When we have a single mass-spring system, the
motion of the mass is described by
, with solution of the form
When you stretch the spring from its equilibrium
position by a distance, A, and then let go, on
average, the mass doesnt go anywhere, it
oscillat
If two objects, with masses, m1 and m2 are
interacting with each other gravitationally, then
the change in potential energy as we move
the two objects relative to each other is given by:
So,
, and we chose to define the gravitational
potential energy of
Work vs Power
Work is the transfer of energy to a body by
the application of a force over a distance.
Power tells you how quickly energy is
transferred to a body.
Why does the rate of energy transfer
matter?
Power = dE/dt
The unit of power is the Watt.
1
The Center of mass of collection of particles is given by
Where
, the total mass of the system.
The total momentum of a collection of particles can be
calculated by multiplying the total mass of the system by
the velocity of the center of mass of the sys
Consider two carts, of masses m and 2m, at rest on an air track (no friction).
And connected by a compressed spring. After release, the spring expands to
its equilibrium length, and the two carts moves apart. After the carts losses
contact with the spring
If you have a pulley system as shown below, and you pull the rope a length d,
how much does the mass raise?
A. d
B. 2d
C. 3d
D. d/2
E. d/3
If you have a pulley system as shown below, and you pull the rope a length d,
how much does the mass raise?
A. d
B.
The formal definition of Newtons universal law of
gravitation is:
If two objects with masses M1 and M2 are a
distance r apart, then M1 will exert an attractive
force on M2, such that
Where is a radial unit vector directed from M 1 to
M2, and G is a consta
Consider two carts, of masses m and 2m, at rest on an air track (no friction).
And connected by a compressed spring. After release, the spring expands to
its equilibrium length, and the two carts moves apart. After the carts losses
contact with the spring
The celestial Sphere
The celestial sphere appears to rotate about the Earth once every 24
hours from east to west on an axis through the north and south poles
of the Earth.
Relative to the stars, the Sun appears to move as
if it is attached to another sph
Explosion example
A 10 g projectile is traveling east at 2 m/s. When it
explodes into 3 pieces. A 3 g fragment is shot due west
at 10 m/s while another 3 g fragment travels 40o north of
east at 12 m/s. What are the speed and direction of the
3rd fragment?
Rubber ball Example,
A 100 g rubber ball is dropped from a height of 2 m (~
6.5 ft) onto a hard floor. The plot below shows the force
that the floor exerts on the ball during contact. How
high does the ball bounce?
Fy
300 N
time
0.008 sec
Law of Conservat
Rubber ball Example,
A 100 g rubber ball is dropped from a height of 2 m (~
6.5 ft) onto a hard floor. The plot below shows the force
that the floor exerts on the ball during contact. How
high does the ball bounce?
Fy
300 N
time
0.008 sec
Now that we have
Consider two carts, of masses m and 2m, at rest on an air track (no friction).
You push each cart for 3 sec, exerting equal force on each. At the end of 3 sec,
the momentum of the light cart is
A. the momentum of the heavy cart.
B. the momentum of the hea
Your friend gives a paperback book a push and sending it sliding along the
table toward you. When the book is halfway between you and your friend
(and still moving), which of the following is NOT a force that is acting on the
book?
A. Gravity
B. Friction
Child on a cart example:
A father places his 20 kg child on a 5 kg cart
to which a 2 m long rope is attached. He
then pushes the cart and child around in a
circle, keeping the rope parallel to the
ground.
If the father kept the tension in the rope at a
co
Wooden box on wood floor example:
I push a 50 kg wooden box across a wood
floor at a steady speed of 2 m/s.
(a) How much force do I have to exert on the
box?
(b) If Istop pushing, how far will the box slide
before coming to rest?
Assume for wooden objects
Mass: Measured in kg, is an intrinsic
property of an object. An object has an
unique mass and it is always the same no
matter where in the universe it is.
