Download Chapter 19 Outline The First Law of Thermodynamics

Kepler’s Laws
• In the 16th century, Nicolaus Copernicus deduced that the
Earth is a planet, and, like all planets, it orbits the Sun.
• In the early 17th century, Johannes Kepler used Tycho
Brahe’s observations of planetary motions to determine
three empirical laws of planetary orbits.
1. Each planet moves in an elliptical orbit, with the sun at
one focus of the ellipse.
2. A line from the sun to a given planet sweeps out equal
areas in equal times.
3. The periods of the planets are proportional to the 32
powers of the major axis lengths of their orbits.
• Newton later discovered the reasons for these laws.
Kepler’s First Law
• Elliptical orbits
• Focal points, 𝑆 and 𝑆′
• Semi-major axis, 𝑎
• Eccentricity, 𝑒, between
zero (circle) and one.
• The sum of the distances
𝑆𝑃 and 𝑆 ′ 𝑃 is constant.
• In our solar system, most
orbits are fairly circular
• Earth: 𝑒 = 0.017
• Most eccentric is
Mercury: 𝑒 = 0.206
Kepler’s Second Law
• The rate at which area is swept out is
called the sector velocity.
𝑑𝐴 1 2 𝑑𝜃
= 𝑟
𝑑𝑡 2 𝑑𝑡
• Triangle area: 𝐴 = 12𝑏ℎ
• The perpendicular component of 𝒗 is
𝑣 sin 𝜙.
• The rate at which 𝜃 is changing is
𝜔=
𝑣 sin 𝜙
r
𝑑𝐴 1
= 𝑟𝑣 sin 𝜙
𝑑𝑡 2
𝑑𝜃
𝑑𝑡
=
Kepler’s Second Law
• Recall that angular momentum is
𝑳 = 𝒓 × 𝑚𝒗.
• 𝐿 = 𝑚𝑟𝑣 sin 𝜙
• So, the sector velocity is simply
𝑑𝐴
𝐿
=
𝑑𝑡 2𝑚
• Kepler’s second law is a natural
consequence of the conservation of
angular momentum.
• There can be no torque because the
force is always central (on a line
between the masses.)
Kepler’s Third Law
• For a circular orbit, we already showed that the period was
proportional to the 32 power of the radius.
• This is also true of an elliptical orbit, but the radius is
replaced by the semi-major axis, 𝑎.
𝑇=
2𝜋𝑎3/2
𝐺𝑀
Escape Velocity Revisited
• We already calculated the escape velocity at the surface of
a planet (or any spherical object).
𝑣escape =
2𝐺𝑀
𝑟
• If the sphere becomes more and more dense, the mass will
remain the same, but the radius decreases, and the escape
velocity increases.
• At what point does this become an issue?
Black Holes
• The maximum speed for anything is the speed of light, 𝑐 =
3.00 × 108 m/s.
• If the escape velocity exceeds this, even light cannot
escape.
• This is a black hole.
Schwarzschild Radius
• For a given mass, what will the radius have to be to form a
black hole?
𝑐=
2𝐺𝑀
𝑅S
2𝐺𝑀
𝑅S = 2
𝑐
• This is the Schwarzschild radius, and it defines the event
horizon.
• Note: To properly derive this, we need to use general
relativity, but the result is the same.
What if the Sun collapsed into a black hole?
• What if the Sun collapsed into a black hole?
• Would the Earth’s orbit be affected?
• What would the radius of the event horizon be?
2𝐺𝑚Sun
𝑅S =
≈ 3 km
2
𝑐
• The Sun’s radius now is 6.96 × 108 m.
Black Hole Detection
• Since no light can escape the event horizon, how can we even
see the black hole?
• When matter falls towards the black hole it is compressed and heated,
emitting radiation.
• This accretion disc is therefore visible evidence.
