ChemPhys 173/273

Unit 9: Circular Motion and Gravitation

Problem Set C

 

Overview:

Problem Set C targets your ability to use mathematical formulas associated with circular and satellite motion to analyze the motion of objects and satellites moving in circles. In a sense, the problems of this set serve as a review of the mathematical principles practiced in Sets A and B. The only new mathematical principles introduced in this set pertain to Kepler's third law of planetary motion. A summary of this law and the other main ideas are presented below. At the end of the discussion, there is a table which lists all relevant formulas from the unit.

 

 

 

 

 

 

Summary of Mathematical Formulas

One difficulty a student may encounter in this problem set is the confusion as to which formula to use. The table below provides a useful summary of the formulas pertaining to circular motion and satellite motion. In the table, many of the formulas were derived from other equations. Thus, there will often be more than one means of determining an unknown quantity. In approaching these problems, it is suggested that you practice the usual habits of an effective problem-solver; identify known and unknown quantities in the form of the symbols of physics formulas, plot out a strategy for using the knowns to solve for the unknown, and then finally perform the necessary algebraic steps required for the solution.

 

To calculate ...

... use the equation(s):

Speed

(v)

v = 2 • pi • R / T

v = SQRT (G • Mcentral / R) for satellites only

Acceleration

(a)

a = v2 / R or a = Fnet / m

a = g = G • Mcentral / d2 for satellites only

Net force

(Fnet)

Fnet = m • aorFnet = m • v2 / R

Fnet = Fgrav = G • msat • Mcentral / d2 for satellites only

Period

(T)

T = 2 • pi • R / v

T2 = 4 • pi2 / (G • Mcentral ) • R3 for satellites only

 

Additional Readings/Study Aids:

The following pages from The Physics Classroom tutorial may serve to be useful in understanding the conceptual and mathematical nature of circular and satellite motion.

Forces | Mass and Weight | Drawing Free-Body Diagrams

Speed and Velocity | Centripetal Acceleration | Mathematics of Circular Motion

Law of Universal Gravitation | Value of g | Kepler's Three Laws | Mathematics of Satellite Motion

 

 

View Sample Problem Set.

 

Problem

Description

Audio Link
1

Use of Kepler's third law as a guide to thinking in order to determine orbital period of a planet's moon.

2

Determine the mass of a planet based upon the orbital radius and period of one of its planets.

3

Use of a force analysis to determine the minimum coefficient of friction for a car making a horizontal turn at a known speed and radius of curvature.

4

Straightforward calculation of tangential velocity for an object with a given rotational frequency and radius.

5

Referring to the previous problem; determine the maximum tangential velocity of a ball on a string if given the maximum tension, mass and radius of the circle.

6

Use of gravitation equations to determine the orbital speed of a satellite about Earth if given its altitude above Earth's surface.

7

Referring to the previous problem; determination of the orbital period.

8

Use of a force analysis to determine the normal force on a car at the top of a bridge with a circular arc if the mass, radius and speed are known.

9

Referring to the previous problem; determine the speed at which the normal force will become 0 N for this same car at the same location.

10

Routine calculation of the mass of a ball which is attached to a string and moving in a horizontal circle; tension, radius, and period are given.

11

Use of a force analysis to determine the coefficient of friction needed to keep a penny of known mass from sliding off a turntable at a known distance from its center and a known speed.

12

Use of Kepler's third law to determine the orbital period of a fictional planet with a known orbital radius using the period and radius of a second fictional planet.

13

Use of a force analysis to determine the scale reading expected on an elevator ride for a specific acceleration.

14

Use of a force analysis to determine the minimum speed which a roller coaster car could have without falling out at the top of the loop.

15

Use of a force and vector analysis to determine the speed of a toy airplane of known mass which is suspended by a string and flies in a circle of known radius at a known angle below the horizontal.

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