nats 1311 - from the cosmos to earth examples: pulling a table cloth out from under a table setting...

NATS 1311 - From the Cosmos to Earth

Examples:

Pulling a table cloth out from under a table setting

The reaction of coffee in a cup when accelerating or decelerating in a car

Tightening of a hammerhead by banging hammer on the ground

Getting ketchup out of a bottle

Not wearing a seatbelt during a head-on car crash

Headrests in a car to prevent whiplash during a read-end collision


Pushing Cart Animation

Newton’s 2nd Law

F=ma

or

a=F/m


Velocity and Acceleration

Newton showed that 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 = Dv/Dt

Differential calculus!

Different cases of acceleration:

Velocity and acceleration are vectors.

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

2. Deceleration (i.e. decreasing speed)

a

v


Newton’s 2nd Law Explains the Feather and the Ball

1 kg on the Earth weighs 9.8 N or 2.2 lbs

F = W = mg

W = 1kg X 9.8 m/s = 9.8 kg m/s = 9.8 N

Take a 1 kg rock and a 10 kg rock and drop them from the same height

a1 = F1/m1 = W1/m1 = 9.8 N/1 kg = 9.8 m/s = g

a2 = F2/m2 = W2/m2 = 98 N/10 kg = 9.8 m/s = g


A body subjected to a force reacts with an equal counter force to the applied force:

That is, action and reaction are equal and oppositely directed, but never act on the same body.

Newton’s Third Law

For every action (force), there is an equal and opposite reaction (force)


Examples of Action/Reaction

Swimming - your hands and the water

Walking - your feet and the ground

Driving - a car’s tires and the road

A bug and a car’s windshield

A falling object - the object and the earth

A person pulling a spring

A deflating balloon - the air rushing out and the balloon

Pushing on the wall - your hand and the wall

Rocket ship - expelled fuel and rocket


apparent weight - weight force that we actually sense not the downward force of gravity, but the normal (upward) force exerted by the surface we stand on

- opposes gravity and prevents us falling to the center of the Earth - what is measured by a weighing scale.

For a body supported in a stationary position, normal force exactly balances earth's gravitational force

- apparent weight has the same magnitude as actual weight.

If no contact with any surface to provide such an opposing force - no sensation of weight (no apparent weight).

- free-fall - experienced by sky-divers and astronauts in orbit who feel "weightless" even though their bodies are still subject to the force of gravity - also known as microgravity.

A degree of reduction of apparent weight occurs, for example, in elevators. In an elevator, a spring scale will register a decrease in a person's (apparent) weight as the elevator starts to accelerate downwards. This is because the opposing force of the elevator's floor decreases as it accelerates away underneath one's feet.

Apparent Weight


Apparent Weight Animation


The Earth is round - its surface drops about 5 m for every 8 km of distance. If you were standing at sea level, you would only see the top of a 5-meter mast on a ship 8000 m away - remember the story of Columbus and the orange.

Given h=1/2gt2, if t=1 s then h = 5 m. So if a projectile is fired horizontally at ~8 km/s, it will fall fast enough to keep “falling around” the Earth - becomes a satellite. So a spacecraft is in free fall around the Earth - free fall is not an absence of gravity. If a satellite is given a velocity greater than 8 km/s, it will overshoot a circular orbit and trace an elliptical path.

Escape velocity - velocity at which gravity can not stop outward motion - 40,000 km/hr for Earth

Cannonball Animation

Orbital Velocity


Momentum is mass times velocity, a vector quantity:Mom=mv

Law of Conservation of MomentumThe total momentum of an isolated system is conserved, I.e., it remains constant.

An outside or external force is required to change the momentum of an isolated system.

The Law of Conservation of Momentum is an alternate way of stating Newton’s laws:

1. An object’s momentum will not change if left alone2. A force can change an object’s momentum, but…3. Another equal and opposite force simultaneously changes some other object’s momentum by same amout

Momentum


Billiard Balls


A Rifle and a Bullet

When a bullet is fired from a rifle, the rifle recoils due to the interaction between the bullet and the rifle.

The force the rifle exerts on the bullet is equal and opposite to the force the bullet exerts on the rifle.

But the acceleration of the bullet is much larger that the acceleration of the rifle - due to Newton’s 2nd law: a = F/m

The acceleration due to a force is inversely proportional to the mass.

The force on the rifle and the bullet is the same but the mass of the rifle is much larger than the the mass of the bullet so the acceleration of the rifle is much less than the acceleration of the bullet.