Chapter 13 Summary
Gravitation
• Newton’s law of gravitation: 𝐹g =
𝐺𝑚1 𝑚2
𝑟2
• Weight and acceleration due to gravity on Earth: 𝑔 =
• Gravitational potential energy: 𝑈g =
• Escape velocity: 𝑣escape =
• Circular orbits
• 𝑣circ =
𝐺𝑀
𝑟
• 𝑇circ =
2𝜋𝑟 3/2
𝐺𝑀
2𝐺𝑀
𝑟
𝐺𝑀𝑚
−
𝑟
𝐺𝑚E
2
𝑅E
Chapter 13 Summary
Gravitation
• Kepler’s laws
• Elliptical orbits
• Sweep equal areas in equal times:
𝑑𝐴
𝑑𝑡
=
𝐿
2𝑚
• Orbital period and semi-major axis: 𝑇 =
• Black holes
• Schwarzschild radius: 𝑅S =
2𝐺𝑀
𝑐2
2𝜋𝑎3/2
𝐺𝑀
Chapter 14 Outline
Periodic Motion
• Oscillations
• Amplitude, period, frequency
• Simple harmonic motion
• Displacement, velocity, and acceleration
• Energy in simple harmonic motion
• Pendulums
• Simple
• Physical
• Resonance
Periodic Motion
• Many types of motion repeat again and again.
• Plucked string on a guitar, a child in a swing, sound waves in a
flute…
• This is called periodic motion, or oscillation.
• Stable equilibrium point
• Displacement from equilibrium leads to a force (or torque) to
return it to equilibrium.
• Kinetic energy leads to overshoot, causing repeat
Periodic Motion Notation
• Amplitude – The maximum
magnitude of displacement from
equilibrium
• Period, 𝑇 – The time for one cycle (s)
• Frequency, 𝑓 – Number of cycle per
unit time (Hz = 1/s)
• Angular frequency, 𝜔 – Equal to 2𝜋𝑓
(rad/s)
• From these definitions,
𝑓=
1
,
𝑇
and 𝑇 =
1
𝑓
Simple Harmonic Motion
• If the restoring force is directly proportional to the
displacement, the resulting oscillation is simple harmonic
motion.
• One example is a mass on spring that obeys Hooke’s law.
𝐹 = −𝑘𝑥
Mass on a Spring
• Using Newton’s second law, in one dimension,
𝐹 = −𝑘𝑥 = 𝑚𝑎
𝑘
𝑎=− 𝑥
𝑚
• Since acceleration is the second time derivative of
position,
𝑑2𝑥
𝑘
=− 𝑥
2
𝑑𝑡
𝑚
Mass on a Spring
• Sinusoidal functions satisfy this differential equation.
𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
• Plugging this into the equation, we can solve for 𝜔,
𝑑𝑥
= −𝜔𝐴 sin(𝜔𝑡 + 𝜙)
𝑑𝑡
𝑑2𝑥
2 𝐴 cos 𝜔𝑡 + 𝜙 = −𝜔 2 𝑥
=
−𝜔
𝑑𝑡 2
• Comparing to
𝑑2𝑥
𝑑𝑡 2
=−
𝑘
𝑥,
𝑚
𝜔=
𝑘
𝑚
Simple Harmonic Motion of a
Mass on a Spring
• Angular frequency
𝜔=
𝑘
𝑚
• Frequency
1 𝑘
𝑓=
2𝜋 𝑚
• Period
𝑚
𝑇 = 2𝜋
𝑘
• Amplitude does not enter the equations as long as the
spring obeys Hooke’s law. (Generally for small
amplitudes.)
Displacement, Velocity and Acceleration
• As we saw earlier, sinusoidal functions satisfy the
differential equation for simple harmonic motion.
𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
• 𝐴 is the amplitude, and 𝜙 is the phase angle (starting position)
• We find the velocity at any time by taking the time
derivative,
𝑑𝑥
𝑣=
= −𝜔𝐴 sin(𝜔𝑡 + 𝜙)
𝑑𝑡
• Likewise, for the acceleration,
𝑑2𝑥
𝑎 = 2 = −𝜔2 𝐴 cos 𝜔𝑡 + 𝜙
𝑑𝑡
SHM Example