Homework 7: Chapter 4

Imagine another solar system, with a star of the same mass as the Sun. Suppose a planet with a mass twice that of Earth (2M earth) orbits at a distance of 1 AU from the star. What is the orbital period of this planet?

1 year (The planet's mass is so small compared to the star's mass that it has essentially no effect on the planet's orbit. (We know this from Newton's version of Kepler's third law.) The fact that the planet has the dame orbital distance of Earth therefore means it must have the same orbital period as Earth.)

Which of the cars is accelerating?

A car going around a circular track at a steady 100 miles per hour.

Each of the following lists two facts. Which pair of facts can be used with Newton's version of Kepler's third law to determine the mass of the Sun?

Earth is 150 million km from the Sun and orbits the Sun in one year. (A single planet's orbital distance and orbital period are all we need to determine the Sun's mass Newton's version of Kepler's third law.)

Suppose that the Sun shrank in size but that its mass remained the same. What would happen to the orbit of the Earth?

Earth's orbit would be unaffected. (The force of gravity between Earth and the Sun, and hence the orbital distance and speed of Earth, depends only on the Sun's mass (and the Earth-Sun distance), not on the Sun's size.)

According to the law of universal gravitation, what would happen to Earth if the Sun were somehow replaced by a black hole of the same mass?

Earth's orbit would not change.

Imagine another solar system, with a star more massive than the Sun. Suppose a planet with the same mass as Earth orbits at a distance of 1 AU from the star. How would the planet's year (orbital period) compare to Earth's year?

The planet's year would be shorter than Earth's. (This is true because the greater mass of the star would mean a stronger force of gravity at any given distance, which in turn would mean a higher orbital velocity.)

Which of the following statements is not one of Newton's Laws of Motion?

What goes up must come down. (This is not one of Newton's laws, and it's not even true. Objects with escape velocity can go up without coming back down.)

Newton's Second Law of Motion tells us that the net force applied to an object equals its ...

mass multiplied by acceleration (We often write this fact simply as F = ma .)

Momentum is defined as ...

mass multiplied by velocity (Notice that because velocity includes direction, momentum also includes direction.)

A net force acting on an object will always cause a change in the object's ...

momentum (Force is actually defined at the rate of change in momentum.)

What does temperature measure?

the average kinetic energy of particles in a substance (For example, air molecules are moving faster on average on a hot day than on a cool day.)

Absolute zero is ...

0 Kelvin (The Kelvin scale starts from absolute zero.)

Suppose you drop a 10-pound weight and a 5-pound weight on the Moon, both from the same height at the same time. What will happen?

Both will hit the ground at the same time. (The acceleration of gravity on the Moon is smaller than it is on Earth, but it still is the same for all objects. Therefore, both objects will fall at the same rate. (And because there is no air on the Moon, they'll hit at the same time no matter what shape or density they have.))

* Newton's theory of gravity has been tested extensively, and while it passed many tests, it did not pass all of them. For example, its prediction for how Mercury's orbit changes with time disagrees slightly with observations. Einstein's general theory of relativity improves on Newton's theory: In most cases the two theories of gravity predict the same results, but in the situations where they differ, Einstein's theory works better than Newton's. *

An addendum to Part D of the following problem: "Each diagram shows a single experimental trial in which you will drop a ball from some height. In each case, the ball's size, mass, and height are labeled..."

Consider the statement "There's no gravity in space." This statement is:

Completely false.

Match the following names and the major roles played in overturning the ancient belief in an Earth-centered universe. Drag the names on the left to the appropriate blanks on the right to complete the sentences.

Copernicus developed a model in which planets move around the Sun along perfectly circular orbits. Tycho provided a large amount of data from astronomical observations and suggested model in which the Sun orbits Earth while all other planets orbit the Sun. Kepler described the laws of planetary motion for the heliocentric model, according to which planets move around the Sun along elliptical orbits.

Which statement must be true for a rocket to travel from Earth to another planet?

