APY1 Review

Gravitation

BIG IDEA 1: Objects and systems have properties such as mass and charge. Systems may have internal structure.
1.C.3.1: The student is able to design a plan for collecting data to measure gravitational mass and to measure inertial mass and to distinguish between the two experiments. [SP 4.2]
BIG IDEA 2: Fields existing in space can be used to explain interactions.
2.B.1.1: The student is able to apply to calculate the gravitational force on an object with mass m in a gravitational field of strength g in the context of the effects of a net force on objects and systems. [SP 2.2, 7.2]
2.B.2.1: The student is able to apply to calculate the gravitational field due to an object with mass M, where the field is a vector directed toward the center of the object of mass M. [SP 2.2]
2.B.2.2: The student is able to approximate a numerical value of the gravitational field (g) near the surface of an object from its radius and mass relative to those of the Earth or other reference objects. [SP 2.2]
BIG IDEA 3: The interactions of an object with other objects can be described by forces.
3.A.2.1: The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. [SP 1.1]
3.A.3.1: The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces. [SP 6.4, 7.2]
3.A.3.3: The student is able to describe a force as an interaction between two objects and identify both objects for any force. [SP 1.4]
3.A.4.1: The student is able to construct explanations of physical situations involving the interaction of bodies using Newton’s third law and the representation of action-reaction pairs of forces. [SP 1.4, 6.2]
3.A.4.2: The student is able to use Newton’s third law to make claims and predictions about the action-reaction pairs of forces when two objects interact. [SP 6.4, 7.2]
3.A.4.3: The student is able to analyze situations involving interactions among several objects by using free-body diagrams that include the application of Newton’s third law to identify forces. [SP 1.4]
3.B.1.2: The student is able to design a plan to collect and analyze data for motion (static, constant, or accelerating) from force measurements and carry out an analysis to determine the relationship between the net force and the vector sum of the individual forces. [SP 4.2, 5.1]
3.B.1.3: The student is able to reexpress a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. [SP 1.5, 2.2]
3.B.2.1: The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. [SP 1.1, 1.4, 2.2]
3.C.1.1: The student is able to use Newton’s law of gravitation to calculate the gravitational force the two objects exert on each other and use that force in contexts other than orbital motion. [SP 2.2]
3.C.1.2: The student is able to use Newton’s law of gravitation to calculate the gravitational force between two objects and use that force in contexts involving orbital motion [SP 2.2]
3.C.2.2: The student is able to connect the concepts of gravitational force and electric force to compare similarities and differences between the forces. [SP 7.2]
3.G.1.1: The student is able to articulate situations when the gravitational force is the dominant force and when the electromagnetic, weak, and strong forces can be ignored. [SP 7.1]
BIG IDEA 4: Interactions between systems can result in changes in those systems.
4.A.2.2: The student is able to evaluate using given data whether all the forces on a system or whether all the parts of a system have been identified. [SP 5.3]

Inertial mass vs. Gravitational Mass

  • Inertial mass. This is mainly defined by Newton's law, the all-too-famous F = ma, which states that when a force F is applied to an object, it will accelerate proportionally, and that constant of proportion is the mass of that object. In very concrete terms, to determine the inertial mass, you apply a force of F Newtons to an object, measure the acceleration in m/s2, and F/a will give you the inertial mass m in kilograms.
  • Gravitational mass. This is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by Newton’ Law of attraction.
  • As it turns out, these two masses are equal to each other as far as we can measure. Also, the equivalence of these two masses is why all objects fall at the same rate on earth

Newton’s Law of Attraction

  • Any two objects will exert an attract force between each other.
  • This is called Newton’s Law of Gravity (Fa)

G is the constant 6.67 x 10-11 N/kg2m2

  • This is called the inverse square law—as the masses are separated the force between them weakens.
  • G is called the universal gravitational constant. Since this number is so small, the gravitational force is inherently weak.

g vs. G

  • The acceleration due to gravity (g) near the Earth’s surface is -9.81 m/s2. This is not a constant. It will change as we go to other planets.
  • G on the other hand is a constant—it is the same wherever you go.
  • The two values are related---if an object is sitting on a planet there is a force of attraction between the person and the planet

m2 is the mass of the planet

r is the radius of the planet

Gravitational Potential Energy

  • The gravitational potential energy between two object is equal to

Satellites

  • Gravitational force can supply the centripetal forces needed to cause an object to move in a circular orbit.

