's gravity is about 1/6 of the weight the rock would have on Earth; the force of the Earth's gravity at that distance is only about 1/3600 of the rock's Earth-weight.)
Gravitational Fields
The influence that a massive body extends into the space around itself, producing a force on another massive body. It is measured in newtons per kilogram
A youngster who has entered a soapbox derby (in which 4-wheel unpowered vehicles roll from rest down a hill) asks if large massive wheels or light weight ones should be used. Also, should the wheel have spokes or be solid?
The light weight solid wheels because they have the least rotational inertia - without spokes (disclike rather than hooplike)
Angular Momentum
The product of Rotational Inertia and Rotational Velocity
Torque
The product of the lever arm and force that tends to produce rotation Torque = lever arm x force
gravitational forces act on all bodies in proportion to their masses. Why, then, doesn't a heavy body fall faster than a light body?
The reason that a heavy body doesn't fall faster than a light body is because the greater gravitational force on the heavier body (its weight), acts on a correspondingly greater mass (inertia). The ratio of gravitational force to mass is the same for every body—hence all bodies in free fall accelerate equally. And it's true not just near the Earth, but anywhere. (This is illustrated in Figures 4.11 and 4.12.)
Einsteins theory of gravitation
Warping of space by mass
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 one doubles. If both masses double, the force is 4 times as much.
What is the shape of the orbit when the velocity of the satellite is everywhere perpendicular to the force of gravity?
When the velocity of a satellite is everywhere perpendicular to the force of gravity, the orbital path is a circle
Why does a typical helicopter with a single main rotor have a second small rotor on its tail? Describe the consequences if the small rotor fails in flight
Without the small rotor on its tail, the helicopter and the main rotor would rotate in opposite directions. The small rotor provides a torque to offset the rotational motion that the helicopter would otherwise have.
As a car speeds up when rounding a curve, does the centripetal force on the car also increase?
Yes - increased speed at the same radical distance means greater centripetal force. If greater centripetal force isn't applied, the car will skid.
Flamingos are frequently seen standing on one leg with the other lifted. Is rotational inertia enhanced with long legs? What can you say about the bird's center of mass with respect to the foot on which it stands?
Yes, rotational inertia is enhanced with long legs. The bird's foot is directly below the bird's center of mass
Why must you bend forward when carrying a heavy load on your back?
You bend forward to shift the center of gravity of you and the weight of the backpack aboce the area bounded by your feet - otherwise you will fall over backward
If our sun shrank in size to become a black hole, show from the gravitational force equation that Earth's orbit would not be affected
~ m1m2/d^2 , where m2 is the mass of the sun (which doesn't change when forming a black hole), m1 is the mass of the orbiting earth, and d is the distance between the center of mass of earth and the sun. None of these terms change, so the force F that holds earth in orbit doesn't change.