It must attain escape velocity from Earth. (If it does not have escape velocity, it will either fall back down or orbit Earth.)

If you want to make a rocket turn left, you need to:

Fire an engine that shoots out gas to the right.

Why is Newton's version of Kepler's third law so useful to astronomers?

It can be used to determine the masses of many distant objects. (We can apply Newton's version of Kepler's third law whenever we observe one object orbiting another; this is the primary way that we measure masses throughout the universe.)

Part A.) Drag each statement into the correct bin based on whether it describes motion that involves acceleration or motion at a constant velocity. Note: For the motions that are on Earth (e.g., car, ball, elevator), ignore any effects of Earth's rotation or orbit. Part B.) Drag each statement into the correct bin based on whether it describes motion in which the object's momentum is changing. Note: For the motions that are on Earth (e.g., car , ball, elevator), ignore any effects of Earth's rotation or orbit. Part C.) Drag each statement into the correct bin based on whether the motion requires the action of a net force. Note: For the motions that are on Earth (e.g., car, ball, elevator), ignore any effects of Earth's rotation or orbit. Part D.) Which of the following statements correctly state general principles or motion? (Assume that the moving object's mass is not changing.) Select all that apply.

Part A.) Constant Acceleration: a car is speeding up after being stopped, a ball is in freefall after being dropped from a high window, a car is slowing down for a stop sign, a planet is orbiting the Sun in an elliptical orbit, a car is holding a steady speed around a curve & a planet is orbiting the Sun in a circular orbit. Constant Velocity: an elevator is going upward at a constant speed, a car is driving 100 km/hr on a straight road & a spaceship is coasting without engine power in deep space. (Acceleration refers to any change in velocity. Because velocity includes both speed and direction, acceleration is occurring whenever there is any change in speed, direction, or both. Constant velocity means that both speed and direction are unchanging.) Part B.) Change in Momentum: same as Constant Acceleration. Constant Momentum: same as Constant Velocity. (Momentum is defined as mass times velocity, so if an object's velocity is changing (that is, if it is accelerating), then its momentum must also be changing.) Part C.) Net Force (nonzero): same as Constant Acceleration. No Net Force: same as Constant Velocity. (The only way to change an object's momentum is to apply a net force to it, so if an object's momentum is changing, then a net force must be acting upon it. For example, in the case of the accelerating cars the net force is from the engine.) Part D.) Accelerated motion includes any motion involving a change in speed, change in direction, or both & An object that is accelerating is also undergoing a change in momentum & An object that is accelerating is also being acted upon by a (nonzero) net force. (As long as an object's mass is not changing, a net force will cause an object to undergo some type of acceleration. because acceleration is a change in velocity, and momentum is mass times velocity, the accelerating object is also undergoing a change in momentum.)

Part A.) The six statements below represent Newton's three laws of motion and Kepler's three laws of planetary motion. Match each statement to the scientist (Kepler or Newton) associated with it. Drag the names in the left-hand column to the appropriate blanks in the right-hand column. The names can be used more than once. Part B.) match the correct laws to the examples in which they apply. Use each law only once. Drag the words in the left-hand column to the appropriate blanks in the right-hand column.

Part A.) Newton: For any force, there is an equal and opposite reaction force & Force = mass x acceleration & An object moves at constant velocity if there is no net force acting upon it. Kepler: More distant planets orbit the Sun at slower average speeds, obeying the precise mathematical relationship p2 = a3 & The orbit of each planet about the Sun is an ellipse with the Sun at one focus & A planet moves faster in the part of its orbit nearer the Sun and slower when farther from the Sun, sweeping out equal areas in equal times. Part B.) Force equals mass times acceleration explains why applying a force to a baseball with your arm can cause the baseball to accelerate from rest to the speed at which it leaves your hand. An object moves at constant velocity if there is no net force acting on it explains why a spaceship with no forces acting on it will continue moving even if it has no fuel. For planets orbiting the Sun, period (p) and orbital distance (a) obey the relation p2 = a3 explains why Earth orbits the Sun at a faster average speed than Mars. The orbit of each planet is an ellipse, with the Sun at one focus explain's why Earth's distance from the Sun varies over the course of the year. A line between a planet and its Sun sweeps out equal areas in equal times explains why Earth's orbital speed varies over the course of each year. For any force, there is an equal and opposite reaction force tells us that, when you are pushing on a table, the table is pushing up on you with a force that precisely balances the force of your push.

