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c) Calculate its angular frequency.
b) Calculate its frequency.
Direction of instantaneous velocity:
Instantaneous Velocity (tangential velocity):
Formula:
Symbol:
Units:
Period:
Cycle:
Uniform Circular Motion:
Circular Motion and Gravity
Units:
Symbol:
Units:
Angular Displacement:
Angular Frequency: Symbol: Formula:
Angular Speed/Velocity:
(Average Speed/Velocity)
Formula:
Frequency:
Symbol: Formula:
Units:
Relationship between period and frequency:
1. A pendulum completes 10 swings in 8.0 seconds.
a) Calculate its period.
Centripetal Force and Acceleration
Centripetal Acceleration:
Symbol: Unit: Formula:
Centripetal Force:
Symbol: Unit: Formula:
1. In each case, what is the direction of the force causing the circular motion?
2. Are these objects accelerating, even if they are moving at a constant speed?
3. In which direction are they accelerating?
d) motorcycle stunt
c) car rounding a curve
b) satellite circling Earth
a) rubber stopper on a string
Draw and label the force causing the circular motion in each case below.
d) What is causing this force?
c) the centripetal force acting on the cart
b) the centripetal acceleration of the cart
a) the time it takes to go around the track once
2. A 1.5 kilogram cart moves on a circular track of 1.3 meter radius at a constant speed of 2.0 meters per second.
Determine the following:
How is the acceleration of a car related to the forces you feel as a rider in the car?
Centrifugal force:
d) the tension in the string
c) the acceleration of the plane
b) the speed of the plane
1. A boy flies a 0.750-kg motorized plane on a 2.3 m string in a circular path. The plane goes around 8.0 times in 12.0 seconds. Determine the following:
NOTE: The phrase “centripetal force” does not denote a new and separate force created by nature. The phrase is merely another name for the net force pointing toward the center of the circular path. The centripetal force always has another name (Fg, FN, FT, Ff, etc.)
3. What is the relationship between centripetal force and radius?
2. What is the relationship between centripetal force and speed?
Graphical Relationships
f) What would happen to the tension in the string if the mass is doubled and the speed is halved?
d) the mass of the stopper is doubled?
e) the speed is doubled and the string’s length is halved?
c) the string’s length is tripled?
b) the speed is halved?
a) the speed is doubled?
1. A student swings a rubber stopper around on a string at a constant speed with a centripetal acceleration of 4.0 m/s2, as shown.
What would happen to the acceleration if:
Proportional Reasoning
b) If the pavement is dry asphalt, will the car be able to safely turn?
3. A 2000. kg car attempts to turn a corner going at a speed of 25 m/s. The radius of the turn is 15 meters.
a) How much friction is needed to negotiate this turn successfully?
4. What is the relationship between centripetal force and mass?
Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and that is inversely proportional to the square of the distance between them.
Formula
Law 3: The square of the orbital period of any planet is proportional to the cube of its average orbital radius.
Law 2: An imaginary line joining any planet to the Sun sweeps out equal areas in equal time intervals.
Law 1: All planets orbit the Sun in elliptical paths with the Sun at one focus.
Kepler’s Three Laws of Planetary Motion
Universal Gravitation
Newton’s Law of Universal Gravitation
Earth
Mean radius = 6.37 x 106 m
Two approximations used in deriving the law:
Sun
Mean radius = 6.96 x 108 m
Mean Earth-Sun distance = 1.50 x 1011 m
1. Masses are considered to be point masses.
Point mass:
2. The force between two spherical masses whose separation is large compared to their radii is the same as if the two spheres were point masses with their masses concentrated at the centers of the spheres.
What is the resultant gravitational force on the Earth from the Sun and Moon, as shown below?
Sun
Mass = 1.99 x 1030 kg
Moon
Mass = 7.36 x 1022 kg
Earth
Mass = 5.98 x 1024 kg
Average Earth-Moon distance = 3.84 x 108 m
Average Earth-Sun distance = 1.50 x 1011 m
Application – “weighing the Sun”
Derivation
What provides the centripetal force for orbital motion?
Newton’s Derivation of Kepler’s Third Law
Gravitational Constant :
Formula
Extended spherical body
Point mass
Re 2Re 3Re 4Re
“g” ratio
2. What is the gravitational field strength at an altitude equal to the radius of Earth?
1. What is the gravitational field strength of the Earth at its surface?
“g” at the surface of the Earth
Extended spherical body
Re 2Re 3Re 4Re
Deriving formula for gravitational field strength at the surface of a planet
Point mass
Symbol:
Formula:
Units:
Type:
Deriving formula for gravitational field strength at any point above the surface of a planet
Gravitational field strength at a point in a gravitational field:
Gravitational Field Strength
b) What is the resultant gravitational force acting on a 1500. kg space probe at this location?
Earth
Mass = 5.98 x 1024 kg
Average Earth-Moon distance = 3.84 x 108 m
Moon
Mass = 7.36 x 1022 kg
4. a) Is there a point where the resultant gravitational field strength of the Earth and Moon is zero? If so, where?
b) What is the resultant gravitational force acting on a 1500. kg space probe at this location?
3. a) What is the resultant gravitational field strength at a point midway between the Earth and Moon?
Moon
Mass = 7.36 x 1022 kg
Average Earth-Moon distance = 3.84 x 108 m
Earth
Mass = 5.98 x 1024 kg
Derivation of gravitational potential energy formula
EP = -100 J
EP = 0
EP = -400 J
Gravitational Potential Energy of a mass at a point in a gravitational field:
Base level:
Old formula for gravitational potential energy:
This difference is path independent.
