USSC2001 Energy
Lecture 2 Dynamics and Statics
Lecture 3 Potential and Kinetic Energy
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email matwml@nus.edu.sg
Tel (65) 6874-2749
CONTENTS
These vufoils contain lectures 2&3 and tutorial 2.
They use the Euclidean (synthetic) and analytic
geometry of space/time developed in lecture 1 and
tutorial 1 to describe Newton’s laws of motion and
the energy concept. They are primarily concerned
with physics as opposed to geometry.
The concept that force was required to move an object
originated before Sir Isaac Newton (1642-1727) [who,
independently with Leibniz (1646-1716) invented The
Calculus], however Newton quantified how an object
moves under the influence of forces by proposing
three laws of motion.
NEWTON’S FIRST LAW
If no force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
Note: force is a vector quantity
– it has both magnitude and direction!
What happens if two or more people
pull on an object? This question leads
to the following more precise statement
If no net force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
STATICS
Why is this object static (not moving) ?


mg
Hint: What are the forces acting on this object?
What is the net force acting on this object?
INERTIAL REFERENCE FRAMES
Newton’s first law does not hold in all reference
frames – however there are frames for which it
holds and these frames are called inertial frames.
Explain why a frame Fm that moves with constant

velocity v with respect to an inertial frame Fi is also
an inertial frame. Hint: what is the relative velocity
with respect to the frame Fmof an object that has

velocity w with respect to the frame Fi
Question: Is the ground an approximate inertial frame?
Suggest a better inertial frame. Are there perfect ones?
MASS
Imagine kicking a soccer ball and a similar sized
stone (we recommend this as a virtual experiment!)
what is the difference in their resulting velocities?
The mass of an object is often called the inertial mass
since the word inertia suggests resistance to change.
This observation leads to the conjecture that the ratio
of the masses of two objects is equal to the inverse of
the ratio of their accelerations when the same force is
applied.
Questions: How we can take the ratio of these vectors?
What happens if a different force is applied? What is
the mass of an object formed by joining two objects?
NEWTON’S SECOND LAW
The net force on a body is equal to the product of
the body’s mass and the acceleration of the body.
Question: how are the net forces on a body along the
horizontal and verticle directions related to the body’s
acceleration?
Question: what constant horizontal force must
be applied to make the object below (sliding on
a frictionless surface) stop in 2 seconds?
v  6m / s
SOME SPECIFIC FORCES
Gravitational Force: direction, weight, near Earth’s
surface and far away, Newton’s law, g and G
Normal Force: surfaces, constraints, orthogonality
Friction: direction, causes, why is heat generated
Tension: direction
TUTORIAL 2
1. Compute the direction of acceleration, normal
force, net force, and acceleration of the object
falling down an inclined frictionless plane shown
below. How long does it take to fall from the top to
the ground if the initial velocity equals zero?
h
θ
NEWTON’S THIRD LAW
When two bodies interact, the forces on the bodies
from each other are always equal in magnitude and
opposite in direction.
Question: how can this fact be used to compute
the ratio of the masses of two objects?
Question: the momentum of a body is the product of
its mass and velocity, the momentum of a system of
bodies is the sum of their momenta, show that when a
system of bodies interacts the momentum of the
system is invariant.
VECTOR ALGEBRA FOR STATICS
The tension forces are

Fl 
 a cos θ 
 a sin θ 
The gravity force is



Fg 

Fr 
 0 
 mg
b cos 
 b sin  
VECTOR ALGEBRA FOR STATICS
Since the object does not move, Newtons’s second law
implies that the net force on the object equals zero

