Units 17, 18, 19, 20

Homework 3 is on the website of the coursehttp://www.astro.wisc.edu/~lazarian/ast103_2014/

Acceleration of a body is its rate of change of

A. MassB. Weight

C. VelocityD. Positions

An object orbiting the sun in a circle can be said to be

A. WeightlessB. Always accelerating

C. Moving at a constant velocityD. Moving under equal and opposite forces

An accelerating body must at all times

A. Have a changing direction of motionB. Have an increasing velocity

C. Be movingD. Have a changing velocity

Which of the following statements about an asteroid moving in a circular orbit around the Sun is untrue?

A. It is moving on a flat planeB. It is moving with constant velocity

C. It is acceleratingD. It is moving with constant speed

• As we saw in Unit 17, we can find the mass of a large object by measuring the velocity of a smaller object orbiting it, and the distance between the two bodies.

• We can re-arrange this expression to get something very useful:

Orbits

G

VdM

2×=

dGM

Vcirc =We can use this expression to determinethe orbital velocity (V) of a small mass orbiting a distance d from the center of a much larger mass (M)

Calculating Escape Velocity

• From Newton’s laws of motion and gravity, we can calculate the velocity necessary for an object to have in order to escape from a planet, called the escape velocity

RGM

Vesc

2=

What Escape Velocity Means

• If an object, say a rocket, is launched with a velocity less than the escape velocity, it will eventually return to Earth

• If the rocket achieves a speed higher than the escape velocity, it will leave the Earth, and will not return!

Escape Velocity is for more than just Rockets!

• The concept of escape velocity is useful for more than just rockets!

• It helps determine which planets have an atmosphere, and which don’t– Object with a smaller mass (such as the

Moon, or Mercury) have a low escape velocity. Gas particles near the planet can escape easily, so these bodies don’t have much of an atmosphere.

– Planets with a high mass, such as Jupiter, have very high escape velocities, so gas particles have a difficult time escaping. Massive planets tend to have thick atmospheres.

Konstantin Tsiolkovsky, pioneer of space exploration

Werner Von Braun --Dark Genius of Rocket Science

Centripetal Force

• If we tie a mass to a string and swing the mass around in a circle, some force is required to keep the mass from flying off in a straight line

• This is a centripetal force, a force directed towards the center of the system

• The tension in the string provides this force.

• Newton determined that this force can be described by the following equation:

d

VmFC

2×=

d

VmFC

2×=

• We know that for planets, the centripetal force that keeps the planets moving on an elliptical path is the gravitational force.

• We can set FG and FC equal to each other, and solve for M!

• Now, if we know the orbital speed of a small object orbiting a much larger one, and we know the distance between the two objects, we can calculate the larger object’s mass!

Masses from Orbital Speeds

G

VdM

2×=

Newton’s Modification of Kepler’s 3rd Law

• Newton applied his ideas to Kepler’s 3rd Law, and developed a version that works for any two massive bodies, not just the Sun and its planets!

• Here, MA and MB are the two object’s masses expressed in units of the Sun’s mass.

• This expression is useful for calculating the mass of binary star systems, and other astronomical phenomena

2

3

YR

AUBA P

aMM =+

The Origin of Tides

• The Moon exerts a gravitational force on the Earth, stretching it! – Water responds to

this pull by flowing towards the source of the force, creating tidal bulges both beneath the Moon and on the opposite side of the Earth

High and Low Tides

As the Earth rotates beneath the Moon, the surface of the Earth experiences high and low tides

The Sun creates tides, too!

• The Sun is much more massive than the Moon, so one might think it would create far larger tides!

• The Sun is much farther away, so its tidal forces are smaller, but still noticeable!

• When the Sun and the Moon line up, higher tides, call “spring tides” are formed

• When the Sun and the Moon are at right angles to each other, their tidal forces work against each other, and smaller “neap tides” result.

The Conservation of Energy

• The energy in a closed system may change form, but the total amount of energy does not

change as a result of any process

• Kinetic Energy is simply the energy of motion

• Both mass (m) and velocity (V) contribute to kinetic energy

• Imagine catching a thrown ball.– If the ball is thrown gently, it hits your hand

with very little pain

– If the ball is thrown very hard, it hurts to catch!

Kinetic Energy

2

2

1VmEK ×=

Thermal Energy

• Thermal energy is the energy associated with heat

• It is the energy of the random motion of individual atoms within an object.

• What you perceive as heat on a stovetop is the energy of the individual atoms in the heating element striking your finger

Potential Energy

• You can think of potential energy as stored energy, energy ready to be converted into another form

• Gravitational potential energy is the energy stored as a result of an object being lifted upwards against the pull of gravity

• Potential energy is released when the object is put into motion, or allowed to fall.

Definition of Angular Momentum

• Angular momentum is the rotational equivalent of inertia

• Can be expressed mathematically as the product of the objects mass, rotational velocity, and radius

• If no external forces are acting on an object, then its angular momentum is conserved, or a constant:

constant=××= rVmL

Conservation of Angular Momentum

• Since angular momentum is conserved, if either the mass, size or speed of a spinning object changes, the other values must change to maintain the same value of momentum– As a spinning figure skater

pulls her arms inward, she changes her value of r in angular momentum.

– Mass cannot increase, so her rotational speed must increase to maintain a constant angular momentum

• Works for stars, planets orbiting the Sun, and satellites orbiting the Earth, too!