# Part 1: Gravity & Satellites

**Part 1: Gravity & Satellites**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1) What is the formula for the force of gravity between two bodies of mass M1 and M2?

Comprehension / (2) Explain the relationship between the gravitational force between two objects and the distance between them?

Application / (3) Given that the universal gravitational constant is given by G = 6.67x10-11 Nm2/kg2, calculate the force due to gravity between Object A with a mass of 1 kg and Object B with a mass of 10 kg, when Object A and Object B are 10 m apart.

- 6.67 x 10-12 N
- 6.67 x 10-11 N
- 1.50 x 10+11 N
- 1.50 x 10+12 N

Analysis / (4) Using the estimate that the acceleration due to gravity at the surface of the Earth is 10 m/s2, G = 6.67x10-11 Nm2/kg2, and that the approximate radius of the Earth is 6400 km, estimate the mass of the earth.

- 6.4 x 1018 kg
- 1.0 x 1024 kg
- 6.4 x 1024 kg
- 1.0 x 1018 kg

Synthesis / (5) A body in a circular orbit about the sun experiences a centripetal acceleration of a = v2/R, where v is the orbital speed and r is the radius of the orbit. Using this relationship derive Kepler’s Third Law for the relationship between the period of the orbit, T, and the radius of the orbit, R.

Evaluation / (6) You find yourself in a space suit in orbit around a black hole close enough to experience tides within your body. What is the optimum orientation for your body relative to the orbital path? Explain the reasons behind your choice.

Instructions: Classify each of the following questions as being either: Knowledge, Comprehension, Application, Analysis, Synthesis or Evaluation.

**Part 2: Potential & Kinetic Energy**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1)What is the potential energy of an object sitting on a table relative to the floor, when the table is a distance c from the floor and the acceleration due to gravity is a?

- mc2
- mac
- zero
- ½ mac2

Comprehension / (2)In your own words, describe the relationship between a body’s Potential Energy and its Kinetic Energy.

Application / (3)Given that the acceleration due to gravity is 10 m/s2, what is the potential energy of a bowling ball with a mass of 7 kg on a table 1 meter high?

- 35 N
- 70 J
- 70 W
- 140 Ws

Analysis / (4)The bowling ball in question (3) drops off the table and hits the floor without bouncing or deforming. How much work does the bowling ball do on the floor, and what are the likely forms it would take?

Synthesis / (5)Given that the relationship between the distance a ball falls from a table h meters high and the time, t, it takes to hit the floor is h = ½ g t2, find the kinetic energy of the ball in terms of the ball’s velocity just as it hits the floor. Show your work for full credit.

Evaluation / (6)A particle slides down a frictionless track forming a loop-D-loop as in the figure at right. Evaluate the minimum height h in terms of the diameter D for the particle to maintain contact at the top of the loop.

Instructions: Classify each of the following questions as being either: Knowledge, Comprehension, Application, Analysis, Synthesis or Evaluation.

**Part 3: Heat & Thermodynamics**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1) Define the Ideal efficiency of a heat engine in terms of the hot reservoir temperature Thot and the cold sink Thot.

- Thot / Tcold
- Thot - Tcold
- (Thot - Tcold) / Thot
- (Thot - Tcold) / Tcold

Comprehension / (2) State the first law of thermodynamics and explain its ramifications.

Application / (3) The hot reservoir of a steam turbine runs at 127 C and exhausts to an ambient temperature of 27 C. What is the Ideal efficiency of this turbine engine?

- 78.7%
- 25.0 %
- 100.0%
- 21.3%

Analysis / (4) Two vessels of equal volume are connected by a pipe, which has a shut off valve at its midpoint. Gas is pumped into one of the vessels and allowed to reach thermal equilibrium with its environment. This “initial” condition is described by P1, V1, and T1. The valve is opened. Describe the new conditions P2, V2, and T2 in terms of the “initial” conditions.

Synthesis / (5) Boyle’s Law is stated as P1V1=P2V2. Charles’ Law is stated as V1/T1=V2/T2. Combine these two laws to obtain the General Gas Law. (See the figure to the right.)

