L1 c
L2 b
1. c
2. b
3. c
4d
5. b
6a
7a
8b
9b
10 d
11 e
12 e
13 c
14 c
15 b
16 d
17 a
18 c
19 c
20 c
21 c
22 d
23 e
24 b
25 c
26 d
27 c
28 a
29 b
30 a
31 d
32 b
33 b
34 b
35 c
36 c
37 a
38 a
39 c
40 b
Clearing up some confusion on vectors
The rela5onship between the magnitude of the average velocity and the displacement
Magnitude of the
displacement
!
r =x i + y j
!
2
2
r = ( x ) + ( y)
!
!
r x y
v avg =
=
i+
j = v avg,x
Elastic Collisions
Lets examine a particular kind of elastic collision: a head-on collision, in which all
of the velocities lie along the same line. In this case we will choose the x-axis.
Before the collision:
After the collision:
mA
mB
mA
mB
v Ai
v Bi
v
and Are Vectors
We have emphasized the fact that and are the rotational equivalent of the translational
velocity v and acceleration a. Another property they have in common with their translational
counterparts is that both and are vectors. We already kno
Rolling without Slipping
We will consider a particular kind of motion called rolling without slipping. The object
rolling could be a disk, a hoop, a sphere, a cylinder, etc, but the important feature of this
motion is that the object is not skidding or sl
Work and Energy
We are now ready to develop some new ideas which describe the
motion of objects. These ideas include:
Kinetic energy
Work
Power
Potential Energy
Total Energy
Probably the most important idea is that of Conservation of Energy. This ide
Dynamics of Rotational Motion
Lets imagine we have a rigid body, which is fixed to an axis about which it can rotate.
Lets also imagine that this object is initally at rest. How do we get the object to rotate?
We need to apply a force. However, there is m
Rotation of a Rigid Body
Lets consider the motion of the object below. It is a rigid (non-deformable) object, which is fixed
to a point about which it can freely rotate. Lets initially focus our attention on the point P, which is
located on the object, at
Center of Mass
Center of Mass:
Up till now, we have dealt with single particles
These were things like a box sliding down a plane, a person running, a ball thrown
through the air
We idealized the motion of these complicated objects
But real objects ar
Work and the Dot-Product
Earlier, we wrote down the expression for work in terms of the component of the force
along the displacement, multiplied by the displacement:
!
m
F
!
d
W = (Fcos)d
There is a more general way to write this, using the Vector Dot-Pr
Conservative Forces
Let us lift an object of mass 10kg up a distance of 5.0m, and then lower back 5.0m,
at constant speed. How much work is done by gravity?
In lifting the object, the work done is:
y
T
d=5.0m
In lowering the object, the work done is:
x
mg
A Block Attached to a Spring
Imagine that we take a block of mass m and attach it to a spring. The block is on a smooth
surface and the arrangement is as shown below. We assume that initially
the block is stationary, and that the spring is ideal - it i
Gravitation
We have already talked about the force of gravity:
mg
Where:
m is the mass of the person
g is a constant
The main assumption needed to use the above expression for the force
of gravity is that the person needed to be very close to the surface
Uniform Circular Motion
Uniform circular Motion
An object moving at constant sped in a circle
The magnitude of the velocity remains constant
The direction of the velocity changes continuously
!
v
!
v
Since acceleration is rate of change of velocity:
!
Measuring Reaction Time: How might you do this?
Have a friend hold a ruler between your open fingers, and let them
drop it. You close your fingers as soon as you notice your friend
drop it. How can this measure reaction time?
To solve this problem:
1) Dra
How to solve 1D constant
acceleration problems
How to solve problems with constant acceleration in
1 dimension:
1) Dont Panic!
2) Draw a simple picture of the problem.
3) Write what you know: x0 = initial position
x = position at the time t
v0 = initial v
General Motion in 2D
General Motion in 2 or 3 Dimensions
Consider 2-dimensions first (easy to generalize to 3-D)
y
(x1, y1 )
!
r
1
!
r2
Path of particle as a function of time = r(t)
(x 2 , y2 )
x
The vector r1 tells you where the object is with respect to
What is Physics
What is Physics
Physics is a attempt by men and women to explain
why the physical world around us behaves in the way
that it does.
Physics provides the foundation upon which other physical sciences are based.
Basic science such as physic
Motion in a vertical plane
Motion in a vertical plane
Weve done this:
v0
Now lets do this:
v0
Question: Two objects, both at the same height H. One is dropped from rest.
Other is thrown horizontally with V0=10m/s. Which hits the ground first?
V0=10m/s
H
B
Force and Motion
Force and Motion
Up to this point, we have simply described the motion of objects.
Now, we will try to determine what made (or is making) the objects
move the way they do. To do this, we need to introduce the ideas
of force and mass.
Forc
Vector Introduction
Vectors
Vector: A quantity that has both magnitude and direction.
Displacement, velocity, and acceleration are examples of vectors
In 1-dimension, the direction is specified by a +/- sign
x1
0
x2
1
2
3
x2
0
4
x(m)
x1
1
2
3
4
x(m)
x =
More on Gravity
More on the Force Due to Gravity
Consider an object in free fall:
y
mg
We said earlier than an object in
free fall has an acceleration downward
of 9.80m/s2. We will now connect this
to the force of gravity:
x
Apply Newtons 2nd Law to
the o
Types of forces
Types of Forces
1. Force of Gravity:
This is often referred to as the weight of an object. It is the attractive force
of the earth. And is always directed toward the center of the earth. It has
a magnitude equal to the mass of the object t
Angular Momentum
We will begin our discussion of angular momentum by first considering an extended object
with moment of inertia I, and rotating with angular velocity .
It turns out that angular momentum can be defined for a single particle relative to an
Friction
Friction
When an object slides over a surface, there is usually some resistance to
this sliding. This is due to a frictional force, and is always directed opposite
the intended direction of motion along the surface. The size of this force
is depe
Special Relativity
Newtonian Mechanics is in general very successful at describing the motion of objects.
It should be noted that Newton's laws of motion are valid in all inertia frames of
reference. This is Newton's Principle of Relativity. An inerti