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Gravity, continued
Reading and Review
Gravitational Force at the Earth’s Surface
The center of the Earth is one Earth radius away, so
this is the distance we use:
g
The acceleration of gravity
decreases slowly with altitude...
...until altitude becomes comparable to the
radius of the Earth. Then the decrease in the
acceleration of gravity is much larger:
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they
are trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled by
the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they are
trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
Astronauts in the space shuttle float because
they are in “free fall” around Earth, just like a
satellite or the Moon. Again, it is gravity that
provides the centripetal force that keeps them in
circular motion.
Follow-up: How weak is the value of g at an altitude of 300 km?
Satellite Motion: FG and acp
Consider a satellite in circular motion*:
Gravitational Attraction:
Necessary centripetal acceleration:
• Does not depend on mass of the satellite!
• larger radius = smaller velocity
smaller radius = larger velocity
Relationship between FG and acp will be important
for many gravitational orbit problems
*
not all satellite orbits are circular!
A geosynchronous satellite is one whose orbital period is equal to
one day. If such a satellite is orbiting above the equator, it will be in
a fixed position with respect to the ground.
These satellites are used for communications and and weather
forecasting.
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
The Moon does not crash into Earth because of its high
speed. If it stopped moving, it would, of course, fall
directly into Earth. With its high speed, the Moon would
fly off into space if it weren’t for gravity providing the
centripetal force.
Follow-up: What happens to a satellite orbiting Earth as it slows?
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
Using the Law of Gravitation:
we find that the ratio is .
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Note the
1/R2 factor
Gravitational Potential Energy
Gravitational potential energy of an object of mass
m a distance r from the Earth’s center:
(U =0 at r -> infinity)
Very close to the Earth’s
surface, the gravitational
potential increases linearly
with altitude:
Gravitational potential energy, just like all
other forms of energy, is a scalar. It
therefore has no components; just a sign.
Energy Conservation
(Remember: gravity is conservative force!)
Total mechanical energy of an
object of mass m a distance r from
the center of the Earth:
This confirms what we already know – as an object
approaches the Earth, it moves faster and faster.
Escape Speed
Escape speed: the initial speed a projectile
must have in order to escape from the Earth’s
gravity
from total energy:
If initial velocity = ve, then velocity at large distance goes to zero. If initial
velocity is larger than ve, then there is non-zero total energy, and the kinetic
energy is non-zero when the body has left the potential well
Maximum height vs. Launch speed
Speed of a projectile as it leaves the Earth,
for various launch speeds
Black holes
If an object is sufficiently massive and
sufficiently small, the escape speed
will equal or exceed the speed of light –
light itself will not be able to escape the
surface.
This is a black hole.
The light itself has mass (in the
mass/energy relationship of
Einstein), or spacetime itself is
curved
Gravity and light
Light will be bent by any
gravitational field; this can be
seen when we view a distant
galaxy beyond a closer galaxy
cluster. This is called gravitational
lensing, and many examples have
been found.
General Relativity
• Previous effects examples
• Clocks run slower in gravitational field
- must be taken into account for GPS systems
to work
- was missed in determining time for neutrinos
to go from CERN to Gran Sasso!
Kepler’s Laws of Orbital Motion
Johannes Kepler made detailed studies of the apparent motions of the planets over
many years, and was able to formulate three empirical laws
1. Planets follow elliptical orbits, with the Sun at one
focus of the ellipse.
Elliptical orbits are stable under inverse-square force law.
You already know about circular
motion... circular motion is just a
special case of elliptical motion
Only force is central gravitational attraction - but for elliptical
orbits this has both radial and tangential components
Kepler’s Laws of Orbital Motion
2. As a planet moves in its orbit, it sweeps out an
equal amount of area in an equal amount of time.
r
v Δt
This is equivalent to conservation of angular momentum
Kepler’s Laws of Orbital Motion
3. The period, T, of a planet increases as its mean
distance from the Sun, r, raised to the 3/2 power.
This can be shown to be a consequence of the
inverse square form of the gravitational force.
For a (near) circular orbit:
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
The weight that a scale reads is the normal force exerted by the
floor (or the scale). At the equator, you are in circular motion, so
there must be a net inward force toward Earth’s center. This
means that the normal force must be slightly less than mg. So the
scale would register something less than your actual weight.
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
By Newton’s Third Law, the forces
are equal and opposite.