Evaluation / (6) A constant volume gas thermometer is shown in the diagram at the right. The flask containing the hydrogen is place in the thermal environment to be measured and the mercury reservoir is raised or lowered until it lines up with the scribe line on the tube. A mercury barometer next to the thermometer reads 76.0 cm. When the flask is placed in ice water (0C) the top of the mercury in the reservoir is 28.5cm above the scribe line. The top of the mercury is 41.4 cm below the scribe line when the bulb is immersed in boiling liquid oxygen. What is the boiling point of oxygen?

**Solutions & Rational for Classification**: Knowledge, Comprehension, Application, Analysis, Synthesis or Evaluation.

**Part 1: Gravity & Satellites**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1) What is the formula for the force of gravity between two bodies of mass M1 and M2?

Rational: This is pure formula memorization.

Solution: F = M1 M2 G / R2, where M1 and M2 are the masses of the two objects, G is the universal gravitational constant, and R is the distance between the masses.

Comprehension / (2) Explain the relationship between the gravitational force between two objects and the distance between them?

Rational: Being able to describe why the force of gravity declines as the square of the distance shows comprehension.

Solution: The force of gravity of gravity extends equally in any radial direction and is equal in magnitude at any given radius. Suppose we have an object a unit distance away and construct a unit square normal to the radial direction and with vertices on the sphere of radius 1. Note that the area of this square is 1. Now extend along the radius for each vertex until the radius is 2. The area between these four vertices is now 4. It is apparent that since the force of gravity must extend equally in all directions, as the distance is doubled the force reduces by a factor of 4, that is the force previously distributed over the square of area 1 is now distributed over a square of area 4. If this is extended as the radius is increased to 3 and then 4, it is apparent that the force decrease with the square of the radius or distance from the object exerting the force.

Application / (3) Given that the universal gravitational constant is given by G = 6.67x10-11 Nm2/kg2, calculate the force due to gravity between Object A with a mass of 1 kg and Object B with a mass of 10 kg, when Object A and Object B are 10 m apart.

- 6.67 x 10-12 N
- 6.67 x 10-11 N
- 1.50 x 10+11 N
- 1.50 x 10+12 N

Solution: Using the formula F = M1 M2 G / R2 and plugging in the values from above we get F = 1 kg *10 kg * 6.67x10-11 Nm2/kg2 / (10m)2 = 6.67x10-12 N.

Analysis / (4) Using the estimate that the acceleration due to gravity at the surface of the Earth is 10 m/s2, G = 6.67x10-11 Nm2/kg2, and that the approximate radius of the Earth is 6400 km, estimate the mass of the earth.

- 6.1 x 1018 kg
- 1.0 x 1024 kg
- 6.1 x 1024 kg
- 1.0 x 1018 kg

Solution: Substituting m for M1 and ME for M2 we get Using the formula F = m a = m ME G / RE2. Solving this equation for the mass of the earth results in ME = a RE2 / G. Plugging in the values from above gives us ME = (10 m/s2) (6.4 x 106 m)2 / 6.67x10-11 Nm2/kg2 = 6.1x1024 kg, i.e. c. The units are correct remembering that 1N=1kgm/s2.

Synthesis / (5) A body in a circular orbit about the sun experiences a centripetal acceleration of a = v2/R, where v is the orbital speed and r is the radius of the orbit. Using this relationship derive Kepler’s Third Law for the relationship between the period of the orbit, T, and the radius of the orbit, R.

Rational: This is an example of synthesis because the student must combine disparate concepts in new ways creating a new relationship.

Solution: Knowing that Kepler’s Third Law involves the planet’s period of revolution T and its average radius R, and knowing that the distance a planet travels in one period is D = 2 π R we write the following equation, D = v T 2 π R = v T or v = 2 π R / T. Plugging this value for the velocity v into the equation supplied and equating the acceleration a to the acceleration due to gravity where MS is the mass of the sun, we get GMS/R2=4π2R/T2. Now both sides multiplying both sides by T2 R2 the equation now becomes GMST2=4π2R3, or T2R3, which is Kepler’s Third Law.

Evaluation / (6) You find yourself in a space suit in orbit around a black hole close enough to experience significant tides within your body. What is the optimum orientation for your body relative to the orbital path? Explain the reasons behind your choice.

Rational: This is an example of evaluation because the student must understand the concept for tides and apply it in a completely new way, thinking about what the affect of these tides on the body might be and deciding which one or ones may be the most important and then come up with a possible solution.

