Uniform Circular Motion
How does a roller coaster do a loop without falling off the track?
How is water removed from a clothes washer?
Does the velocity of an object change when it is moving in a circle?
Is the object accelerating when it is moving in a circle?
Uniform Circular Motion
V- tangential velocity
A- acceleration
What is the direction?
Uniform Circular Motion
v=v2-v1
a = v/t
Uniform Circular Motion
Centripetal acceleration (ac )
“Center-seeking” acceleration
What force causes an object to have centripetal acceleration?
Centripetal force - net force necessary to cause centripetal acceleration.
Dynamics of Uniform Circular Motion
Centripetal force is directed toward the center of the circle.
Centripetal vs. Centrifugal Force
If centripetal means “center seeking”, what does centrifugal mean?
Does centrifugal force really exist?
Centripetal vs. Centrifugal Force
If there was a centrifugal force, what direction would the ball travel when the string broke?
If there was a centrifugal force, the ball would fly outward
Frames of Reference Video
How does a roller coaster do a loop without falling off the track?
How is water removed from a clothes washer?
What agent exerts the centripetal force that keeps a roller coaster from going in a straight line?
What is the force called?
What agent exerts the centripetal force that keeps a the clothes from going outside the washer bucket?
What is the force called?
What agent exerts the centripetal force that keeps a car going around a bend in the road?
What is the force called?
Free body diagram of a car going around a curve
What agent exerts the centripetal force on the moon?
What is the force called?
Think about it!
Does the earth do work on the moon? Explain.
Centripetal acceleration
rvaC
2
Practice Problems 3-5
rmvF c
2
Circumference = 2 π r
Which pulls harder, the earth on the moon, or the moon on the earth?
Think about it!
Newton wondered “Why does an apple fall and the moon doesn’t?”
Newton’s Law of Universal Gravitation
Because the moon has tangential velocity, so it stays up.
Newton’s Law of Universal Gravitation
But, the moon is held in orbit by a centripetal force, which he called “gravity”.
As the mass of an object increases, the force of gravity _______________.
Fg mass
What do you know about gravity?
What do you know about gravity?
As the distance from the earth decreases, the force of gravity (Fg) _______________.
Fg 1/d
Newton’s Law of Universal GravitationIf Fg decreases as you increase distance from the earth’s center, how does g change?
Newton was able to calculate the acceleration of the moon toward the earth. How?He knew the orbital period (T) of the moon
Newton’s Law of Universal Gravitation
T
rrT
ac 2
2 24)/2(
Newton calculated that the centripetal acceleration of the moon toward the earth was 1/3600 the acceleration of an object at the earth’s surface.
g/3600 = ac of the moon
Newton’s Law of Newton’s Law of Universal GravitationUniversal Gravitation
Newton knew the moon was 60 earth radii away from the center of the earth.Since 602 = 3600…
Fc 1/distance2
Inverse square law
Newton’s Law of Universal Gravitation
Newton concluded that the gravitational force of the earth decreases as the square of the distance from the earth increases.
Centripetal Acceleration and Gravity
Newton’s Law of Universal Gravitation
G = 6.67x10-11N-M2/kg2
221
r
mmGFG
Newton’s Law of Universal Gravitation
The value of G was measured in 1798 by Henry Cavendish
Problems 6 and 7
Calculate the mass of the earth Problem 8
The force of Earth’s gravity on the space shuttle in orbit is zero or nearly zero.
Think About It! (True or False)
““Weightlessness”
In free fall, an object undergoes apparent weightlessness.
Measuring g
How could we measure the acceleration due to gravity?
Pendulums are used to keep time, measure g, and to show that the earth spins on its axis.
Geophysical Applications
The value of g can vary on the earth’s surface because of
(a)variations in elevation
(b) rock density.
How is the gravitational force between two planets altered if the mass of one planet doubles?
Think About It!
When the mass of both planets doubles?
When they are three times as far apart?
Planetary Motion
Copernicus – 1543
Proposed that the earth and other planets orbit the sun in perfect circles.
Planetary MotionBrahe – 1570 Precise measurements of planetary and stellar motion. Copernican model did not explain the data
Planetary Motion
Kepler – 1595
Circular orbits don’t work but elliptical orbits do!
Kepler’s Laws of Planetary Motion1. The paths of all planets are ellipses with the sun at one focus.
Kepler’s Laws of Planetary Motion
Kepler’s Laws of Planetary Motion2. Each planet moves such that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal time intervals. (Planets move faster closer to the sun)
Kepler’s Laws of Planetary Motion
Kepler’s Laws of Planetary Motion3. The square of a planet’s orbital period (T2) proportional to the cube of the average distance between the planet and the sun (r3), or T2 r3
Kepler’s Laws of Planetary Motion
Kepler’s third law can be used to find orbital period and orbital speed.
Rotational Motion Center of mass – the point at which all mass of the body can be considered to be concentrated.
Rotational Motion It’s the point at which an object will rotate if only gravity is acting on it.
Summary Translational quantity
Rotational quantity
Inertia Moment of inertia
MomentumP=mv
Angular momentum= moment of inertia x ang. velocity
ForceF=ma
TorqueTorque = mass x ang. acceleration
Summary Translational quantity
Rotational quantity
Inertia Moment of inertia
MomentumP=mv
Angular momentum= moment of inertia x ang. velocity
ForceF=ma
TorqueTorque = mass x ang. acceleration
Rotational Motion Inertia – resistance to translational motion; depends on mass
Moment of inertia – resistance to rotational motion; depends on mass and distribution of mass around the axis of rotation
Rotational Motion As the farther the mass from the axis of rotation the greater the moment of inertia.
The greater the moment of inertia the ______ it is to rotate the object. (flip pencil)
Conservation of Angular Momentum
Gyroscope video
Summary
Translational quantity
Rotational quantity
Inertia Moment of inertia
MomentumP=mv
Angular momentum= moment of inertia x ang. velocity
ForceF=ma
TorqueTorque = mass x ang. acceleration
Rotational Motion - Dynamics
Torque is the product of the force times the length of the lever arm
Torque = force x lever arm
Unit : N-m
Rotational Motion - Dynamics
The greater the lever arm, the ______ the torque for the same amount of force. The greater the torque the ____ the angular acceleration.
Summary
Translational quantity
Rotational quantity
Inertia Moment of inertia
MomentumP=mv
Angular momentum= moment of inertia x ang. velocity
ForceF=ma
TorqueTorque = mass x ang. acceleration
Rotational Motion - DynamicsWhere can you apply the least amount of force to open or close a door?
Where can you hold a baseball bat to get the greatest acceleration of the ball?
There is a gravitational force of 1.02 x 10-8N between a 75.0 kg student and a 85.0 kg student. How far apart are they?
Numbered Heads Review
Calculate the acceleration due to gravity 15.6 km above the surface of the earth.
Numbered Heads Review
By how many Newtons does the weight of a 52 kg person change when he goes from sea level to an altitude of 12 km?
Numbered Heads Review
Mercury has a radius of 2.44 x 103 km and a mass of 3.30 x 1023 kg. Calculate the acceleration due to gravity on the surface of Mercury in m/s2 and g’s.
Numbered Heads Review
What is the force of gravity acting on a 4500 kg spacecraft when it is 3 Earth radii from the Earth’s center?
Numbered Heads Review
A ball is whirled around on a string that has a 0.55 m radius. The ball takes 0.32 seconds to make one revolution. What is the centripetal acceleration of the ball?
Numbered Heads Review