Energy Stored In An Elastic Material

CONTENT

• Force in a bar
• Energy stored in a wire
• Energy stored per unit area

FORCE IN A BAR

When a bar is heated and then prevented from contracting as it cools, a considerable force is exerted at the end of the bar. Given a bar of a young modulus E, a cross sectional area A, a linear expansivity of magnitude α and a decrease in temperature of Ө,then

E = stress/strain = F/A/e/L

=F/A х L/e

E = FL/Ae

F = EAe/L ………………………….1

Recall, Linear expansivity α  = change in length/original length х temperature change

α = e/L х Ө

e = αLӨ

F = EAαeLӨ/eL

F = EAαӨ ……………………………2

EVALUATION

1. Show that F = EAαӨ when a bar is heated and then prevented from contracting as it cools
2. A steel rod of cross sectional area 2cm² is heated to 100˚C and then prevented from contracting when it cooled to 10˚C. find the force exerted on the steel = 12 х 10^-6/K and young modulus is 2 х10¹¹N/m²

ENERGY STORED IN A WIRE

The application of force on any wire provided the elasticity limit is not exceeded is proportional to the extension provided.

Consequently, the force in the wire has increase from zero to F

Average force = F + O/2 = F/2

Recall work done = Average force х distance

W = f/2 x e = ½ Fe ………………………….3

Substitute eqn 1 into 3

W = EAe²/2L……………………………….4

ENERGY STORED PER UNIT VOLUME

W_V =½ X Stress X Strain…………………..5

This implies that Wv = Ee²/2L²

Substance which lengthens considerable and undergo plastic deformation until they break are known as ductile substance.  E.g lead, copper, wrought iron.

Substance which break after the elastic limit is reached are known as brittle substance e.g glass and high carbon steel. It should be noted  that brass, bronze and many alloys appears to have no yield point. By this we mean that this material increase in length beyond the elastic limit as the lead is increased without the sudden appearance of a plastic stage.

EVALUATION

1. A uniform steel wire of length 4m and are of cross section 3×10^-6m² is extended by 1mm. Calculate the energy stored in the wire if the elastic limit is not exceeded (young modulus = 2×10¹¹n/m²)
   1. A spiral spring is compressed by 0.02m. Calculate the energy stored in the spring if the force constant is 400n/m²

READING ASSIGNMENT

Read more on elasticity – New school physics (pg96-98)

GENERAL EVALUATION

1. A body moving with uniform acceleration has two points [5,15 and 20,60] on the velocity time graph of its motion. Calculate its acceleration.
2. A body of mass 50Kg is moving in a circular path of radius 4m with a uniform speed 20m/s. determine its centripetal force

WEEKEND ASSIGNMENT

1. Young modulus of elasticity is the  ratio of tensile  stress to tensile strain  provide the load does not exceed the (a)  breaking point (b)  elastic limit (c) yield  point (d)  stress limit
2. A piece of rubber 10cm long stretches 6mm when a load of 100N is hung from it. What is the strain? (a) 60 (b) 0.6 (c) 6×10-² (d) 6×10-³
3. A force of 1.8N extends a wire by 0.4cm. What force will extend the wire by 1.25cm if the elastic limit is not exceeded. (a) 5.625N  (b)4.500N  (c) 3.125N (d) 2.25N
4. Calculate the work done to stretch a string by 40cm if a force of 10N produces an extension of 4cm in it. (a) 0.2J (b) 2.0J (C) 20.0J (d) 200.0J
5. Which of the following is not the consequence of a force field? (a) weight (b) surface tension (c) gravitational pull (d) magnetic force

THEORY

1. A steel rod of cross sectional area 2cm² is heated to 100˚C and then prevented from contracting when it cooled to 10˚C. find the force exerted on the steel = 12 х 10^-6/K and young modulus is 2 х10¹¹N/m²
2. A uniform steel wire of length 4m and are of cross section 3×10^-6m² is extended by 1mm. Calculate the energy stored in the wire if the elastic limit is not exceeded (young modulus = 2×10¹¹n/m².