Solution: Tides are caused by the difference between the forces due to gravity on the parts an object separated by some distance. It is not necessary to try to calculate the forces required since the force is stated to be significant. One can imagine tides affecting the circulatory system of our body in space. For example, if we were orbiting the black hole feet first, the blood would tend to pool in our feet and our head due to the tides. If the force were high enough, we could die from lack of oxygen to the brain. To limit this affect, we would want to keep our head, arms, heart, and feet at the same radius during our orbit, sort of like superman flying. This would also provide the minimum tides for all the other objects in our body assuming our navel pointed along the radius to the black hole.

**Solutions & Rational for Classification**: Knowledge, Comprehension, Application, Analysis, Synthesis or Evaluation.

**Part 2: Potential & Kinetic Energy**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1)What is the potential energy of an object sitting on a table relative to the floor, when the table is a distance c from the floor and the acceleration due to gravity is a?

- m c2
- m a c
- zero
- ½ m a c2

Solution: You have a for the acceleration instead of the normal g and c instead of h, but this is still just memorization and PE = m g h or m a c.

Comprehension / (2)In your own words, describe the relationship between a body’s Potential Energy and its Kinetic Energy.

Rational: Since this requires a rewording of the relationship in the student’s own words this is comprehension.

Solution: At any given point in time the sum of the potential energy of a body and its kinetic energy is a constant that does not change. If the potential energy is decreasing, it must be transforming into kinetic energy under the law of conservation of energy and thus the kinetic energy is increasing.

Application / (3)Given that the acceleration due to gravity is 10 m/s2, what is the potential energy of a bowling ball with a mass of 7 kg on a table 1 meter high?

- 35 N
- 70 J
- 70 W
- 140 Ws

Solution: PE = m g h = 7 kg * 10 m/s2 * 1 m = 70 kg m2 / s2 = 70 N m = 70J (or 70 W s since 1 W = 1 J /s.)

Analysis / (4)The bowling ball in question (3) drops off the table and hits the floor without bouncing or deforming. How much work does the bowling ball do on the floor, and what are the likely forms it would take?

Rational: This is an example of analysis because the student must breakdown the problem into its parts before solving the problem.

Solution: Since the ball does not bounce or deform, all of its potential energy is converted to work. So the amount of work done by the ball on the floor is 70 J the same as its potential energy. The forms that the work is most likely to take is cracking or deforming the floor and raising the temperature of the floor under the ball.

Synthesis / (5)Given that the relationship between the distance a ball falls from a table h meters high and the time, t, it takes to hit the floor is h = ½ g t2, find the kinetic energy of the ball in terms of the ball’s velocity just as it hits the floor. Show your work for full credit.

Rational: This is an example of synthesis because the student is deriving the

Equation for kinetic energy in a novel manner.

Solution: Just before the ball hits the floor, its kinetic energy will be just equal to the potential energy, m g h and thus KE = m g h. Substituting the provided equation we obtain KE = ½ m g2 t2. Recognizing that v = a t or here v = g t and v2 = g2 t2, and substituting we get the standard equation for the kinetic energy of a body in terms of its mass and velocity KE = ½ m v2.

Evaluation / (6)A particle slides down a frictionless track forming a loop-D-loop as in the figure at right. Evaluate the minimum starting height, h, in terms of the diameter, D, of the loop for the particle to maintain contact with the track at the top of the loop.

Rational: This is an example of evaluation since the student must choose which relationships to utilize, how they apply to the problem, and where to apply them.

Solution: The critical point is when the particle is at the highest point of the loop to loop. The Kinetic Energy at this point is equal to the change in Potential Energy so KE = ½ m v2 = m g (h – D). In addition we know that if the particle is just in contact with the loop at its apex the centripetal acceleration must be equal to the acceleration due to gravity or a=v2/(D/2)= 2 v2/D = g. Rearranging this equation we get v2= g D / 2. Substituting this into the energy equation above we get ¼mgD=mghmgD or h = 1¼ D. This results in a non intuitive answer since one might expect that the particle will return to the height from which it was released and answer h = D.

