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