  
0

Fnet  Fl  Fr  Fg 
0
 a cos   b cos   0  0
Therefore
hence
a sin   b sin   mg  0
mg cos 
mg cos 
a
,b
cos()
sin( )
DEFINITIONS OF ENERGY
[1] The American Heritage Dictionary of the English
Language, Houghton Mifflin, Boston, 1992.
1 The capacity for work or vigorous activity, strength
2 Exertion of vigor or power
‘a project requiring a great deal of time and energy’
3 Usable heat or power
‘Each year Americans consume a high percentage
of the world’s energy’
4 Physics. The capacity of a physical system to do
work -attributive. energy – conservation, efficiency
ENERGY-WORK-TOOL CONCEPT
[1] Appendix: PIE http://www.bartleby.com/61/roots/IE577.html
(old form 5.5-7ky) Werg – to do
(suffixed form) Werg-o
derivatives handiwork,boulevard,bulwark, energy, erg, ergative,-urgy;
adrenergic,allergy,argon,cholinergic,demiurge, dramaturge,endergonic,
endoergic,energy,ergograph,ergometer, ergonomics,exergonic,exergue,
exoergic,georgic,hypergolic,lethargy,liturgy,metallurgy,surgery,synergids
ynergism,thaumaturge,work
Greek: ergon  energos  energeia  Latin: energia  French:energie
Germanic: werkam  Old High German: werc, Old English: weorc,werc
(zero-grade form) Wig
derivatives wrought, irk, wright
(o-grade form) Worg
derivatives organ, organon (= tool), orgy
LIFTING AS WORK - BALANCE
Lifting mass is a form of work. It requires energy.
One source of this energy is to lower another mass.
3kg
1kg
1m
arm or lever
3m
frulcrum
These ‘toys’ for children are examples of reversible
machines – they can be used to lift and then lower the
heavier weights using an arbitrarily small extra force
that is sufficient to overcome the friction.
LIFTING AS WORK - PULLEY
In the balance shown below, the heavier/lighter mass
may be lifted by lowering the lighter/heavier mass.
1m
2kg
1kg
2m
Here, as in the balance, the objects move in opposite
directions by distances that are inversely proportional
to their masses ?
TUTORIAL 2
2. Compute the mass of the object on the side of the
block that has length 2m.
? kg
3kg
1m
2m
GRAVITATIONAL POTENTIAL ENERGY
Consider a set of objects numbered 1,2,…,N
having weights
and heights
gm1 , gm 2 ,..., gm N
y1, y 2 ,..., y N
and initially at rest. If these objects interact so
the total effect only changes their heights, then
the weighted
sum of heights
i1
N
remains
gm i y i unchanged.
The gravitational potential energy is conserved.
ELASTIC POTENTIAL ENERGY
Consider the reversible machine that uses a spring
to lower a weight by sliding it to the left
compressible spring
Initially, the two weights are placed on each side of
the fulcrum so as to balance the lever.
What happens as either weight is moved to the left?
Where did the gravitational potential energy go?
WORK POTENTIAL ENERGY
To do work on a static system (consisting of massive
objects and springs), such as lifting objects or
compressing springs, means to increase the net
potential energy. This requires force. The work, which
measures the increase in potential energy, is related to
the force and distance (for one dimensional motion) by
Work (energy ) 
x final
x
initial
Force( x )dx
ELASTIC ENERGY IN A SPRING
The figure below illustrates a spring being compressed
k = spring constant
xi
x xf
Initial (Relaxed) State
Compressed State
Hook’s Law states that F( x )  k ( x  x i ) hence
xf
E elast ic 

xi
2
k
k ( x  x i )dx  ( x f  x i )
2
POTENTIAL AND KINETIC ENERGY
Theorem: For a dropping weight, the total energy
2
1
mgy  my is conserved (constant function of t)
2 1
2
my is called the kinetic energy.
The quantity
2
Proof. Let E = E(t) denote the total energy. Then
dE( t ) dt  mgy  myy  0
since
y  g
and the fundamental theorem of calculus implies that
E ( T )  E ( 0) 
t  T dE( t )
t  0
dt
dt  E (0)
TUTORIAL 2
3. Compute the required spring constant of a spring
gun that is is to be compressed by 0.1m and capable
of shooting a 0.002kg projectile to a height of 100m.
Assume that the mass of the spring is zero and that
no frictional forces are present.
4. Compute the energy required to compress 1 cubic
meter of gas to one half of its original volume at
constant temperature if the original pressure equals
101300N / square meter. Hint: use the fact that the
pressure is inversely proportional to the volume
(and therefore increases as the gas is compressed).
HARMONIC OSCILLATIONS
For an object attached to a spring
that moves horizontally, the total
2
energy E  1 kx 2  1 mx

x

x
0
2
2
is conserved, therefore x ( t )  a cos ( t
where
is the amplitude
a  2E k
 k m
R
2

T

is the angular frequency
is the phase, and
is the period.
 )
HARMONIC OSCILLATIONS
Consider a pendulum - an object on
a swinging lever. Then for small θ

2
2
Lm

E
gθ  Lθ
2
θ

 θ( t )  a cos (t  )
a
2E ,
Lmg

g
,
L
R
L