Review
Newton’s law of universal gravitation
force of gravity between any two objects is attractive, acting
on a line between the two objects
• a sphere can be treated as a point mass at the sphere’s center
• above the surface of a planet (i.e., earth):
•
Orbital motion problems
•balance force of gravity with necessary centripetal acceleration
Gravitational potential energy
•chose U=0 point to be at r -> infinity
•escape
velocity: K >= U
•total
energy conserved
Force Vectors
A planet of mass m is a distance d from Earth.
Another planet of mass 2m is a distance 2d from
Earth. Which force vector best represents the
direction of the total gravitation force on Earth?
2d
e
d
d
a
m
b
c
2m
Force Vectors
A planet of mass m is a
distance d from Earth.
Another planet of mass 2m
is a distance 2d from Earth.
Which force vector best
represents the direction of
the total gravitation force on
Earth?
The force of gravity on the Earth
due to m is greater than the
force due to 2m, which means
that the force component
pointing down in the figure is
greater than the component
pointing to the right.
2d
2m
e
d
d
a
b
c
m
F2m = GME(2m) / (2d)2 =
GMm / d 2
Fm = GME m / d 2 = GMm / d 2
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
The weight that a scale reads is the normal force exerted by the
floor (or the scale). At the equator, you are in circular motion, so
there must be a net inward force toward Earth’s center. This
means that the normal force must be slightly less than mg. So the
scale would register something less than your actual weight.
Oscillations about
Equilibrium
Periodic Motion
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called
the Hertz:
Simple Harmonic Motion
A spring exerts a restoring force that is
proportional to the displacement from
equilibrium:
Displaced,
at rest
Moving, past
equilibrium point
Displaced,
at rest
Moving, past
equilibrium point
Displaced,
at rest
This is called “Simple
Harmonic Motion”
Simple Harmonic Motion
A mass on a spring has a displacement as a
function of time that is a sine or cosine curve:
Here, A is called
the amplitude of
the motion.
Simple Harmonic Motion
If we call the period of the motion T – this is the
time to complete one full cycle – we can write the
position as a function of time:
Time
Position
t =0
x=A
t=T
x = Acos(2π) = A
t = T/2 x = Acos(π) = -A
T
t = T/4 x = Acos(π/2) = 0
The position at time t +T is the same as the
position at time t, as we would expect.
Sine vs Cosine
x at t=0 : A
v at t=0 : 0
x at t=0 : 0
v at t=0 : >0
Harmonic Motion I
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass after a time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
Harmonic Motion I
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass after a time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
In the time interval T (the period), the mass goes through
one complete oscillation back to the starting point. The
distance it covers is A + A + A + A = (4A).
Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is the
net displacement of the mass after a
time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is the
net displacement of the mass after a
time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
The displacement is x = x2 – x1. Because the
initial and final positions of the mass are the
same (it ends up back at its original position),
then the displacement is zero.
Follow-up: What is the net displacement after a half of a period?
The Pendulum
A simple pendulum consists of a mass m (of
negligible size) suspended by a string or rod of
length L (and negligible mass).
The angle it makes with the vertical varies with
time as a sine or cosine.
How a pendulum is like the mass on a spring
Looking at the forces
on the pendulum bob,
we see that the
restoring force is
proportional to sin θ
The restoring force for
a spring is proportional
to the displacement
This is the condition for
simple harmonic motion
Approximation for sin θ
However, for small angles, sin θ and θ are
approximately equal.
θ (deg)
θ (rad)
sin(θ)
1
0.01745 0.01745
5
0.08727 0.08716
10
0.1745
0.1736
20
0.3491
0.3420
Pendulum for small angles = simple harmonic
for small angles of the
pendulum bob, the
restoring force is
proportional to θ
F = -mg θ = -mg s / L
The restoring force for
a spring is proportional
to the displacement
F= -kx
So: the motion of the angle of the pendulum is the same as
the motion for the mass on a spring, with k
mg/L
Uniform Circular Motion and Simple Harmonic
Motion
An object in simple
harmonic motion has the
same motion as one
component of an object in
uniform circular motion
Assume oscillation of mass on the spring has
the same period T as the circular motion of the
peg on the “record player”
Position of Peg in Circular Motion
Here, the object in circular motion has
an angular speed of:
where T is the period of motion
of the object in simple harmonic
motion.
The position as a function of time:
... just like the simple harmonic motion!
