Elasticity: Hooke’s law, Tensile stress, Tensile strain and Young Modulus

LEARNING OBJECTIVES
By the end of the lesson students should be able to:
 Analyse the term ‘Elasticity’
 State Hooke’s law
 Distinguish between the following terms: Elastic limit, Yield Point, Breaking
Point, Maximum load, Tensile stress, strain and Young Modulus
 Differentiate between Tensile stress and strain
 Perform an experiment to establish the relationship between force F and
extension, e

INTRODUCTION
ELASTICITY
This is the ability of a substance to regain its original shape and size after being
distorted by an external force.
Elastic Material is one that regains its original shape and size after the
distorting force has been removed e.g. spring, string, Rubber
The shape and size of an elastic material can be distorted by stretching or
compressing it.
Hooke’s Law
When a force is applied to an elastic spring or string, the length of the material
increases by a certain amount. If the force is removed, the length returns to its
former value.
Thus, Hooke’s law states that provided the elastic limit of an elastic material is
not exceeded, the extension e of the material is directly proportional to the load
or applied force, F
Activity: Experimental Demonstration of Hooke’s Law
Aim: Experimental investigation on Hooke’s law
Apparatus: A steel spring, a 100g mass hanger, 12, 100 g masses, a retort
stand, a boss and clamp, a clamp, a metre rule, an s-hook, a pointer,
safety goggles, a slotted base.

Method
1. Set up apparatus as shown in the diagram. Use a slotted base to secure the
metre stick and ensure that it is vertical.
2. Attach the mass hanger s -hook and pointer to the lower end of the spring.
The pointer should just touch the metre rule.
3. Read the pointer value from the metre rule. Record this length in a suitable
table. This is the initial length of the spring for zero mass. We can neglect
the mass of the hanger.
4. Add a 100g slotted mass to the hanger. Record the mass in kg in the table.
5. Read the new position of the pointer on metre rule. This is the stretched
length of the spring. Record this length in the table.
6. Calculate the stretching force = weight of masses: W = mg.
7. Calculate: extension = stretched length – original length.
 8. Repeat the procedure by adding 100g masses in steps of 100g up to 1200g.
Record the new stretched length each time by reading the position of the
pointer on the metre rule. Subtract the original length from the new
stretched length to calculate each extension.
Error
The main cause of error in this experiment is reading the stretched length of the
spring.
The metre rule scale should be read at eye level directly opposite the pointer.
Use the slotted base to ensure that the metre rule is vertical.
Results
Initial length of the spring = X cm.

Mass in Stretching force (F in Stretched length in Extension e in

kg N) cm cm

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Graph
Plot a graph of stretching force, F in N on the y-axis, against extension, e in cm
on the x-axis.
Join the points with a line of best fit.
Conclusion
We can see from the graph that as the stretching force increases the extension
of the spring also increases. This agrees with our prediction.
In fact, since the line of best fit is a straight line through the origin, up to a
certain point, we can be even more precise.
We can say that the stretching force F is directly proportional to the extension e
up to a limit known as the limit of proportionality.
This is known as Hooke’s law.
Equation
Stretching force F = spring constant, k x extension,e
F = ke
stretching force f
The gradient of the graph = =spring constant , k
extension e
Q: Calculate the gradient of the line.
If we continue increasing the load until the spring no longer returns to its original
length when the load is removed, a graph of extension e against load l will be
shown
Elastic limit is the limit of force beyond which the stretched wire does return to
its original length when the stretching force is removed.
Yield Point is the point beyond the elastic limit in which the elastic material has
yielded all its elasticity permanently and has become plastic.
The change from elastic to plastic region is shown by a sudden and rapid
increase in the extension for any slight increase in load. The maximum
extension is reached at the breaking point where the material may finally snap
or break entirely.
According to Hooke’s law, the force f required to stretch or compress an elastic
material is directly proportional to the extension or compression e.
Thus:
f ∝e
f =k e
Where k = constant of proportionality called elastic constant or force
constant or stiffness of the material. If f is in newtons and e is in metres the
elastic constant is given by:
f
k= (N m )
−1
e
Elastic constant or stiffness of an elastic material is the force required to give
unit extension.
The working of the spring balance is based on Hooke’s law. Here f = mg, the
weight of the body which is proportional to e, the extension of the spring.
ILLUSTRATION
Q: A force of 0.8 N stretches an elastic spring by 2 cm. Find the elastic constant
of the spring
Solution
From Hooke’s law:
f =k e
k × 2m
0.8 N= =k × 0.02 m
100
0.8 −1
k= =40 N m
0.02
Q: A force of 2 N stretches an elastic material by 30 mm. What additional force
will stretch the material 35 mm? Assume that the elastic limit is not exceeded
Solution
f =k e
30
2=k ×
1000
 2000
k=
30
Let the force stretching the material 35 mm be F
35 2000 35
f =k × = × =2.33 N
1000 30 1000
∴ additional force=2.33 N −2.00 N=0.33 N
Young’s Modulus
Let a wire of length l (m) and cross-sectional area A (m2) be extended through e
F
(m) by a force F (N). The ratio of the force to the area is called the stress or
A
tensile stress of the material. The unit is Nm-2.
F
Stress=
A