Fc = mv2/R

(known as orbital speed)

  • The speed is independent of the orbiting mass---it just depends on the mass of the planet and the distance between the two.
  • Orbits of satellites and planets can be elliptical.

Escape Speed

  • Sometime you want the speed necessary for a satellite to escape the gravitational orbit.
  • In this case, we will use the conservation of energy.
  • We will compare the point when the satellite is on the planet ready to launch and the point when the satellite is at infinity.
  • When the satellite is at infinity, the gravitational potential energy is zero (look at the equation) and theoretically the satellite comes to a stop.

At Earth At Infinity

KE + PE = KE + PE

½ msv2 + = 0

½ msv2 =

Angular Momentum (L)

  • Angular Momentum is defined to be L = mrvt where vt is the tangential component of velocity.

L = mrvt

  • As a satellite moves in a circular orbit, then the satellite’s angular momentum is conserved.

mrv = mrv

The object will move faster when it is closer to the focus.

Multiple Choice

Identify the letter of the choice that best completes the statement or answers the question.

____1.The mass of Planet X is one-tenth that of the Earth, and its diameter is one-half that of the Earth. The acceleration due to gravity at the surface of Planet X is most nearly (1984 M 20)

a. / 2m/s2 / b. / 4m/s2 / c. / 5 m/s2 / d. / 7 m/s2 / e. / 10 m/s2

____2. A satellite travels around the Sun in an elliptical orbit as shown above. As the satellite travels from point X to point Y. which of the following is true about its speed and angular momentum? (1988 M 20)

SpeedAngular Momentum

a. / Remains constant Remains constant
b. / Increases Increases
c. / Decreases Decreases
d. / Increases Remains constant
e. / DecreasesRemains constant

____3.A newly discovered planet, "Cosmo," has a mass that is 4 times the mass of the Earth. The radius of the Earth is Re. The gravitational field strength at the surface of Cosmo is equal to that at the surface of the Earth if the radius of Cosmo is equal to (1988 M 21)

a. / ½Re / b. / Re / c. / 2Re / d. /

____4.Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively, as shown above. If R2 = 2R1, the accelerations a2 and a1 of the two satellites are related by which of the following? (1988 M 22)

a. / a2 = 4a1 / b. / a2 = 2a1 / c. / a2 = a1 / d. / a2 = a1/2 / e. / a2 = a1/4

____5.The radius of the Earth is approximately 6,000 kilometers. The acceleration of an astronaut in a perfectly circular orbit 300 kilometers above the Earth would be most nearly (1988 M 28)

a. / 0 m/s2 / b. / 0.05 m/s2 / c. / 5 m/s2 / d. / 9 m/s2 / e. / 11 m/s2

____6.A satellite moves in a stable circular orbit with speed vo at a distance R from the center of a planet. For this satellite to move in a stable circular orbit a distance 2R from the center of the planet, the speed of the satellite must be (1988 M 35)

a. / / b. / / c. / vo / d. / / e. / 2vo

____7.A ball that is tossed straight up from the surface of a small, spherical asteroid with no atmosphere. The ball rises to a height equal to the asteroid's radius and then falls straight down toward the surface of the asteroid. What forces, if any, act on the ball while it is on the way up? (1998 M7)

a. / Only a decreasing gravitational force that acts downward
b. / Only an increasing gravitational force that acts downward
c. / Only a constant gravitational force that acts downward
d. / Both a constant gravitational force that acts downward and a decreasing force that acts upward
e. / No forces act on the ball.

