USSC2001 Energy Lecture 2 Dynamics and Statics Lecture 3 Potential and Kinetic Energy Wayne M....
-
Upload
tyrone-riley -
Category
Documents
-
view
223 -
download
0
Transcript of USSC2001 Energy Lecture 2 Dynamics and Statics Lecture 3 Potential and Kinetic Energy Wayne M....
USSC2001 Energy Lecture 2 Dynamics and Statics
Lecture 3 Potential and Kinetic EnergyWayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected] (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 objectmoves 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 velocitycannot change; that is, the body cannot accelerate.
What happens if two or more people pull on an object? This question leads to the following more precise statement
Note: force is a vector quantity – it has both magnitude and direction!
If no net force acts on a body, then the body’s velocitycannot 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.
Question: Is the ground an approximate inertial frame?Suggest a better inertial frame. Are there perfect ones?
Explain why a frame that moves with constant velocity with respect to an inertial frame is also an inertial frame. Hint: what is the relative velocitywith respect to the frame of an object that has velocity with respect to the frame w
v mF
iF
mF
iF
MASSImagine kicking a soccer ball and a similar sized stone (we recommend this as a virtual experiment!) what is the difference in their resulting velocities?
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.
The mass of an object is often called the inertial mass since the word inertia suggests resistance to change.
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: what constant horizontal force must be applied to make the object below (sliding on a frictionless surface) stop in 2 seconds?
Question: how are the net forces on a body along the horizontal and verticle directions related to the body’s acceleration?
s/m6v
SOME SPECIFIC FORCES
Gravitational Force: direction, weight, near Earth’s surface and far away, Newton’s law, g and G
Friction: direction, causes, why is heat generated
Normal Force: surfaces, constraints, orthogonality
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?
θh
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
mg
0F
g
θsina
θcosaF
l
sinb
cosbF
r
The gravity force is
VECTOR ALGEBRA FOR STATICS
Since the object does not move, Newtons’s second lawimplies that the net force on the object equals zero
)sin(cosmg
)cos(cosmg
b,a
0mgsinbsina
00cosbcosa
0
0FFFF
grlnet
Therefore
hence
DEFINITIONS OF ENERGY
1 The capacity for work or vigorous activity, strength2 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
[1] The American Heritage Dictionary of the English Language, Houghton Mifflin, Boston, 1992.
ENERGY-WORK-TOOL CONCEPT
(old form 5.5-7ky) Werg – to do
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,synergidsynergism,thaumaturge,work
[1] Appendix: PIE
(suffixed form) Werg-o
Greek: ergon energos energeia Latin: energia French:energie Germanic: werkam Old High German: werc, Old English: weorc,werc
http://www.bartleby.com/61/roots/IE577.html
(zero-grade form) Wigderivatives wrought, irk, wright
(o-grade form) Worgderivatives 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.
These ‘toys’ for children are examples of reversiblemachines – they can be used to lift and then lower theheavier weights using an arbitrarily small extra force that is sufficient to overcome the friction.
arm or lever
frulcrum
1m
3kg
3m1kg
In the balance shown below, the heavier/lighter mass may be lifted by lowering the lighter/heavier mass.
Here, as in the balance, the objects move in opposite directions by distances that are inversely proportional to their masses ?
LIFTING AS WORK - PULLEY
2kg
2m
1m
1kg
TUTORIAL 2
kg3
2. Compute the mass of the object on the side of the block that has length 2m.
kg?
m1 m2
GRAVITATIONAL POTENTIAL ENERGY
Consider a set of objects numbered 1,2,…,N
N21 gm,...,gm,gmhaving weights
and heights N21 y,...,y,yand initially at rest. If these objects interact so the total effect only changes their heights, then
N
1i iiygm remains unchanged.
the weighted sum of heights
The gravitational potential energy is conserved.
ELASTIC POTENTIAL ENERGY
Consider the reversible machine that uses a springto lower a weight by sliding it to the left
compressible spring
Initially, the two weights are placed on each side ofthe 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
final
initial
x
xdx)x(Force)energy(Work
ELASTIC ENERGY IN A SPRING
The figure below illustrates a spring being compressed
2
if
f
i
i )xx(dx)xx(kE2k
x
x
elastic
Initial (Relaxed) State Compressed State
fxx
Hook’s Law states that )xx(k)x(Fi
ix
hence
k = spring constant
POTENTIAL AND KINETIC ENERGY
is conserved (constant function of t)
Theorem: For a dropping weight, the total energy2ym
21mgy
2ym21 The quantity is called the kinetic
energy.Proof. Let E = E(t) denote the total energy. Then
)0(ETt
0tdt
dt
)t(dE)0(E)T(E
0yymymgdt)t(dE since gy and the fundamental theorem of calculus implies that
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 springthat moves horizontally, the total energy 2
2
12
2
1 xmkxE )t(cosa)t(x
xx 0
kE2a is conserved, thereforewhere
mkR
2T
is the angular frequency
is the phase, and
is the period.
is the amplitude
HARMONIC OSCILLATIONS
Consider a pendulum - an object ona swinging lever. Then for small
222
Lm θLθgE
)t(cosa)t(θ
θ Lθ
R,L
g,
LmgE2a