**Solutions & Rational for Classification**: Knowledge, Comprehension, Application, Analysis, Synthesis or Evaluation.

**Part 3: Heat & Thermodynamics**

(Senior Year Physics)

Questions by Dave Nelson (Class of 2005)

Type(s) of Reasoning / QuestionKnowledge / (1) Define the Ideal efficiency of a heat engine in terms of the hot reservoir temperature Thot and the cold sink Thot.

- Thot / Tcold
- Thot - Tcold
- (Thot - Tcold) / Thot
- (Thot - Tcold) / Tcold

Solution: (Thot - Tcold) / Thot, c.

Comprehension / (2) State the first law of thermodynamics and explain its ramifications.

Rational: This requires comprehension since the law is stated in their own words and the ramifications or interpretations of this law require a basic understanding of the law.

Solution: The first law of thermodynamics can be stated as follows: the heat added to a system is equal to the increase in internal energy of the system plus the work done by the system. This is another way of stating the conservation of energy (energy cannot be created or destroyed.)

Application / (3) The hot reservoir of a steam turbine runs at 127 C and exhausts to an ambient temperature of 27 C. What is the Ideal efficiency of this turbine engine?

- 78.7%
- 25.0 %
- 100.0%
- 21.3%

Solution: To apply the Ideal efficiency equation, (Thot - Tcold) / Thot, first convert the temperatures from Centigrade to Kelvin. Thus Thot=127+273K=400K and Thot=27+273K=300K. Plugging these into the equation gives Efficiency = (400 – 300)/400 = ¼ or changing to percentage, 25%, b.

Analysis / (4) Two vessels of equal volume are connected by a pipe, which has a shut off valve at its midpoint. Gas is pumped into one of the vessels and allowed to reach thermal equilibrium with its environment. This “initial” condition is described by P1, V1, and T1. The valve is opened. Describe the new conditions P2, V2, and T2 in terms of the “initial” conditions.

Rational: The student needs to break down the problem into its components so to understand which of the gas laws is appropriate before applying it.

Solution: Since the system is in thermal equilibrium, the temperature does not change and so T2 = T1. This is the condition under which Boyle’s Law applies, P1V1=P2V2. Since the volume is doubled when the valve is opened, V2 = 2 V1 and by Boyle’s law P2 = ½ P1.

Synthesis / (5) Boyle’s Law is stated as P1V1=P2V2. Charles’ Law is stated as V1/T1=V2/T2. Combine these two laws to obtain the General Gas Law. (See the figure to the right.)

Rational: This is an example of synthesis because the student is deriving the

Equation for the general gas law applying the other two laws one at a time sequentially.

Solution: We first apply Boyle’s Law along the T1 isotherm resulting in P1V1=P2Va. Then we apply Charles’s Law from the T1 isotherm to the T2 isotherm horizontally (constant pressure). This results in Va/T1=V2/T2. Solving the second equation for Va and plugging it into the first equation results in P1V1/T1 = P2V2/T2.

Evaluation / (6) A constant volume gas thermometer is shown in the diagram at the right. The flask containing the hydrogen is place in the thermal environment to be measured and the mercury reservoir is raised or lowered until it lines up with the scribe line on the tube. A mercury barometer next to the thermometer reads 76.0 cm. When the flask is placed in ice water (0C) the top of the mercury in the reservoir is 28.5cm above the scribe line. The top of the mercury is 41.4 cm below the scribe line when the bulb is immersed in boiling liquid oxygen. What is the boiling point of oxygen?

Rational:

Solution: With this type of thermometer, as its name implies, the volume of the gas is kept at a constant. This is a very unusual thermometer in that it requires only one temperature for calibration if the temperature is referred to absolute zero; i.e. need to use the Kelvin scale. From the information in the question we know that at 0 C or 273 K, the pressure in the thermometer is 76.0 + 28.5 cm = 104.5 cm of mercury (Interestingly for this problem we can leave out the conversion of the weight of the mercury to pressure and just keep it in cm. You can prove this to yourself if you want using the equation for pressure, g h.) The gas equation P V = n R T, when V is a constant, becomes T = Constant * P. Thus with T = 273 K and P = 104.5 cm of mercury, Constant = 2.6. Now applying this to the case of the boiling oxygen where with the mercury 41.4 cm below the scribe line the pressure is just P = 76 – 41.4 cm = 34.6 cm and plugging it into our equation we get T = 2.6 * 34.6 = 90 K, or -183 C.

Assignment 1 – p. 1