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For the next TWO questions: An object is moving in a clockwise direction around a

circle at constant speed. Use your understanding of the concepts of velocity and

acceleration to answer the next four questions. Use the diagram shown at the right.

1. Which vector below represents the direction of the velocity vector when the

object is located at point B on the circle?

2. Which vector below represents the direction of the acceleration vector when the object is

located at point C on the circle?

Whenever an object is accelerated there must be a ……

This is known as centripetal force, Fc. This is not a new force, it is simply the net force that

accelerates an object towards the center of its circular path.

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A 900.0 kg car moving at 10.0 m/s takes a turn

around a circle with a radius of 25.0 m. Determine

the acceleration and the net force acting upon the

car.

The coefficient of friction acting upon a 945-kg car

is 0.850. The car is making a 180-degree turn around

a curve with a radius of 35.0 m. Determine the

maximum speed with which the car can make the

turn.

A plane makes a complete circle with a radius of 3622

m in 2.10 min. What is the speed of the plane?.

Worksheet #1

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Anna Litical is riding on The Demon at Great America. Anna experiences a downward acceleration of 15.6 m/s2 at the

top of the loop and an upward acceleration of 26.3 m/s2 at the bottom of the loop. Use Newton's second law to

determine the normal force acting upon Anna's 864 kg roller coaster car at the top and bottom of the loop.

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There is a wealth of sporting events

where participants lean into the turn

in order to momentarily move in a

circle.

- A downhill skier makes her

turn by leaning into the snow

- A football player makes his

turn by leaning into the

ground

- A cyclist makes his turn in a

similar manner as he leans at

an angle to the horizontal

Bonnie is ice-skating at the Olympic Games. She is making

a sharp turn with a radius of 22.6 m and with a speed of

16.1 m/s. Use Newton's second law to determine the

acceleration and the angle of lean of Bonnie's 55.0-kg

body.

Worksheet #2

b) at the bottom?

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Three objects are placed in a line 5.00 x 10-1 m apart. B and C have a mass of 15.0 kg and A has a mass of 10.0 kg. What i s the net gravitational force on the center object due to the other two objects?

Worksheet #3

mEarth = 5.98 x 1024 kg mMoon = 7.35 x 1022 kg rEarth = 6.37 x 106 m

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Worksheet #4

Example: What is the gravitational field strength on the surface of the Moon? mmoon = 7.35 x 1022 kg

rmoon = 1.74 x 106 m

We will discuss 3 types of fields in

this class

1. Gravitational Fields

2. Electric Fields

3. Magnetic Fields

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Kepler’s Laws of Planetary Motion

Johannes Kepler (1571 – 1630) discovered three laws that describe the behaviour of every

satellite that orbits a central object such as the sun.

i) 1st Law (The Law of Orbits)

States that the paths of the

planets are elliptical with the

central object (such as the Sun)

being located at a focal point.

ii) 2nd Law (The Law of Area’s)

States that when an imaginary line is drawn out from the

focal point (Sun) to the orbiting object, it sweeps out equal

areas (A) during equal time periods. In general, orbiting

objects travel faster when closer to the central objects.

iii) 3rd Law (The Law of Periods – Harmonic Law)

States the ratio of the squares of the periods of any two objects orbiting around the

same object is equal to the ratio of the cubes of the distances away from the central

object.

3

2

1

2

2

1

d

d

T

T where

T1 is the time (period) needed for object #1 to complete one orbit

T2 is the time (period) needed for object #2 to complete one orbit

d1 is the distance object #1 is from the centre of the central object

d2 is the distance object #2 is from the centre of the central object

Circular Motion and Gravitation Notes

6 - Mathematics of Satellite Motion

Orbital Period Equation

The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's

third law.

There is an important concept evident in all three of these equations - the period, speed and the

acceleration of an orbiting satellite are not dependent upon the mass of the satellite.

Orbital Speed Equation The following equation is for the velocity of a satellite moving about a central body in circular motion

where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the satellite orbits, and

R is the radius of orbit for the satellite.

Orbital Speed Equation The following equation is for the velocity of a satellite moving about a central body in circular motion

where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the satellite orbits, and

r is the radius of orbit for the satellite.

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Practice Problems

1. A satellite wishes to orbit the earth at a height of 100.0 km (approximately 60 miles) above the surface of the earth.

Determine the speed, acceleration and orbital period of the satellite.

Worksheet #5

A geosynchronous satellite is a satellite that orbits the earth with an orbital period of 24 hours, thus matching the

period of the earth's rotational motion. A special class of geosynchronous satellites is a geostationary satellite. A

geostationary satellite orbits the earth in 24 hours along an orbital path that is parallel to an imaginary plane drawn

through the Earth's equator. Such a satellite appears permanently fixed above the same location on the Earth. If a

geostationary satellite wishes to orbit the earth in 24 hours (86400 s), then how high above the earth's surface must

it be located?

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Worksheet #1

For the next THREE questions: An object is moving in a clockwise direction around a circle at constant

speed. Use your understanding of the concepts of velocity, acceleration and force to answer the next

five questions. Use the diagram shown at the right.

1) Which vector below represents the direction of the velocity vector when the object is located at point C on the

circle?

2) Which vector below represents the direction of the acceleration vector when the object is located at point A on

the circle?

3) Which vector below represents the direction of the force vector when the object is located at point A on the

circle?

