Physics I
95.141
LECTURE 10
3/3/10
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Exam Prep Problem (Conical Pendulum)
• A small mass (m) suspended on a cord (l=1m) revolves in a circle of
radius r. (θ=30º)
– A) (10pts) Draw a free body diagram for the mass, labeling your
coordinate system
– B) (5pts) What is the acceleration of the ball, and in what direction?
– C) (5pts) What is the tangential velocity of the ball?
– D) (5pts) (5pts) If you want the frequency of oscillation to double, what
could you change the cord length to, assuming a constant angle?


m
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Exam Prep Problem (Conical Pendulum)
• A small mass (m) suspended on a cord (l=1m) revolves
in a circle of radius r. (θ=30º)
– A) (10pts) Draw a free body diagram for the mass, labeling your
coordinate system


m
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Exam Prep Problem (Conical Pendulum)
• A small mass (m) suspended on a cord (l=1m) revolves
in a circle of radius r. (θ=30º)
– B) (5pts) What is the acceleration of the ball, and in what
direction?


m
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Exam Prep Problem (Conical Pendulum)
• A small mass (m) suspended on a cord (l=1m) revolves
in a circle of radius r. (θ=30º)
– C) (5pts) What is the tangential velocity of the ball?


m
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Exam Prep Problem (Conical Pendulum)
• A small mass (m=2kg) suspended on a cord revolves in
a circle of radius r=0.5m. (θ=30º)
– D) (5pts) If you want the frequency of oscillation to double, what
could you change the cord length to, assuming a constant
angle?


m
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Banked Curves
• Can a car make a turn on a banked frictionless surface
without skidding? For speed v, radius R, what angle is
required?
• Coordinate system!!
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Example Problem
• A car goes around an unbanked curve (R=100m) at a
speed of 50m/s. The concrete/tire interface has a
coefficient of static friction of 1. Can the car make this
turn?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Example Problem
• A car goes around an banked curve (R=100m) at a
speed of 50m/s. Ignoring friction, what angle should the
curve be banked at to allow the car to make the curve?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Outline
•
•
•
Newton’s Law of Universal Gravitation
Weightlessness
Kepler’s Laws
•
What do we know?
–
–
–
–
–
–
–
–
–
–
–
–
–
Units
Kinematic equations
Freely falling objects
Vectors
Kinematics + Vectors = Vector Kinematics
Relative motion
Projectile motion
Uniform circular motion
Newton’s Laws
Force of Gravity/Normal Force
Free Body Diagrams
Problem solving
Uniform Circular Motion
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Law of Universal Gravitation
• We know that falling objects accelerate.
• We also know that if an object accelerates, there
must be a force acting on it.
• The Force that accelerates falling bodies is
gravity.
• But what exerts this force?
• Since all falling objects fall towards the center of
the Earth, Newton suggested that it is the Earth
itself which is exerting this Force.
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Law of Universal Gravitation
• What form does this Force take?
• 1) Dependence on distance
– Newton knew that the moon orbited the Earth.
– We know that a circular motion requires an inward radial
acceleration
v2
a Rmoon  
R
R  384,000km
T  27.3days
R  3.84 108 m
v moon  1022 m s
T  2.36 106 s
amoon  2.72 103 m s 2  2.78 104 g
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Law of Universal Gravitation
a moon
1