1. Same ”EP between any two points no matter what in a gravitational field:
Base level:
“Old” formula for gravitational potential energy:
This difference is path independent.
1. Same ΔEP between any two points no matter what path is taken between them.
2. Work done in moving a mass between two points in a gravitational field is independent of the path taken.
3. ΔEP is zero between any two points at the same level no matter what path is taken.
4. ΔEP is zero for any closed path (a path that begins and ends at same point).
Difference in gravitational potential energy between any two points in a gravitational field:
Gravitational Potential Energy
Gravitational potential at a point in a gravitational field:
Gravitational Potential
Formulas:
Difference in gravitational potential:
Symbol:
Units:
Gravitational Potential
vs. distance
c) How much does the potential energy of the satellite increase when it is put into this orbit?
Symbol:
Units:
Type:
What is the gravitational potential energy of a 5000 kg satellite:
b) orbiting the Earth at an altitude of 200 km?
a) on the surface of the Earth?
Type:
Gravitational potential at a point:
Potential energy vs. distance:
Formula:
3. Derive an expression for the gravitational potential at the surface of a planet in terms of the gravitational field strength.
2. What is the net gravitational potential at a spot midway between the Earth and the Sun?
Earth
Sun
c) What is the change in potential in moving from the surface to this new location?
d) What is the minimum amount of energy needed to lift a 5000 kg satellite to this location?
b) At a location three Earth radii from the center of the Earth?
a) at the surface of the Earth?
1. What is the gravitational potential due to the Earth’s gravitational field:
Note:
1.
2.
3.
Derivation:
1. What is the escape speed for Earth?
2. If the Earth became a black hole, how large would it be?
Assumptions:
Travel to infinity
Just make it – means velocity is zero at infinity – means EK is zero at infinity as well as EP
Escape Speed:
Planet
Escape Speed
Artificial Satellites
Natural Satellites
“Weightlessness”
Orbital motion
Deep space
Free fall
3. What would happen to a satellite if it encountered appreciable air resistance?
2. What happens to the required orbital speed if:
a) the mass of the satellite increases?
b) the satellite is boosted into a higher orbit?
1. Compare the motion of satellites A and B.
Orbital speed of a satellite:
Period of a satellite:
Acceleration of a satellite:
Satellite Motion
Total Energy
Kinetic Energy
e) What is the minimum amount of energy needed to lift the satellite from the surface of the Earth to this altitude?
d) What is the orbital speed of the satellite?
A 1500 kg satellite is to be put into orbit around the Earth at an altitude of 200 km.
b) How much kinetic energy will the satellite need to orbit at this altitude?
a) How much potential energy will the satellite have at this altitude?
c) What is the total amount of energy the satellite has at this altitude?
Energy Derivations
Comparisons:
RE
Graphs of the energies of an orbiting satellite
Gravitational Potential Energy
Kinetic Energy
Total Energy
Gravitational Potential Energy
Compare the energies of the two orbiting satellites.
Energy of Orbiting Satellites
two point masses
1. The gravitational force does no work as a mass moves on along equipotential surface.
2. The work done in moving a mass between equipotential surfaces is path independent.
3. The work done in moving a mass along a closed path is zero.
one point mass
Equipotential surface:
Comparisons
What is the average gravitational field strength between equipotential surfaces A and B if they are 5.0 m apart?
-80 J/kg
-70 J/kg
A
B
On the diagram at right:
a) Sketch the gravitational field around the point mass.
b) Sketch equipotential surfaces around the point mass.
Formula
gravitational potential gradient:
gradient:
Gravitational Potential Gradient
What is the relationship between the gravitational field and the equipotential surfaces?
2. a) Calculate the strength of the gravitational field of the Sun at a location one million kilometers from the Sun’s center.
b) Explain why the net gravitational field strength at the surface of the Earth can be approximated as due solely to the Earth’s gravitational field.
3. a) Calculate the strength of the Sun’s gravitational field at the surface of the Earth.
b) What is the Sun’s gravitational force at this point?
b) Compare this to the gravitational force that the Earth exerts on the Sun.
1. a) Calculate the gravitational force the Sun exerts on the Earth.
Practice Questions
d) How much gravitational potential energy does the satellite have at this altitude?
e) What is the minimum energy needed to lift the satellite to this altitude?
c) What is the gravitational potential at orbiting altitude?
b) How much gravitational potential energy does the satellite have on the surface of Mars?
a) What is the gravitational potential at the surface of Mars?
Mass of Mars: 6.42 x 1023 kg
Mean planetary radius: 3.37 x 106 m
Mars
5. A 5000kg satellite orbits Mars at a distance of 1000 km.
b) Compare the contributions from the Sun and the Earth to this resultant field.
c) What is the gravitational force acting on a 5000 kg space probe at this location?
4. a) Calculate the resultant gravitational field at a spot midway between the Earth and Sun.
g) What is the minimum amount of energy needed to lift the satellite into this orbit?
f) Calculate the total energy of the satellite.
e) Calculate the kinetic energy of the satellite.
d) Calculate the gravitational potential energy of the satellite.
c) Calculate the orbital period of the satellite.
b) Calculate the satellite’s orbital speed.
a) If the altitude is sufficiently low, what is the approximate radius of the satellite’s orbit?
6. A 5000. kg satellite is placed in a low altitude orbit.
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