USC2001 Energy Lecture 2 Elastic Energy and Work

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sgTel (65) 6874-2749

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.

COMPRESSION OF A SPRING

show that if an object is placed on the top end of a vertically positioned spring whose bottom end isfixed, the spring will be compressed by a distance that is proportional to the mass of the object

Empirical measurements

Md kgM

INCREMENTAL COMPRESSION

If a small amount of mass is added on to the masson the spring, then the spring compresses by asmall (incremental) amount, the gravitationalpotential energy has been reduced by

Md kg

dMM M

ΔdMΔdΔMM

INCREMENTAL COMPRESSION

If a small amount of mass is added on to the masson the spring, then the spring compresses by asmall (incremental) amount, the gravitationalpotential energy has been reduced by

Md kg

dMM M

ΔdΔMM

INCREMENTAL DECOMPRESSION

If an small amount of mass is subtracted from themass on the spring, then the spring decompressesby a small (incremental) amount, the gravitationalpotential energy has been increased by

Md kg

dMM M

ΔdM

dM

REVERSIBILITYIf mass M is divided into N equal small masses and

added incrementally to the top of the spring the total change in gravitational potential energy is

2dMNNdMNΔdM 2 If mass M is divided into N equal small masses and

subtracted incrementally from the top of the spring the total change in gravitational potential energy is 2dMNNdMΔdM)1( 2 N

The net change in gravitational potential energy is

0ΔM as 0ΔM MdΔN kg

ELASTIC POTENTIAL ENERGY

The empirical facts together with the precedingargument shows that a compressed spring can be used to create, through decompression,

gravitational potential energy exactly equal to the gravitational potential energy required to compress it by slowly adding matter to the top of the spring. The amount of gravitational potential energy thatcan be obtained from decompressing a spring is called the elastic energy.

FORCE AND WORK

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

WORK TO COMPRESS A SPRING

The figure below illustrates a spring being compressed

2)()( 2 if

f

i

xxdxxFE k

x

x

elastic

Initial (Relaxed) State Compressed State

fx

Hook’s Law states that )()(ixxkxF

ix

therefore

k = spring constant

WORK AND FULCRUMS

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

fulcrum

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 ?

WORK AND PULLEYS

2kg

2m

1m

1kg

TUTORIAL 2

1. Show that the elastic energy of a spring having stiffness k that is compressed by a distance d is

22k

elastic dE 2. Explain the elastic energy in an elastic band that

is stretched. What happens if it is compressed ? How high can you shoot an elastic band ?

TUTORIAL 2

kg3

3. Compute the mass of the object on the side of the block that has length 2m. Hint: use that fact that if one object is moved down and the other is moved up the total gravitational potential energy remains the same.

kg?

m1 m2

TUTORIAL 2

4. 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.

5. 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).