____8. A ball that is tossed straight up from the surface of a small, spherical asteroid with no atmosphere. The ball rises to a height equal to the asteroid's radius and then falls straight down toward the surface of the asteroid. The acceleration of the ball at the top of its path is (1998 M 8)

a. / at its maximum value for the ball's flight
b. / equal to the acceleration at the surface of the asteroid
c. / equal to one-half the acceleration at the surface of the asteroid
d. / equal to one-fourth the acceleration at the surface of the asteroid
e. / zero

____9.A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. The acceleration due to gravity on the other planet is most nearly (1998 M 10)

a. / g/4 / b. / g/2 / c. / g / d. / 2g / e. / 4g

____10.A satellite of mass M moves in a circular orbit of radius R with constant speed v. True statements about this satellite include which of the following? (1998 M 11)

I. Its angular speed is v/R.

II. Its tangential acceleration is zero.

III. The magnitude of its centripetal acceleration is constant.

a. / I only / b. / II only / c. / I and III only / d. / II and III only / e. / I, II, and III

Experimental Question

2005 M2

A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass MS of Saturn. Assume the orbits of these moons are circular.

(a)Write an algebraic expression for the gravitational force between Saturn and one of its moons.

(b)Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbital period T of a moon as a function of its orbital radius R.

(c)Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn’s mass?

(d)Complete the data table by calculating the two quantities to be graphed. Label the top of each column, including units.

(e)Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate the scale.

(f)Using the graph, calculate a value for the mass of Saturn.

Short Answer

Marty is an astronaut who is preparing to go on a mission in orbit around the Earth. For health reasons, his mass needs to be determined before take-off and while he is in orbit. The morning of the launch, Marty sits on one pan of a two-pan scale and 94 kg of mass is needed to balance him.

(a)State and explain whether the two-pan scale registered Marty’s gravitational mass or inertial mass.

(b)After a few days in orbit Marty is again to determine his mass. Explain why the two-pan scale used before launch cannot be used to measure his mass while in orbit.

(c)To determine Marty’s mass in orbit he is to sit in a chair of negligible mass that is attached to a wall by a spring that has a force constant, k. Consequently, the chair freely vibrates back and forth with a period, T when displaced sideways a distance, x. Explain how the spring-mounted chair can be used to determine Marty’s mass, m. Give relevant measurements and equation(s).

(d)If Marty has lost mass while in orbit, what specific change would occur when he sits in the chair and starts it oscillating?

(e)Explain why this spring-mounted chair measures Marty’s inertial mass.

Qualitative/Quantitative

1992 Mech 3

A spacecraft of mass 1,000 kilograms is in an elliptical orbit about the Earth, as shown above. At point A the spacecraft is at a distance rA = 1.2 x 107meters from the center of the Earth and its velocity, of magnitude VA = 7.1 x 103 meters per second, is perpendicular to the line connecting the center of the Earth to the spacecraft. The mass and radius of the Earth are ME = 6.0 X 1024kilograms and rE =6.4 X 106meters, respectively.

Determine each of the following for the spacecraft when it is at point A .

a.The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is zero at an infinite distance from the Earth.

b.The magnitude of the angular momentum of the spacecraft about the center of the Earth.

Later the spacecraft is at point B on the exact opposite side of the orbit at a distance rB =3.6 X 107meters from the center of the Earth.

c.Determine the speed vBof the spacecraft at point B.

Suppose that a different spacecraft is at point A, a distance rA =1.2 X 107meters from the center of the Earth. Determine each of the following.

d.The speed of the spacecraft if it is in a circular orbit around the Earth

e.The minimum speed of the spacecraft at point A if it is to escape completely from the Earth

Paragraph

Your fiend says that an object weights less on Jupiter than on Earth as Jupiter is far away from the center of Earth. Do you

______agree ______disagree

Write a full paragraph whether you agree or disagree with statement above. Be sure include the physics concepts involved in this process.

APC Gravitation

Answer Section

MULTIPLE CHOICE

1.ANS:B

2.ANS:D

3.ANS:C

4.ANS:E

5.ANS:D

6.ANS:B

7.ANS:A

8.ANS:D

9.ANS:A

10.ANS:E

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