Transcript r - mrdsample

UNIT 5 (end of mechanics)
Universal Gravitation, energy, orbits, &
SHM
2 point masses will attract one another with equal and
opposite force given by:
Gm1m2
Fg 
2
r
Gravity acts as an
inverse square law
where ‘r’ is distance
from center to center
G = 6.67x10-11 Nm2/kg2
Consider a particle at the center of a spherical planet
of uniform density…
What is the net gravitational force on the particle
due to the planet?
What would you weigh at the center of the
planet?
What if the particle was moved halfway to the surface…
What is the net gravitational force
on the particle due to the planet?
Analysis of gravity inside a planet of mass M and
radius R. Assume uniform density.
Consider a particle of mass, m, located a
distance, r, from the center of the planet.
r
R
Gravity as a function of distance for a planet
of mass M and radius R.
What if the density of a planet is NON-uniform?
Consider a spherical planet whose mass density is given by
ρ = br (where b = constant in kg/m4) and has radius, a.
Calculate the gravitational force on an object of mass, m, a
distance r (where r < a) from the center of the planet.
For the outside of the planet the gravitational force
would be
G-field
The region around a massive
object in which another
object having mass would
feel a gravitational force of
attraction is called a
gravitational field. The
gravitational field's strength
at a distance r from the
center of an object of mass,
M, can be calculated with the
equation.
M
Gravitational Potential Energy (Ug)
We must derive an expression that relates the potential
energy of an object to its position from Earth since ‘g’ is NOT
constant as you move away from Earth. All previous problems
dealt with objects ‘close’ to the earth where changes in ‘g’
were negligible.
Consider a particle of mass, m, a very far
distance, r1 ,away from the center of the earth (M).
r1
r1
r2

r

r

F
All values of the potential energy are negative, approaching
the value of zero as R approaches infinity. Because of this
we say that the mass is trapped in an "energy well" - that
is, it will have to be given additional energy from an
external source if it is to escape gravity's attraction.
Ug between rod and sphere (non-point mass)
A thin, uniform rod has length, L, and mass, M. A small uniform
sphere of mass, m, is placed a distance, x, from one end of the
rod, along the axis of the rod. Calculate the gravitational
potential energy of the rod-sphere system.
The potential energy of a system can be thought of as the
amount of work done to assemble the system into the
particular configuration that its currently in. So, to compute the
potential energy of a system, we can imagine assembling the
system piece by piece, computing the work necessary at each
step. The potential energy will be equal to the amount of work
done in setting the system up in this way.
dM
Kepler's 1st Law: The Law of Elliptical Orbits
All planets trace out elliptical
orbits. When the planet is
located at point P, it is at the
perihelion position. When the
planet is located at point A it is
at the aphelion position. The
distance PA = RP + RA is called
the major axis.
Semi-major axis is half of the major axis. Eccentricity is
a measure of how "oval" an ellipse is.
Kepler's 2nd Law: Law of Areas
A line from the planet to the sun
sweeps out equal areas of space
in equal intervals of time where
dA/dt = constant
At the perihelion, the planet’s
speed is a maximum.
At the aphelion, the planet’s
speed is a minimum.
Angular momentum is conserved
since net torque is zero. Gravity
lies along the moment arm or
position vector, r.
vARA = vPRP
Kepler's 3rd Law: Law of Periods
The square of the period of any planet is proportional
to the cube of the avg distance from the sun.
Since the orbits of the planets in our
solar system are EXTREMELY close to
being circular in shape (the Earth's
eccentricity is 0.0167), we can set the
centripetal force equal to the force of
universal gravitation and,
Energy Considerations for circular Satellite Motion
Changing the orbit of a satellite
Calculate the work required to move an earth satellite of mass
m from a circular orbit of radius 2RE to one of radius 3RE.
How was the energy distributed in changing the orbit?
The initial and final
potential energies were:
The kinetic
energies were:
Escape Velocity
If an object of mass m is launched vertically upwards with
speed vo we can use energy to determine minimum speed to
escape planet’s gravitational field. Our parameters are when
the object can just reach infinity with zero speed.
Example: Two satellites, both of mass m, orbit a planet of
mass M. One satellite is elliptical and one is circular.
Elliptical satellite has energy E = -GMm/6r.
a) Find energy of circular satellite
M
r
b) Find speed of elliptical satellite at closest approach
c) Derive an equation that would allow you to solve
for R but do not solve it.
R
We can relate periodic motion with an object
undergoing uniform circular motion. Consider a
mass sitting on top of a rotating platform where
lights cast a shadow of mass on wall.
Shadow of rotating mass
will move back and forth on
wall exhibiting periodic
motion.
As the mass rotates
with a constant angular
speed, ω through an
angle θ in a time t, we
can describe the linear
movement across the
screen as
Since θ = ωt we can relate
the position of the object with
respect to time and angular
movement as
Trig Calculus
d
(sin x)  cos x
dx
d
(cos x)   sin x
dx
d
d
(cos ax)   a sin ax
(sin ax)  a cos ax
dx
dx
 sin xdx   cos x  C  cos xdx  sin x  C
1
 sin( ax)dx   a cos ax  C
1
 cos(ax)dx  a sin ax  C
Expressions for velocity and
acceleration
SIMPLE HARMONIC MOTION
A special case of periodic
When periodic motion possesses a restoring force
that is directly proportional to the negative of
displacement, this is said to be SHM.
Consider a mass oscillating on
a frictionless surface
Fs
v
Energy considerations for SHM
Example: A 0.500-kg mass is vibrating in a system in
which the restoring constant is 100 N/m; the amplitude
of vibration is 0.20 m. Find:
a. the energy of the system
b. the maximum kinetic energy and maximum velocity
c. the PE and KE when x = 0.100 m
d. the maximum acceleration
e. the equation of motion if x = A at t =0
a) E = ½
kA2
= 2.0J
b) Kmax = E = 2.0J
b) vmax = 2.83m/s
c) U = ½ kx2 = 0.5J
K = E - U = 1.5J
d) amax =| -ω2 A |= 40m/s2
e) x(t) = 0.20 cos (14.1t)
Does a vertical spring obey SHM due to
gravitational force?
If a mass is at rest, suspended from
a spring, the mass is in static
equilibrium. The only forces acting
on the mass are a gravitational
force down, mg, and the spring
force up, kxo, where xo is the
stretch of the spring at equilibrium.
A block is moving undergoing SHM having amplitude
A0. At instant block passes thru equilibrium, putty of
mass M is dropped vertically onto block and it sticks.
Find new amplitude and period
k
Is a simple pendulum a S.H.O.?
Physical pendulum
Any rigid body that is suspended from a fixed
point that does not pass through the COM
and oscillates about that point.
Pivot
d
A uniform rod of mass M and length L is pivoted about
one end and oscillates in a vertical plane. Find the
period of oscillation if the amplitude is small.
Torsional Pendulum
A mass suspended to a fixed
support by a thin wire can be
made to twist about its axis. This
is known as a torsional
pendulum. The mass attached to
the wire rotates in the horizontal
plane where θ is the angle of
rotation. It is the twisting of the
wire that creates a restoring
torque due to the resistance of
the wire to the deformation.