Today’s Lectures:Important Physical

Quantities,The Laws of Gravity

• Speed, velocity, acceleration, force

• Mass, momentum and inertia• Newton’s three laws of motion• The Force of Gravity

Assigned Reading

• Chapter 5.1, up to page 92, not including orbital velocity.

Before Newton• Kepler understood the phenomenology of

planetary motions, I.e. the three laws, but never understood *why* planets move the way they do– He thought of magnetic force from the Sun– He even thought (and dismissed) that Angels pull

planets along their orbits

• Galileo understood motion and its causes, like inertia, force and acceleration very well…

• …but fell short of either understand or explain the causal relationship between force and motion, i.e. acceleration. He also did not propose a universal law for the force of gravity, i.e. he did not recognize gravitation as a universal interaction.

• For that, humanity had to wait for Newton

Describing Motion

• Motion occurs when the position of an object relative to some reference frame changes with time

• If position does not change, the object is at rest• To describe motions we need a reference frame

and we need to monitor position and time• The rate at which an objects covers a given

amount of space in a given amount of time is called speed

– v = s/t

Vectors,or things with a sense of

direction• There are physical quantities in nature

for which only one number, their intensity, tells the whole story: e.g. temperature, mass, luminosity, color.– We call them SCALARS

• Other quantities need both an intensity and a direction to be fully described: e.g. speed, force, acceleration.

• These quantities with a “direction” are called VECTORS

Velocity and Acceleration

Acceleration (a) is the change of a body’s velocity (v) with time (t):

1. Acceleration in the conventional sense (i.e. increasing speed)

a = v/t

Different cases of acceleration:

Velocity and acceleration are directed quantities (vectors)!

3. Change of the direction of motion (e.g., in circular motion)

2. Deceleration (i.e. decreasing speed)

a

v

Vocabulary Words• Speed is the distance traveled in each

second – it’s measured in meters per second (m/s).

• Velocity is the speed in a particular direction – it’s measured in meters per second (m/s).

• Acceleration is the change in velocity in each second – it’s measured in meters per second per second (or m/s2)

More On Acceleration

• An acceleration is a change in velocity.

• Acceleration occurs when either the magnitude or direction of the velocity (or both) are altered.

• Uniform Circular Motion is Accelerated Motion

If an object does not accelerate, that means that the total sum of forces acting on it is zero. It does not mean that

there are no forces on it. Just the they cancel out

Which of the following does NOT describe an acceleration:

• a car traveling with constant speed around a bend

• a car decreasing speed on a straight road

• a car traveling with constant speed on a straight road

• a planet traveling around a Sun

Momentum• But Velocity is not the whole story, two objects of different

mass, e.g. a train and bike, might move with the same velcity, but they carry different “quantity of motion”.

• Momentum is a quantitative way to describe an object’s “quantity of motion”.

• It also describes the tendency to continue to move in the same way it was moving previously (“… keep your momentum…”).

• An object’s momentum is simply its mass times its velocity.

P = m·v

Two objects of different mass subject to the same acceleration acquire the same speed, but different momentum.

Don’t forget that velocity is a vector, so momentum is a vector: the direction of motion is very important!

Galileo’s Law of Inertia

• momentum P = m·v

• An object maintains its “quantity of motion” (momentum), unless a force is acted upon it

• Only a force can change the “quantity of motion” (momentum) of an object, either by changing its speed, or by changing its direction of motion, or both

Galileo and the Concept of Inertia

Aristotle held that objects at rest remained at rest unless a force acted on them, but that objects in motion did not remain in motion unless a force acted constantly on them: F = v WRONG.

Galileo concluded that an object in a state of motion possesses an ``inertia'' that causes it to remain in that state of motion unless an external force acts on itRIGHT.

Vectors: when the force is not parallel to velocity (momentum)

• Earth and moon attract each other through gravitation, i.e. there is a force between them.

Earth

Moon

v v’

v

F

• Earth’s gravitational force constantly accelerates the moon towards Earth.

• This acceleration is along the direction of the force, i.e. along the radius•The velocity is perpendicular to the force•This force is constantly changing the moon’s velocity.

•But it only changes the direction of

the velocity vector, not its intensity

(speed).•So, when the force is perpendicular to the velocity, velocity varies in direction not in strength

Sir Isaac Newton and the Unification of Physics & Astronomy

• Newton was by many standards the most important figure in the development of modern science.

• He demonstrated that the laws that govern the heavens are the same laws that govern the motion on the surface of the Earth: the universality of gravity, and hence of physical laws.

• Newton's Three Laws of Motion. • Theory of Universal Gravitation

(1642-1727)

Newton’s First Law of Motion(really, Galielo’s Inertia Law)

• In the absence of a net force, an object moves with constant velocity, or it conserves its momentum.

• (paraphrased) If nothing acts on an object, the object will keep moving in a straight line and at a constant speed.

• The same is true for zero velocity.

Newton’s Laws of Motion (1)

1. A body continues at rest or in

uniform motion in a straight line

unless acted upon by some

net force.An astronaut floating in space will continue to float forever in

a straight line unless some external force is accelerating

him/her.

Newton’s Laws of Motion (2)

• A body subject to a force, changes its momentum, i.e. accelerates, and the acceleration a of a body is:1. inversely

proportional to its mass m,

2. directly proportional to the net force F,

3. and in the same direction as the net force. a = F/m F = m a

Newton’s Laws of Motion (3)

3.To every action, there is an equal and opposite reaction.

The same force that is accelerating the boy forward, is accelerating the skateboard backward.