4) Calculate the centripetal force acting on a 925 kg car as it rounds an unbanked curve with a radius of 75 m at a speed of 22 m/s (6.0 x 103 N)

5) A small plane makes a complete circle with a radius of 3282 m in 2.0 min. What is the centripetal acceleration of the plane? (9.0 m/s2)

6) An electron (m = 9.11 x 10-31 kg) moves in a circle whose radius is 2.00 x 10-2 m. If the force acting on the electron is 4.60 x 10-14 N, what is its speed? (3.18 x 107 m/s)

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7) A 925 kg car rounds an unbanked (flat) curve at a speed of 25 m/s. If the radius of the curve is 72 m, what is the minimum coefficient of friction between the car and the road required so that the car does not skid? (0.88)

8) A 2.7 x 103 kg satellite orbits the Earth at a distance of 1.8 x 107 m from the Earth's centre at a speed of 4.7 x 103 m/s. What force does the Earth exert on the satellite? (3300 N)

9) A 932 kg car is traveling around an unbanked turn with a radius of 82 m. What is the maximum speed that this car can round this curve before skidding

a) if the coefficient of friction is 0.95? (28 m/s)

b) if the coefficient of friction is 0.40? (18 m/s) **Note how much slower you SHOULD be going if the coefficient of friction is lower (for example if it’s icy)

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Worksheet #2

1) A 2.2 kg object is whirled in a vertical circle whose radius is 1.0 m. If the time of one revolution is 0.97 s, what is

the tension in the string (assume uniform speed)

a) at the top? (71 N [down]) b) at the bottom? (110 N [up])

2) Determine the centripetal force acting upon a 40.0 kg child who makes 10 revolutions around the

Cliffhanger in 29.3 seconds. The radius of the barrel is 2.90 meters. (Fnet = 533 N [inward])

3) A 1.50-kg bucket of water is tied by a rope and whirled in a circle with a radius of 1.00 m. At the

bottom of the circular loop, the speed of the bucket is 6.00 m/s. Determine the acceleration, the net

force and the individual force values when the bucket is at the bottom of the circular loop.

m = 1.5 kg

a = ________ m/s/s

Fnet = _________ N

[ ]

[ ]

[ ]

[ ]

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Worksheet #3 mEarth = 5.98 x 1024 kg mMoon = 7.35 x 1022 kg rEarth = 6.37 x 106 m

1) Two students are sitting 1.50 m apart. One student has a mass of 70.0 kg and the other has a mass of

52.0 kg. What is the gravitational force between them? (1.08 x 10-7 N)

2) What gravitational force does the moon produce on the Earth if their centers are 3.84 x 108 m apart? (1.99 x 1020 N)

3) If the gravitational force between two objects of equal mass is 2.30 x 10-8 N when the objects are 10.0 m apart what is the mass of each object? (186 kg)

4) Calculate the gravitational force on a 6.50 x 102 kg spacecraft that is 4.15 x 106 m above the surface of the Earth. (15 100 N)

5) The gravitational force between two objects that are 2.1 x 10-1 m apart is 3.2 x 10-6 N. If the mass of one object is 55 kg, what is the mass of the other object? (38 kg)

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mEarth = 5.98 x 1024 kg mMoon = 7.35 x 1022 kg rEarth = 6.37 x 106 m

6) If two objects, each with a mass of 200.0 kg, produce a gravitational force of 3.7 x 10-6 N, what is the distance between them? (0.85 m)

7) What is the gravitational force on a 70.0 kg object standing on the Earth's surface? (688 N)

8) Three 10.0 kg objects are placed in a straight line 5.00 x 10-1 m apart. What is the net gravitational force on the center object due to the other two objects? (0 N) Three objects A, B, and C are placed 0.50 m apart along a straight line. A and B have masses of 10.0 kg and C has a mass of 15.0 kg, what is the net gravitational force on B due to A and C? (1.3 x 10-8 N)

10) Three objects A, B, and C are placed 0.50 m apart along a straight line. A and B have masses of 10.0 kg

and C has a mass of 15.0 kg, what is the net gravitational force on C due to A and B? (5.0 x 10-8 N)

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11) The force of gravity between two small masses A and B when placed very near each other is 3.24 x 10-7 N. What will the force between these objects be if both of their masses are doubled and the distance between them is tripled? (1.44 x 10-7 N)

Worksheet #4 mEarth = 5.98 x 1024 kg mMoon = 7.35 x 1022 kg rEarth = 6.37 x 106 m

1) Calculate the gravitational field strength on the surface of Mars. Mars has a radius of 3.43 x 106 m and a mass of 6.37 x 1023 kg. (3.61 N/kg)

2) At what distance from Earth's surface is the acceleration due to gravity 7.33 m/s2 (7.37 x 106 m)

3) On the surface of Planet X an object has a mass of 22.5 kg and a weighs 63.5 N. What is the gravitational field strength on the surface? (2.82 N/kg)

4) On the surface of Planet Y, which has a mass of 4.83 x 1024 kg, a 30.0 kg object weighs 50.0 N. What is the radius of the planet? (1.39 x 107 m)

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Worksheet #5

1) Use the following information to help determine the orbital velocity and orbital period of a satellite at a location

of 15000 miles (2.41 x 107 m) above the surface of the earth. (3620 m/s; 5.30 x 104 s or 14.7 h)

mEarth = 5.98 x 1024 kg radius of Earth = 6.37 x 106 m 1609 m = 1.00 mi

2. The period of the moon is approximately 27.2 days (2.35 x 106 s). Determine the radius of the moon's orbit and

the orbital speed of the moon.