Universal Gravitation

• Sir Isaac Newton first deduced that the force responsible for making objects fall on Earth is the same force that keeps the moon in orbit.

• This idea is known as the law of universal gravitation.

• Gravitational force exists between all objects that have mass.

• The strength of the gravitational force depends on the mass of the objects and the distance between them.

What is the law of universal gravitation?

The force of attraction between two objects is directly related to the masses of the objects and indirectly related to the distance between them

What is the law of universal gravitation?

1. Any object with mass exerts a force on any other object with mass. This force is called __________________.

2. How would increasing the mass of one of the objects affect the force of gravity between the objects?

3. How would increasing the distance between the objects affect the force of gravity between the objects?

4. Calculate the force between two objects that have masses of 70 kilograms and 2,000 kilograms. Their centers of gravity are separated by a distance of 1.00 meter.

How do we define mass, weight, and gravity?

mass weight gravityMass is a measure of the amount of matter in an object. Mass is not related to gravity.

The mass of an object does not change when it is moved from one place to another.

Mass is commonly measured in grams or kilograms.

Weight is a measure of the gravitational force between two objects.

The weight of an object does change when the amount of gravitational force changes, as when an object is moved from Earth to the moon.

Weight is commonly measured in newtons or pounds.

The force that causes all masses to attract one another. The strength of the force depends on the size of the masses and their distance apart.

Gravity• The force of gravity on an object is called

weight. • At Earth’s surface, gravity exerts a force of

9.8 N on every kilogram of mass.

Weight vs. mass• Weight and mass are not the same.• Mass is a fundamental property of matter

measured in kilograms (kg). • Weight is a force measured in newtons (N).• Weight depends on mass and gravity.

Weight depends on mass and gravity

A 10-kilogram rock has the same mass no matter where it is in the universe. On Earth, the10 kg rock weighs 98 N. On the moon, the same rock only weighs 16 N.

Calculating weight• The weight equation can be rearranged into three

forms to calculate weight, mass, or the strength of gravity.

EXAMPLE

• Calculate the weight (in newtons) of a 5.0-kilogram backpack on Earth (g = 9.8 m/s2).

m = 5.0 kg, g = 9.8 m/s2 FW = ?

FW = m(g)

FW = (5.0 kg) (9.8 m/s2)

FW = 49 kg-m/s2 = 49 N

EXAMPLE

• The same backpack weighs 8.2 newtons on Earth’s moon. What is the strength of gravity on the moon?

FW = 8.2 newtons, m = 5.0 kg

g = ?g= FW /m

g = 8.2 N / 5.0 kg = 8.2 kg-m/s2/5.0 kgg= 1.6 m/s2

1. An astronaut weighs 104 newtons on the moon, where the strength of gravity is 1.6 newtons per kilogram.

a) What is her mass on Earth?

1. An astronaut weighs 104 newtons on the moon, where the strength of gravity is 1.6 newtons per kilogram.

b) What is her weight on Earth?

1. An astronaut weighs 104 newtons on the moon, where the strength of gravity is 1.6 newtons per kilogram.

c) What is her mass on Mars?

2. What is the weight of a 15 kg box on the earth?

3. What is the mass of a ball that weighs 14.5 newtons?

4. A 12 kg box weighs 19.6 N on the moon, What is the acceleration due to gravity on the moon?

Free fall

• An object is in free fall if it is accelerating due to the force of gravity and no other forces are acting on it.

Free fall• Falling objects increase their speed by 9.8 m/s

every second, or 9.8 m/s2 The letter “g” is used for acceleration due to gravity.

Acceleration due to gravity is known to be 9.8 meters/second/second or 9.8 m/s2 and is represented by g. Three conditions must be met before we can use this value:(1) the object must be in free fall(2) the object must have negligible air resistance,

and(3) the object must be close to the surface of Earth.

Here is the equation for solving for velocity:

final velocity = initial velocity + (the acceleration due to the force of gravity × time)

OR

vf = vi + gt

Here is the equation for solving for distance.

distance = the acceleration due to the force of gravity × time× time2

OR

y = ½ gt2

Example 1: How fast will a pebble be traveling 3.0

seconds after being dropped? vi = 0 m/s, g = - 9.8 m/s2, t = 3.0 s

vf = ?

vf = vi + gt

vf = 0 m/s + (- 9.8 m/s2) (3.0 s) = (- 9.8 m/s2) (3.0 s)

vf = - 29 m/s

(Note that gt is negative because the direction is downward.)

Example 2: A pebble dropped from a bridge strikes the

water in 4.0 seconds. How high is the bridge?g = 9.8 m/s2, t = 4.0 s

y = ?

y = ½gt2

y = ½(9.8 m/s2)(4.0 s)2

y = 78.4 m

1. A penny dropped into a wishing well reaches the bottom in 1.50 seconds. What was the velocity at impact?

2. A pitcher threw a baseball straight up at 35.8 meters per second. What was the ball’s velocity after 2.50 seconds? (Note that, although the baseball is still climbing, gravity is accelerating it downward.)

3. In a bizarre but harmless accident, a watermelon fell from the top of the Eiffel Tower. How fast was the watermelon traveling when it hit the ground 7.80 seconds after falling?

4. A water balloon was dropped from a high window and struck its target 1.1 seconds later. If the balloon left the person’s hand at –5.0 meters per second, what was its velocity on impact?

5. A stone tumbles into a mine shaft and strikes bottom after falling for 4.2 seconds. How deep is the mine shaft?

6. A boy threw a small bundle toward his girlfriend on a balcony 10. meters above him. The bundle stopped rising in 1.5 seconds. How high did the bundle travel? Was that high enough for her to catch it?

Curved motion

• Circular motion is another type of curved motion.

• An object in circular motion has a velocity vector that constantly changes direction.

Centripetal Force

• Any force that causes an object to move in a circle is called a centripetal force.

• A centripetal force is always perpendicular to an object’s motion, toward the center of the circle.

Projectile Motion

• Any object that is moving through the air affected only by gravity is called a projectile.

• The path a projectile follows is called its trajectory.

Motion and the Velocity Vector

• The distance a projectile travels horizontally is called its range.

Curved motion

• A soccer ball is an example of a projectile. .

• The path of the ball makes a bowl-shaped curve called a parabola.

Projectile Motion

• When we drop a ball from a height we know that its speed increases as it falls.

• The increase in speed is due to the acceleration gravity, g = 9.8 m/sec2.

• Projectile motion has vertical and horizontal components

Horizontal motion

• The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force.

• Since there is no force, the horizontal acceleration is zero (ax = 0).

• The ball will keep moving to the right at 5 m/sec.

Vertical motion

• The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second.

• After one second has passed, vy of the ball will be 9.8 m/sec.

• After 2 seconds have passed, vy will be 19.6 m/sec and so on.