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Basic Mechanics
Units
• To communicate the result of a measurement for a quantity, a unit
must be defined
• Defining units allows everyone to relate to the same fundamental
amount
• Dimensional Analysis:
– Both sides of an equation must have the same dimensions
– Can be used to verify equations, answers
• Example v = d/t
Velocity and Acceleration
• Speed: Time rate of change of position.
• Velocity: Speed in a specific direction. Velocity is
specified by a Vector.
• Acceleration: The time rate of change of velocity (speed
and/or direction). Acceleration is also a vector.
10 m/s2
Gravitational acceleration of the Earth = 9.8 m/s2
Constant velocity  Zero acceleration
Constant acceleration in the same direction as v  Increasing velocity
Constant acceleration opposite of v  Decreasing velocity
• Velocity: the average velocity over an infinitesimal (very
short) time interval
æ Dx ö dx
lim
vº
ç ÷º
Dt ® 0 è Dt ø dt
• Acceleration: the average acceleration
infinitesimal (very short) time interval
over
æ Dv ö dv
lim
aº
ç ÷º
Dt ® 0 è Dt ø dt
an
Instantaneous Velocity
Instantaneous velocity is
slope of tangent at any point
1D Kinematics Equations for Constant Acceleration
chosing t0 = 0;
Dx x - x0
v=
=
Þ
Dt
t
Dv v - v0
a=
=
Þ
Dt
t
v0 + v
v=
2
x = x 0 + vt
v = v 0 + at
(average velocity)
Substituting v = v 0 + at
v 0 + v 2v 0 + at
v=
=
Þ v = v 0 + 12 at
2
2
Now substituting v in x = x 0 + vt gives Þ
x = x 0 + v 0t + at
1
2
2
1D Kinematics Equations for Constant Acceleration
x = x 0 + vt
v0 + v
v=
2
(average velocity)
v - v0
Þt=
a
Combining equations :
æ v + v 0 öæ v - v 0 ö
1 2 2
x = x0 + ç
֍
÷ = x 0 + (v - v 0 ) Þ
è 2 øè a ø
2a
v - v = 2a(x - x 0 )
2
2
0
0
x (meters)
v2
200
20
150
15
100
10
- v0 = 2a x
0
0
5
10
t (seconds)
15
20
0
a (m/s )
20
2
150
15
1.5
100
10
1
50
5
0.5
10
t (seconds)
20
5
10
t (seconds)
15
20
5
10
t (seconds)
15
20
2
v (m/s)
200
5
15
5
50
2
x (meters)
0
10
t (seconds)
(linear)
0
0
5
v (m/s)
x = v0t + 1/2 at2 (parabolic)
v = v0 + at
0
15
20
0
0
5
10
t (seconds)
15
20
0
0
Momentum and Angular Momentum
• Momentum (운동량) is the combination of both mass and
velocity. Mathematically:
p = mv
• Momentum is a vector, and it is always conserved (can be
neither created nor lost, only transferred).
• Angular Momentum (각운동량) is momentum of
spinning/rotating or revolving. It is also a vector and is also
conserved. Mathematically:
L = mv  r
m
m
v
v
2m
m
0.5v
v
m
0.5m
v
2v
Momentum and Angular Momentum
• Momentum (운동량) is the combination of both mass and
velocity. Mathematically:
p = mv
• Momentum is a vector, and it always conserved (can be
neither created nor lost, only transferred).
• Angular Momentum (각운동량) is momentum of
spinning/rotating or revolving. It is also a vector and is also
conserved. Mathematically:
L = mv  r = mvr sinq
Angular momentum: m(vxr)=mvr sinq
Newton's Laws of Motion
(뉴턴의 운동 법칙)
1. If the sum of the forces on an object is zero, then the
object's velocity will remain constant. Since neither speed
nor direction of motion changes that we always have
motion in a straight line when there are no forces.
2. Acceleration experienced by an object is directly
proportional to the amount of force exerted on it.
F = ma
3. Action and Reaction: When two bodies interact, they create
equal and opposite forces on each other.
First law
V: constant
Second law
Third law
Action and Reaction
• The Newton's law tells us that if there are no forces acting on an
object then its motion is in a straight line. So how do we produce
circular motion?
• We have to apply a force. The force constantly change the direction
of of the object's motion without altering the speed (not velocity!) of
the object. If we swing the object around in a circle above our heads,
we can feel the tension (장력) as we swing the object. If we were to
suddenly cut the string the object would continue off on a tangential
(접선) line from the spot it was released.