Velocity of Peg in Circular Motion
Linear speed v = Aω
x component:
Acceleration of Peg in Circular Motion
Linear acceleration a = Aω2
x component:
Summary of Simple Harmonic Motion
The position as a function of time:
From this comparison with circular motion, we can see:
The angular frequency:
The velocity as a function of time:
The acceleration as a function of time:
Speed and Acceleration
A mass on a spring in SHM has
a) x = A
amplitude A and period T. At
b) x > 0 but x < A
what point in the motion is v = 0
c) x = 0
and a = 0 simultaneously?
d) x < 0
e) none of the above
Speed and Acceleration
A mass on a spring in SHM has
a) x = A
amplitude A and period T. At
b) x > 0 but x < A
what point in the motion is v = 0
c) x = 0
and a = 0 simultaneously?
d) x < 0
e) none of the above
If both v and a were zero at
the same time, the mass
would be at rest and stay at
rest! Thus, there is NO
point at which both v and a
are both zero at the same
time.
Follow-up: Where is acceleration a maximum?
The Period of a Mass on a Spring
For the mass on a spring:
Substituting the time dependencies of a and x gives
and the period is:
The Period of a Mass on a Spring
Vertical Spring
Fs=kx
What if the mass hangs from a vertical spring?
new equilibrium position: x= -d = -mg/k
total force as a function of x:
d
x=0
x=-d
W=mg
with
Looks like the same spring, with a different
equilibrium position (x’=0 -> x = -d)
Simple harmonic motion is unchanged from the horizontal case!
Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces,
the total mechanical energy is conserved. For a
mass on a spring:
Since we know the position and velocity as
functions of time, we can find the maximum
kinetic and potential energies:
Energy Conservation in Oscillatory Motion
As a function of time,
So the total energy is constant; as the kinetic
energy increases, the potential energy decreases,
and vice versa.
0
Period of a Pendulum
A pendulum is like the mass on a spring, with k=mg/L
Therefore, we find that the period of a pendulum
depends only on the length of the string:
Physical Pendula
A physical pendulum is a
solid mass that oscillates
around its center of mass, but
cannot be modeled as a point
mass suspended by a
massless string. Examples:
Period of a Physical Pendulum
In this case, it can be shown that the period
depends on the moment of inertia:
Substituting the moment of inertia of a point mass a
distance l from the axis of rotation gives, as expected:
Damped Oscillations
In most physical situations, there is a
nonconservative force of some sort, which will
tend to decrease the amplitude of the oscillation,
and which is typically proportional to the speed:
This causes the amplitude to decrease
exponentially with time:
Damped Oscillations
This exponential decrease is shown in the
figure:
“underdamped” means that there is
more than one oscillation
Damped Oscillations
The previous image shows a system that is
underdamped – it goes through multiple
oscillations before coming to rest.
A critically damped system is one that relaxes
back to the equilibrium position without
oscillating and in minimum time;
an overdamped system will also not oscillate
but is damped so heavily that it takes longer to
reach equilibrium.
Driven Oscillations and Resonance
An oscillation can be driven by an oscillating driving
force; the frequency of the driving force may or may
not be the same as the natural frequency of the
system.
Driven Oscillations and Resonance
If the driving frequency
is close to the natural
frequency, the amplitude
can become quite large,
especially if the damping
is small. This is called
resonance.
Energy in SHM I
A mass on a spring oscillates in
simple harmonic motion with
amplitude A. If the mass is
doubled, but the amplitude is not
changed, what will happen to the
total energy of the system?
a) total energy will increase
b) total energy will not change
c) total energy will decrease
Energy in SHM I
A mass on a spring oscillates in
simple harmonic motion with
amplitude A. If the mass is
doubled, but the amplitude is not
changed, what will happen to the
total energy of the system?
a) total energy will increase
b) total energy will not change
c) total energy will decrease
The total energy is equal to the initial value of the
elastic potential energy, which is PEs = kA2. This
does not depend on mass, so a change in mass will
not affect the energy of the system.
Follow-up: What happens if you double the amplitude?
Spring on the Moon
A mass oscillates on a vertical
spring with period T. If the whole
setup is taken to the Moon, how
does the period change?
a) period will increase
b) period will not change
c) period will decrease
Spring on the Moon
A mass oscillates on a vertical
spring with period T. If the whole
setup is taken to the Moon, how
does the period change?
a) period will increase
b) period will not change
c) period will decrease
The period of simple harmonic motion depends only on the
mass and the spring constant and does not depend on the
acceleration due to gravity. By going to the Moon, the value of
g has been reduced, but that does not affect the period of the
oscillating mass–spring system.