Weight: Measured in N, changes depending
on the strength of gravity and whether or
not the object is
1-D kinematics example
Consider an object whose position on the x-axis varies
with time according to the equation:
m 2
x 20m (50 2 )t
S
a) Find the objects displacement between ti = 1 sec
and tf = 2 sec.
b) Find the object average velocity during the time
An object moves with simple harmonic motion. If the amplitude
and the period are both doubled, the objects maximum speed is,
A.
B.
C.
D.
E.
Quadrupled
Doubled
Unchanged
Halved
Quartered
An object moves with simple harmonic motion. If the amplitude
and the
Lets look at a system other than mass-on-aspring that execute simple harmonic motion.
Lets consider a Simple Pendulum:
is assumed to be small (< 25o)
Is the force
constant?
on
the
bob
is assumed to be small (< 25o)
Is the force on the bob constant?
Answ
, where S is the displacement from the equilibrium position.
is assumed to
be small (< 25o)
= 0 and is at its maximum
at the end points
is at its maximum and = 0
at the equilibrium position
You can make any solid object into
pendulum by pivoting it at
Consider a point particle of mass m, located a distance r
from the origin of the coordinate system that we are
using
r
r
If we apply a force, to the particle, then the torque the
particle experiences relative to the origin of our
coordinate system would b
The linear wave equation
,where
So,
But before we describe what kind of a velocity actually
represents, we note that:
T is really a material property of the string because it
depends on the elasticity of the string. The stiffer the
string, the larger the
In physics,
Work means the transfer of energy to a
body by the application of a force over a
distance.
Lets consider a force F, of magnitude |F|,
acting on a particle of mass m as the particle
is constrained to move along the x-axis from
an initial positi
The potential energy of a certain system as a
function of position is given blow:
U(x)
x1
x2
x3
x4
x
The potential energy of a certain system as a
function of position is given blow:
U(x)
F
F
F
F
x1
x2
x3
x4
x
The potential energy of a certain system as a
A force that changes magnitude with position:
The force from a spring, when the spring is
anchored at one end.
Suppose you have a spring whose equilibrium
length is ro. This is the length of the spring when it
is neither pushing nor pulling.
Lets define r
Potential energy is conformational, it
describes the energy state of a system
based on the relative position of
different parts of a system.
Kinetic energy is motional, it describes
the energy state of a system based on the
relative motion of different
Many important application of rotational dynamics
involve objects, such as pulleys that are connected via
ropes or belts to other objects
If we say the rope turns on the pulley
without slipping, then that means
there is no relative motion between
the rim
Which of the following is a representation of the vector shown?
y
4
A.-2+-2
3
B.-2+2
2
1
C.2+2
D.-2+3
E. None of the above
-5 -4 -3 -2 -1
-2
-3
1 2 3
x
Which of the following is a representation of the vector shown?
y
4
A.-2+-2
3
B.-2+2
2
1
C.2+2
D.-2+3
E
Swing Rod Example:
A 1 m long, 200 grams uniform rod is hinged at one end
and connected to a wall. It is held out horizontally, then
released. What is the speed of the tip of the rod as it hits
the wall?
Assume the width and thickness of the rod is neglig
2015 version of notes
Michael Cone gave the first two lectures on this topic.
I will start with a couple of clicker questions on that
material to act as a reminder and then pickup where
Cook left off.
For completeness, I keep the entire set of notes here.
Last lecture I did not really draw the Etot line on this
curve. I need to do it right.
Kinetic Energy
Potential Energy
Total Energy
Draw above graph and show
where these are
Chapter 15 Lecture notes
It turns out that solving this simple equation
F= - k x
Chapter 10 Lecture notes:
The angle is expressed in
radiens
The average angular speed, avg, of a rotating rigid
object is the ratio of the angular displacement to the
time interval.
The instantaneous angular speed is defined as the
limit of the average sp
Potential Energy:
Ask Class, what is potential energy?
Potential energy is energy determined by the
configuration of a system in which the components of the
system interact by forces.
Common misconception: Potential energy does not refer
to something that
Chapter 13 Lecture Notes
Have students write down some of the things Newton
was famous for:
The will probably write, Calculus, mechanics, gravity.
I then say Invisible Force Fields
Newton's law of gravity,
G is determined by the Cavendish experiment
Wikip