Part A.) Drag words indicating what happens to the first blank and indicating the reason to the second blank. Part B.) Drag words indicating what happens to the first blank and indicating the reason to the second blank. Part C.) As the cloud shrinks in size, its central temperature ... as a result of its ... Drag words indicating what happens to the first blank and indicating the reason to the second blank. Part D.) Suppose that the Sun were to collapse from its current radius of about 700,000 km to a radius of only about 6000 km (about the radius of Earth). What would you expect to happen as a result? Part E.) Suppose that two asteroids are orbiting the Sun on nearly identical orbits, and they happen to pass close enough to each other to have their orbits altered by this gravitational encounter. If one of the asteroids ends up moving to an orbit that is closer to the Sun, what happens to the other asteroid?

Part A.) The video shows a collapsing cloud of interstellar gas, which is held together by the mutual gravitational attraction of all the atoms and molecules that make up the cloud. As the cloud collapses, the overall force of gravity that draws the cloud inward gradually becomes stronger because the strength of gravity follows an inverse square law with distance. (The force of gravity between any two particles increases as the particles come closer together. Therefore, as the cloud shrinks and particles move closer together, the force of gravity strengthens. This will tend to accelerate the collapse before the internal gas pressure builds up. (Once the gas pressure builds up, the outward push of the pressure can counteract the inward pull of gravity, which is why the cloud eventually stops contracting.) Part B.) As the cloud shrinks in size, its rate of rotation speeds up because its total angular momentum is conserved. (The law of conservation of angular momentum states that, in the absence of external influences (torques), the angular momentum of an object or a system of objects stays constant. Since the angular momentum of an object depends on both its size and rate of rotation, the cloud's rate of rotation will increase as its size (or radius) decreases in order to conserve angular momentum.) Part C.) As the cloud shrinks in size, its central temperature increases as a result of its gravitational potential energy being converted to thermal energy. (As the cloud shrinks in size, its gravitational potential energy decreases. Because energy cannot simply disappear, the "lost" gravitational potential energy must be converted into some other form. Some of it is converted into thermal energy, which raises the temperature of the gas cloud. The rest is mostly converted into radiative energy, which is released into space as light.) Part D.) A tremendous amount of gravitational potential energy would be converted into other forms of energy, and the Sun would spin much more rapidly. (The dramatic shrinkage of the Sun would mean the loss of a huge amount of gravitational potential energy. Because energy is always conserved, this "lost" gravitational potential energy must reappear in other forms, such as heat (thermal energy) and light (radiative energy). Meanwhile, conservation of angular momentum would ensure that the collapsed Sun would spin much faster.) Part E.) It will end up on an orbit that is farther from the Sun. (Total energy must be conserved, so if one asteroid loses energy and moves to a closer orbit, the other must gain energy and move to a more distant orbit.)

Part A.) Suppose you drive from home to your child's school, pick her up, and drive back home, covering a total distance of 25 miles in 1 hour. Which of the following statements are true? Select all the true statements. Part B.) Complete the sentences correctly. Drag the words in the left-hand column to the appropriate clanks in the sentences in the right-hand column. You may use the same words more than once. Part C.) Why are astronauts (and other objects) weightless inside the International Space Station as it orbits Earth?