Newton’s First Law can also be stated as:

• In the absence of a net force, the acceleration of an object must be zero

• If an object is being accelerated, there must be a net force exerting on it

Newton’s Second Law Of Motion

• Acceleration is caused by force but also related to the mass of the object

Force = Mass x Acceleration

F = m·a

Or a = F/m

Gravity is such that acceleration is constant in any given planet, not the force. So, should a heavy object

fall faster than a light one, as Aristotle believed?

Bigger force

Smaller force

Why di they fall with the same speed? The it is not true that F = v…

The Falling Body Experiment on the Moon

The crew of the Falcon L.E.M., of the Apollo 15 mission, repeating Galileo’s experiment on the Moon

No air on the Moon, so drag by air not a factor in the experiment

Survey QuestionTwo identical spacecraft are to be accelerated by

rockets. The first rocket fires with a force 4 times as great as that of the second rocket. The acceleration of the first rocket is _____ times as great as the acceleration of the second rocket.

1) 1/4

2) 1/2

3) the same

4) 2 times

5) 4 times

Newton’s Third Law Of Motion

• For every ACTION (I.e. application of a force) there is an equal and opposite re-ACTION

F1 = -F2

Newton’s Third Law

For any force, there always is an equal and opposite reaction force.

Table pushesback onbox

Box pushes downon table due to gravity.

Example: The Jet Engine

• What makes a jet-propelled airplane move forward?

The Third Law demonstrated: without external force, momentum is

conserved If one isolated system (i.e. skate-board + human) is at

rest (I.e. v=0) is suddenly separated into two parts (i.e. the human jumps off), one moves in one direction and the other must move in the opposite direction!

Before separation: P1 + P2 = 0

After separation: P1 = m1·v1 P2 = m2·v2

CONSERVATION: P1 + P2 = m1·v1 + m2·v2 = 0

THUS: m1·v1 = -m2·v2

THUS:To the force that created the first motion…

there is an opposite-acting force that creates the second motion

Gravity

• What force is responsible for motions in the universe?

• What force makes objects fall?• What keeps us on the rotating Earth?• Why don’t planets move in straight lines,

but orbit around the Sun instead?

Gravity• We can summarize the universal law of

gravitation with the following statements:– Every mass attracts every other mass through

the force of gravity.– If mass #1 exerts force on mass #2, and

mass#2 exxerts force on mass#1, the force must depend o both masses, namely:

– The force of attraction is directly proportional to the product of the two masses.

– The force of attraction is inversely proportional to the square of the distance between the masses.

How did Newton figure out Gravity: if the apple was initially at rest and then started to fall, there must be a force acting on the apple. What force?

More importantly, if the force of gravity reaches to the tree, might it not reach even further? In particular, might it not reach all the way to the orbit of the Moon?

Newton calculated what the Earth’s gravity would be at the position of the Moon. And found that it is what would be required to keep the Moon in its orbit

The Universal Law of Gravity

• Any two bodies are attracting each other through gravitation, with a force proportional to the product of their masses and inversely proportional to the square of their distance:

F = - G Mm

r2

(G is the Universal constant of gravity.)

To calculate Earths’ gravityon the Moon, Newton assumed that Gravity obeys the inverse square law

How did Newton calculate Earth’s gravity at the position of the Moon?

F ~ 1/D2

The Law of Gravity

221

d

MMGFg

d

M1M2

G = 6.67x10-11 m3/kg/s2

2

21

2

9.8m/s

d

MGg

gMFg

Near Earth’s surface

Mass and Weight• Mass is a measure of how much

material is in an object.• Weight is the force exterted by gravity

on a massive body (body with mass), e.g. placed on the surface of Earth

• Weight is a measure of the gravitational force exerted on that material.

• Thus, mass is constant for an object, but weight depends on the location of the object.

• Your mass is the same on the moon, but your weight on the surface of the moon is smaller

Question: How do objects accelerate dueto the force of gravity?

Remember these two very, very important concepts:

1)Your mass decides the total force of gravity acting on you• Both you and a truck have the same distance to the

center of earth. • Yet, the force of gravity on the truck is bigger

2)The gravity of Earth decides your acceleration. In other words, when object fall under gravity, they all have the same acceleration, regardless of their mass (remember the two bodies falling from Pisa’s leaning tower?)

The gravitational force on an object nearthe surface of Earth is:

Fgrav = m·g (g = 9.8m/s2)

221

d

MMGFg Survey Question

Two equal masses, m, separated by a distance, d, exert a force, F, on each other due to their gravitational attraction. How large is the gravitational force between an object of mass m and an object of mass 2m separated by the same distance d?

¼ F

½ F

F

2F

4F

221

d

MMGFg Survey Question

Two equal masses, m, separated by a distance, d, exert a force, F, on each other due to their gravitational attraction. How large would their gravitational attraction be if the distance between them was doubled?

1) ¼ F

2) ½ F

3) F

4) 2F

5) 4F

221

d

MMGFg Survey Question

If aliens magically turned our Sun into a black hole of the same mass but 10 times smaller in diameter, what would change about the Earth’s orbit?

1) it would be 10 times smaller in radius

2) it would spiral into the black hole

3) nothing would change

4) it would spiral away from the black hole

5) it would be 10 times larger in radius

… so why don’t planets just fall into the sun?

M1M2

… because they miss (that is, they have enough tangential

velocity to always miss)

M1M2

v

This is the concept of an orbit.

Fg Fg

Why doesn't the earth fall to the sun?

• It has a velocity and it has inertia!

• Force of gravity causes change in the direction of velocity --- acceleration.

• The earth is falling towards the sun all the time!

V=8km/s

Why don't they fall?

They are circuling Earth at a speed of 8 km/s!

Quiz Astronauts inside the space shuttle

float around because ____they are falling in the same way as the space shuttle.

If you are in a free-falling elevator, you are massless. (true or false)false