units 17, 18, 19, 20 homework 3 is on the website of the course lazarian/ast103_2014
TRANSCRIPT
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Units 17, 18, 19, 20
Homework 3 is on the website of the coursehttp://www.astro.wisc.edu/~lazarian/ast103_2014/
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Acceleration of a body is its rate of change of
A. MassB. Weight
C. VelocityD. Positions
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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
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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
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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
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• 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)
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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=
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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!
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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.
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Konstantin Tsiolkovsky, pioneer of space exploration
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Werner Von Braun --Dark Genius of Rocket Science
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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×=
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• 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×=
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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 =+
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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
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High and Low Tides
As the Earth rotates beneath the Moon, the surface of the Earth experiences high and low tides
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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.
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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
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• 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 ×=
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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
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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.
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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
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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!