Circular motion and Gravitation Chapter 6 1Physics Chapter 6.

Physics Chapter 6 1

Circular motion and Gravitation

Chapter 6

Physics Chapter 6 2

Dynamics of Circular Motion

Remember that when an object moves in a circle with a constant speed, its acceleration is always directed toward the center of the circle.

R

varad

2

2

24

T

Rarad

Physics Chapter 6 3

Circular motion

The acceleration in circular motion is caused by a net force, just like any acceleration.

If this force is removed, the object will continue along a straight path with a constant velocity.

Physics Chapter 6 4

Centripetal force

The net force that causes centripetal acceleration.

Not a separate force Does not appear in the free body

diagram.

Physics Chapter 6 5

Newton’s second law

maF

R

vmF

2

Physics Chapter 6 6

Example

A hockey puck with mass 0.500 kg revolves in a uniform circle on the frictionless ice. It is attached to a 0.400 m long cord nailed into the ice. It makes one revolution per second.

What is the force, F, exerted by the cord on the puck?

Physics Chapter 6 7

Example You have a summer job as part of an

automobile design team. You are testing new prototype tires to see whether or not the tires perform as well as predicted. In a skid test, a new BMW 530i was able to travel at a constant speed in a circle of radius 45.7 m in 15.2 s without skidding.• What was its speed, v?• What is the acceleration?• Assuming air drag and rolling friction to be negligible,

what is the minimum value for the coefficient of static friction between the tires and the road?

Physics Chapter 6 8

Example

A curve of radius 30 m is banked at an angle q. Find q for which a car can round the curve at 40 km/h even if the road is covered with ice so that friction is negligible.

Physics Chapter 6 9

Vertical circles

Be careful about weight Apparent weight – what you feel like you

weigh Apparent weight = normal force

Physics Chapter 6 10

Normal Force

Can be equal to, less than, or greater than weight

If contact with the surface is lost, normal force is zero.

Physics Chapter 6 11

Example

You swing a cup of water with mass m in a vertical circle of radius r. If its speed is vt at the top of the circle, find• The force exerted on the water by the cup at

the top of the circle

• The minimum value for vt for the water to remain in the cup.

Physics Chapter 6 12

On your own

What is the force exerted by the cup on the water at the bottom of the circle, where the pail’s speed is vb?

Physics Chapter 6 13

Universal Law of Gravitation

A gravitational force acts between every pair of particles in the universe.

Gravitational forces are always attractive.

Published by Newton in 1687.

Physics Chapter 6 14

Universal Law of Gravitation

The m’s are the masses of the two objects.

r is the distance between their centers of mass.

G is a fundamental physical constant called the gravitational constant.

221

r

mGmFg

Physics Chapter 6 15

Value of G

Newton didn’t have sensitive enough equipment to measure G.

In 1798, Henry Cavendish used a torsion balance to measure G.

In SI units, G is 6.67 x 10-11 N-m2/kg2

Physics Chapter 6 16

Spherical objects

The gravitational interaction between two objects having spherical symmetry is the same as though all the mass was concentrated at the center.

So, we can treat them as particles.

Physics Chapter 6 17

Superposition of Forces

If each of two masses exerts a force on a third, the total force on the third mass is the vector sum of the individual forces from the first two.

Physics Chapter 6 18

Example

Particle 1 has a mass of 6.0 kg and is located at the origin. Particle 2 has a mass of 4.0 kg and is located at (0.0 , 2.0) cm. Particle 3 has a mass of 4.0 kg and is located at (-4.0 , 0.0) cm. Find the net gravitational force on particle 1.

4.1 x 10-6 N @ 104°

Physics Chapter 6 19

Gravitational Forces

Between ordinary, household objects, they are small.

Between astronomical objects they are large.

Gravity is what keeps the universe running – orbits, energy output of stars, etc.

Physics Chapter 6 20

Weight

According to the Universal law of gravitation, an object of mass m on the surface of the earth would have the following weight:

2E

Eg R

mGmFw

Physics Chapter 6 21

Weight

Setting this equal to mg,

2E

E

R

mGmmg

2E

E

R

Gmg

Physics Chapter 6 22

Weight

If an object is a distance (r-RE) above the surface of the earth, then it is at a distance r above the center of the earth, and

2r

Gmg E

Since r > RE, g < 9.8 m/s2

Physics Chapter 6 23

Escape speed

In order for a space shuttle to leave the earth, it must have enough speed to stay in the air long enough that the Earth curves away from it faster than it falls.

We can calculate the minimum velocity required to do this.

Physics Chapter 6 24

Motion of Satellites

If a satellite is traveling in a circular orbit (which most of them do), the only force acting on it is gravity.

maF

r

vm

r

mGmE2

2

r

Gmv E

Physics Chapter 6 25

Motion of satellites

This tells us that if you want a satellite to orbit with a certain speed, it must be at a certain radius.

Doesn’t depend on mass – apparent weightlessness of astronauts.

Physics Chapter 6 26

Period of circular orbits

For a circular orbit,

T

rv

2

If you set this equal to the velocity equation we just found,

EE Gm

rT

Gm

rrT

2/32or 2

Physics Chapter 6 27

Satellites

Not always manmade Don’t always orbit Earth Moons Rings of Saturn, Uranus, and Neptune

Physics Chapter 6 28

You try

You want to place a communications satellite into a circular orbit 300 km above the earth’s surface. What must be its speed, its period, and its radial acceleration? The earth’s radius is 6.38 x 106 m and its mass is 5.98 x 1024 kg.