Universal Law of Gravity
• One day Newton was out sitting under an apple tree
looking at the Moon.
• When he saw an apple fall, he wondered
if the force that was keeping the Moon in
orbit around the Earth was the same
force that made the apple fall to the
ground. It would mean that the Moon is
always falling toward Earth in the way
we described above. He also then
considered that it must be gravity that
holds planets in orbit about the Sun.
If the Moon is falling a little towards the Earth, just like an apple dropped on the surface, why
does the Moon travel around the Earth in an orbit instead of falling onto it?
The way to answer this question is to consider what would happen if there was no gravity acting:
Q: How far would the Moon travel in a straight line in 1 sec if there were no gravity acting?
A: About 1000 meters.
At the same time, the Moon's motion along this straight-line path would also cause it to move
away from the Earth.
Q: How far away from the Earth would the Moon move in 1 second if no gravity were acting?
A: About 0.00136 m
In round numbers, the amount the Moon falls towards the Earth due to gravity is just enough to
offset the straight-line path it would take if gravity were not acting to deflect it. This balance
effectively closes the loop.
We have therefore reached a conclusion:
The Moon is really falling around the Earth!
Universal Law of Gravity
• He studied Kepler's law of planetary motion and his own
laws of motion to come up with a Universal Law of
Gravitation.
• Given two masses, M1 and M2 the force between them is
given by
M1 M 2
F = -G 2
r
where r denotes the distance between two masses, and G
(=6.67x10-11 m3 kg-1 s-2) means a gravitational constant.
Example (1)
• The Fall of an Apple
Stand on the Earth and drop an apple.
• What is the force of the Earth on the apple?
F = GMearthMapple / Rearth2
• What is the apple's acceleration (Newton's 2nd Law of Motion):
aapple = F / Mapple = GMearth / Rearth2
= 6.710-11  (6.01024) / (6.4107)2
= 9.8 m/s2
• Note that the mass of the apple (Mapple) had divided out of the equation.
This means that the acceleration due to gravity is independent of the
mass of the apple.
Mapple
REarth
MEarth
Example (2)
• Newton's 3rd Law of Motion states that all forces come in equal yet opposite pairs.
What force does the the apple apply in return upon the Earth?
F = GMearthMapple / Rearth2
• How much does the Earth accelerate towards the apple?
aearth = F / Mearth = GMapple / Rearth2
• This can be rewritten to give the acceleration of the Earth in terms of the
acceleration of the apple towards the Earth as
aearth = aapple x (Mapple/Mearth)
where aapple=9.8 m/s2, and the ratio of the Mass of the apple to the Mass of the
Earth is very small number. For a typical 200g apple, this works out to be about
10-25 m/s2.
Kepler's Three Law
1. The orbit of every planet is an ellipse with the sun at one of the foci.
2. A line joining a planet and the sun sweeps out equal areas during equal
intervals of time as the planet travels along its orbit. This means that the
planet travels faster while close to the sun and slows down when it is far
from the sun.
a
3. The squares of the orbital periods of planets are proportional to the cubes
of the semi-major axes (the "half-length" of the ellipse) of their orbits.
a3=K T2
ellipse
Diameter
Mass
Semi-major
axis (AU)
Rotational
period
Sun
Terrestrial
planets
Gas
planets
Dwarf
planets
Orbital
period (yr)
-
Mercury
0.38
0.06
0.39
59
0.24
Venus
0.95
0.82
0.72
-243
0.62
Earth
1
1
1
1
1
Mars
0.53
0.11
1.52
1.03
1.88
Jupiter
11.2
318
5.20
0.41
11.9
Saturn
9.45
95
9.54
0.43
29.5
Uranus
4.01
14.6
19.22
-0.72
84.3
Neptune
3.88
17.2
30.06
0.67
165
Ceres
0.08
0.0002
2.77
0.38
4.60
Pluto
0.19
0.002
39.48
-6.39
248
a3=K T2
Energy
• Kinetic Energy (운동 에너지): the energy of motion for
any object with mass, m, and velocity, v, its kinetic energy,
KE, is given by
1 2
K E = mv
2
• Potential Energy (위치 에너지): the amount of work that
can potentially be done. For an object, with mass, m, under
the gravitational attraction of another with mass, M, and a
distance, r, away the potential energy, PE, is
mM
K E = -G
r
• Total Mechanical Energy = KE+PE
m
m
v
v
2m
m
0.5v
v
2m
m
v
1/3v
2/3v
칠석(七夕)
Altair
Vega