Angular Momentum

Momentum associated with rotational or orbital motionangular momentum = mass x velocity x radius


Torque and Conservation of Angular Momentum

Conservation of angular momentum - like conservation of momentum -in the absence of a net torque (twisting force), the total angular momentum of a system remains constant

Torque - twisting force


A spinning skater speeds up as she brings her arms in and slows down as she spreads her arms because of conservation of angular momentum


The law of universal gravitation.


The force on a body of mass m1 is:

(Newton’s Second Law)

If this force is due to gravity, then:

m1 cancels out, and:

€

F =m1a

€

m1a=Gm1m2

d2

€

a=Gm2

d2

Newton’s 2nd Law and the Acceleration Due to Gravity


The acceleration due to the force of gravity is called g, so:

Mass of the Earth (m2) = 5.97 X 1024 kgRadius of Earth (d) = 6.378 X 106 mG= 6.67 x 10-11 Nm2/ kg2

g= (6.67 x 10-11 Nm2/ kg2) X (5.97 X 1024 kg)/(6.378 X 106 m)2

g= 9.79 m/s2

g does not depend on the mass of the body m1 - so the feather falls at the same speed as the steel ball - Galileo learned this by experimentation (the Leaning Tower of Pisa experiment) - Newton showed why.

Weight is the result of the force of gravity on a body of mass m1:

Therefore all objects on earth having the same mass have the same weight.

€

g=Gm2

d2

€

W =m1g


The acceleration of gravity and therefore a person’s weight is dependent on a planet’s mass and radius.

Planetary Mass, Radius and Weight


Newton’s Formulation of Kepler’s Laws

As a planet moves around its orbit, it sweeps out equal areas in equal times - a planet moves slower when it is farther from the Sun and faster when it is closer

Kepler’s Laws were based on observation (experimentation). Newton’s lawsexplained Kepler’s Laws

Kepler’s Second Law


€

F =m1a

For a circular orbit: (r = radius of orbit)

€

a=v2

r

€

F =m1

v2

r=Gm1m2

r2

€

v2 =Gm2

r

Substitute (2) into (1):F is the force of gravity:

Cancel m1and r; then

(1)

(2)

(3)

(4)

The smaller the radius, the greater the speed.The orbital speed is independent of the mass of the orbiting body (m1). As the radius (the distance to the orbiting body) increases, the orbital speed decreases.

When you swing a ball around, the string exerts a force that pulls the ball inward (gravity for orbiting body). The acceleration is also inward.


The square of any planet's period of orbital revolution, P, is proportional to the cube of its mean distance, r, from the sun.

Kepler’s 3rd Law

Orbital Period vs Distance Animation


€

v2 =Gm2

r

€

v=2πrP

€

v2 =4π 2r2

P2

From Kepler’s Second Law (previous slide):

Speed around orbit:Circumference (2r)/ timeP=period, time of 1 orbit

(1)

(2)

(3)

(4)

(5)

Combine (1) and (3):

Rearrange terms:

€

Gm2

r=

4π 2r2

P2

€

P2 =4π2

Gm2

r3

Square both sides:


A more complex derivation of this equation yields:

€

P2 =4π 2

G(m1 +m2)r3

From this equation, if one knows the mass of the orbiting body (m1), the mass of the central body (m2) may be calculated.

What is the mass of the Sun?

MSun (m1) >> MEarth (m2) so: m1 + m2 m1

M1 = 42r3/GP2

G = 6.67 x 10-11 Nm2/ kg2

r = 1.5 x 1011mP = 3.15 x 1011 s

So:

Msun = 2 x 1030 kg


Geosynchronous/Geostationary OrbitsA geosynchronous orbit has a period the same as the rotational speed of the Earth - e.g., it orbits in the same amount of time that the Earth rotates - 1 sidereal day. A geostationary orbit is a geosynchronous orbit at the equator - it always stays above the same place on the Earth - communications satellites, satellite TV, etc…

What is the altitude of a geostationary orbit?

From Newton’s formulation of Kepler’s 3rd Law:

MEarth (m1) >> MSatellite (m2) so:

r = (GMEarth P2/42)1/3

G = 6.67 x 10-11 Nm2/ kg2

P = 3.15 x 1011 s

MEarth = 5.97 X 1024 kgSo:

R = 42,000 km above the center of the Earth and the altitude is about 35,600 km

€

P2 =4π 2

G(m1 +m2)r3

nats 1311 - from the cosmos to earth examples: pulling a table cloth out from under a table setting...

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