Part A.) Your velocity is different on the return home than it is on the way to school & Your average speed for the trip is 25 miles per hour & You must accelerate when you reach the school. (Your average speed is 25 miles per hour because you travel a total distance of 25 miles in one hour. Your velocity is different on the way home than on the way to school because you are going in a different direction, and velocity includes direction. You must accelerate when you reach the school because you must slow down to stop there, then turn around (which is a change in direction), and them speed up to start on your way home; all three of these (slowing, change of direction, speeding up) represent changes in velocity and therefore are examples of acceleration.) Part B.) On Earth, the acceleration of gravity tells us that the velocity of a falling object increases by about 10 m/s for each second it falls. If you stand on a scale on the moon, your weight will be different than it is on Earth but your mass will be the same. If you are in free-fall, then your weight will be zero. An objects momentum is its mass times its velocity, and we say that it has angular momentum if it is rotating or turning on a curved path. If your momentum is changing, then a(n) net force must be acting on you. Part C.) they are in free-fall (Note that this question is nearly identical to the one answered within the video.)

Part A.) Each diagram shows a single experimental trial in which you will drop a ball from some height. In each case, the ball's size, mass, and height are labeled. Note that two diagrams show a basketball, one diagram shows a bowling ball of the same size but larger mass, and one diagram shows a much smaller marble with the same mass as the basketball. You have a timer that allows you to measure how long it takes the ball to fall to the ground. Which pair of trials will allow you to test the prediction that an object's mass does not affect its rate of fall? Check exactly two of the following diagrams. Part B.) Assume you have completed the two trials chosen in Part A. Which of the following possible outcomes from the trials would support Newton's theory of gravity. Neglect effects of air resistance. Part C.) Consider again the experimental trials from Part A. This time, you wish to test how the size of an object affects the rate of its fall. Which pair of trials should you compare? Check exactly two of the following diagrams. Part D.) If you actually performed and compared the two trials chosen in Part C, you would find that, while the basketball and marble would hit the ground at almost the same time, it would not quite be exact: The basketball would take slightly longer to fall to the ground than the marble. Why? Part E.) Einstein's theory, like Newtons, predicts that, in the absence of air resistance, all objects should fall at the same rate regardless of their masses. Consider the following hypothetical experimental results. Which one would indicate a failure of Einstein's theory?

Part A.) size = bowling ball, mass = 5.0 kg, height = 20 m & size = basketball, mass = 0.5 kg, height = 20 m (The simplest way to test the effects of mass is to compare the results of two trials that are identical except for the mass of the balls. In the language of experimental design, we say that the mass is the "variable of interest" for this experiment, and we therefore hold the other variables (size and height) constant so that they cannot affect the results.) Part B.) Both balls fall to the ground in the same amount of time. (Newton's theory of gravity predicts that, in the absence of air resistance, all objects on Earth should fall with the same acceleration of gravity, regardless of mass. This means that balls dropped from the same height should take the same amount of time to reach the ground.) Part C.) size = marble, mass = 0.5 kg, height = 30 m & size = basketball, mass = 0.5 kg, height = 30 m (The variable of interest is now size, so appropriate trials to compare are those in which size differs but other variables are constant.) Part D.) Because air resistance has a greater effect on the larger ball. (The larger size and lower density of the basketball means it will encounter more air resistance than the marble, so it will take slightly longer to reach the ground.) Part E.) Scientists dropping balls on the Moon find that the balls of different mass fall at slightly different rates. (Dropping the balls on the Moon removes any potential effects due to air resistance, so a result in which mass affects the rate of fall would directly contradict the prediction of Einstein's (as well as Newton's) theory.)

The planets never travel in a straight line as they orbit the Sun. According to Newton's second law of motion, this must mean that ...

a force is acting on the planets (Because the planets are not traveling in straight lines, the planets are always accelerating, and Newton's second law tells us that a force must be acting to cause the acceleration.)

According to the universal law of gravitation, if you triple the distance between two objects, then the gravitational force between them ...

decreases by a factor of 9 (Gravity follows an inverse square law, so the force goes down withe the square of the distance; in this case, increasing the distance by a factor of 3 causes the force to decrease by a factor of 3^2 = 9.)

Compared to their values on Earth, on another planet ...

your mass would be the same but your weight would be different.