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 matwml@nus.edu.sgTel (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