g
3600
Rmoon  3.84 108 m
REarth  6.38 106 m
REarth 1

RMoon 60
• So if the Force causing the moon to orbit the Earth, is
the same force which causes object to accelerate at the
surface of the Earth, then this Force goes as the inverse
square of the distance from the center of the Earth
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Law of Universal Gravitation
• The other thing we should note, is that the Force due to
gravity produces the same acceleration for ALL
OBJECTS, regardless of mass. So this is a Force which
must scale with the mass of the object.
• Symmetry also suggests that this Force must depend on
the mass of the Earth, or the second body.
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Law of Universal Gravitation
• Finally, Newton argues that if this is the Force
causing the moon to orbit the Earth, perhaps it is
also the Force causing the planets to Orbit the
sun. In fact, perhaps every mass exerts a
gravitational force on every other mass in the
universe.
• And we can write this force as
Fg  G
M object1 M object 2
R2
G  6.67 1011 Nm
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
2
kg 2
Direction of Gravitational Force
• Force is a vector, and therefore has a magnitude and
direction.
• Direction is along line connecting two masses.
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Example Problem
• Imagine 3 Blocks, of equal mass, placed at three
corners of a square. Draw the gravitational
Force vectors acting on each block.
1
2
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
3
Gravitational Attraction Between Two People
• Tom Cruise (160lbs) and Katie Holmes (118lbs)
are dancing (about 0.5 m apart). What is
attractive Force of Gravity between them?
160lbs  73kg
118lbs  54kg
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Gravity at Earth’s Surface
• If this law is correct, what should we get for Fg at
the Earth’s surface?
M EARTH  5.98  10 24 kg
REARTH  6.38  106 m
G  6.67  10 11 Nm
2
kg 2
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Satellites
• Imagine I throw a ball with some horizontal velocity vo.
• In previous chapters, we studied projectile motion, which
tells us the ball will accelerate towards earth and
eventually fall to Earth.
• But this is an approximation…the Earth is not flat, and
the Force of gravity is not “downward”, but towards the
center of the Earth.
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Satellites
• In order for an object to travel with uniform
circular motion, a radial Force is required.
• What speed would the ball need to have to travel
in a circular path at the surface of the Earth?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Satellites
• The uniform circular motion due to gravitation is
known as an “orbit”, and an orbiting object is
often referred to as a “satellite”.
• We can calculate the speed of a satellite for a
given orbital radius
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Geosynchronous Orbit
• A geosynchronous orbit, is one that orbits at the same
speed the Earth rotates, so that the satellite stays at the
same position with respect to Earth as it orbits. How can
we calculate the height of a satellite of mass M in
geosynchronous orbit?
T  1day  24hrs  86,400 s
2R
mv 2 GM E m
v
F

T
R
R2
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Weightlessness
• Remember, we define weight as the magnitude of the
Force of gravity acting on an object.
• At the surface of Earth this is mg.
• But we measure weight, by measuring the Force a mass
exerts on a scale.
• Imagine we are weighing a mass in
an elevator.
• If the elevator is at rest, or moving
at a constant velocity, what does
the scale read?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Weightlessness
• Now, if the elevator is accelerating upwards with a=g/2.
• What is the mass’ apparent weight?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Weightlessness
• Now, if the elevator is in freefall (a=-g).
• What is the mass’ apparent weight?
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Weightlessness
• The “weightlessness” one experiences in orbit is
exactly the same one would feel in a freely
falling elevator.
• Remember, the Force causing a satellite to orbit
is the Force of Gravity….in essence, the satellite
is freely falling.
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Kepler’s Laws
• Kepler’s laws of planetary motion
– Empirical (Experimental)
• Kepler’s 1st Law: The path of each planet around the sun
is an ellipse with the sun at one focus
a  semi-major axis
b  semi-minor axis
e  eccentricity
e  es/a
es
a
b
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Kepler’s Laws
• Kepler’s 2nd Law: Each planet moves so that an
imaginary line drawn from the sun to the planet sweeps
out equal areas in equal periods of time
t
t
t
t
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Kepler’s Laws
• Kepler’s 3rd Law: The ratio of the squares of the periods
of any two planets revolving around the sun is equal to
the ratio of the cubes of their semi-major axes.
2
3
T12 T22
 T1   s1 
 3
    
3
s1
s2
 T2   s2 
s2
s1
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Newton’s Synthesis
• Newton was able to derive Kepler’s
experimental laws from his Universal Law of
Gravity.
• Perturbations
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Kepler’s 3rd Law: Circular Orbit
• Earth’s orbit around sun has e=0.017, almost
circular. Can we prove Kepler’s third law for
circular orbits?
 F  ma
v2
ar 
R
2R
T
v
95.141, F2010, Lecture 10
Department of Physics and Applied Physics
Example Problem
• If we know the Earth’s distance from the sun,
can we determine the sun’s mass from Kepler’s
laws?
rES  1.5 1011 m
TE  1 yr  3.15 107 s
95.141, F2010, Lecture 10
Department of Physics and Applied Physics