The tensile strain is defined as the ratio of the extension to original length ( el ). It
has no unit since e and l is the same unit. Strain is a measure of deformation and
the stress describes the distorting force.
e
Strain=
l
Stress is proportional to strain provided elastic limit is not exceeded. The
constant of proportionality in this statement of Hooke’s law is known as Young’s
modulus of elasticity (Y)
' stress
Youn g s Modulus=
strain
F
A
Y = (N m−2 )
e
l
Case Study: Hooke's Law
 A material obeys Hooke’s Law if its extension is directly proportional
to the applied force (load)
 The Force v Extension graph is a straight line through the origin (see
“Extension and Compression”)
 This linear relationship is represented by the Hooke’s law equation

Hooke’s Law
  The constant of proportionality is known as the spring constant k
Worked Example
A spring was stretched with increasing load.
The graph of the results is shown below.

Q: What is the spring constant?

The Spring Constant, K

 k is the spring constant of the spring and is a measure of
the stiffness of a spring
o A stiffer spring will have a larger value of k
  It is defined as the force per unit extension up to the limit of
proportionality (after which the material will not obey Hooke’s law)
 The SI unit for the spring constant is N m-1
 Rearranging the Hooke’s law equation shows the equation for the spring
constant is

Spring constant equation

 The spring constant is the force per unit extension up to the limit of
proportionality (after which the material will not obey Hooke’s law)
 Therefore, the spring constant k is the gradient of the linear part of a
Force v Extension graph

Spring constant is the gradient of a force vs extension graph

EVALUATION
1. Which one of the following substances is not elastic?
A. Iron B. Copper C. Brass D. Modelling clay
2. Which one of the following Substances is not plastic?
A. Butter B. Iron C. Plasticine D. Wax
3. When the distance between any two particles of a body remains
constant when subjected to a deforming force of any magnitude, the body
is said to be_________
(A) An Elastic Body (B) A Rigid Body (C) A Plastic Body (D) An
isotropic body
4. When a body undergoes continuous deformation when subjected to
external deforming forces and retains its deformed size or shape when the
deforming forces are removed, the body is said to be_________
(A) A Rigid Body (B) An Elastic Body (C) An Isotropic Body (D) a
plastic body
5. With rise in temperature, the Young s modulus of elasticity of a
material_______
(A) Increases (B) Decreases (C) Does Not Change (D) may increase
or decrease
6. Which one of the following does not affect the elasticity of a
substance______?
(A) Hammering (B) Adding Impurity In The Substance (C) Changing
the Dimensions (D) Change of temperature
7. According to Hooke’s law of elasticity, within elastic limits if the stress
is increased, the ratio of stress to strain________
(A) Increases (B) Decreases (C) Becomes Zero (D)
remains constant
8. When a body resists any change in its size or shape, when subjected to
external deforming forces and tries to regain the original size and shape,
when the deforming forces are removed, the body is said to be a__________
(A) A Plastic Body (B) An Elastic Body (C) A Rigid Body (D) an
isotropic body
9. Which one of the following substances possesses the highest elasticity?
(A) Rubber (B) Steel (C) Glass (D) Aluminium
10. The dimensional formula for stress is__________
(A) [Stress] = [L‐¹ M¹ T-² (B) [Stress] = [L-² M-¹ T-²] (C) [Stress =
[L¹ M¹ T¹] (D) [Stress] = [L² M¹ T-¹]
11. The dimensional formula for stress is the same as that for_________
(A) Work (B) Power (C) Pressure (D) force
12. Stress is the___________
(A) Applied Force per Unit Area of Cross Section(B) Internal Restoring
Forces per Unit Area of Cross Section (C) Deformation Produced In the
Body per Unit Area
(D) Compression of the body per unit area
13. The dimensional formula for shearing strain is___________
(A) L⁰ M¹ T-¹ (B) L¹ M⁰ T-² (C) L⁰ M⁰ T⁰ (D) L²M¹ T-¹
14. Strain has no units and dimensions because_______
(A) It Is A Ratio (B)It Is A Ratio Of Two Similar Quantities (C) It Is A
Constant Number (D) both size and shape of the body are not changed
during deformation
15. The S.I. unit of stress is_________
(A) Newton/Metre (B) Newton²/Metre (C) Newton/Metre² (D)
Newton/metre³
16. Shearing strain is given by________
(A) Deforming Force (B) Shape of The Body (C) Angle of Shear (D)
Change in volume of the body
17. The Young's modulus for a plastic body is________
(A) One (B) Zero (C) Infinity (D) Less than one
18. When the intermolecular distance increases due to tensile force,
then________
(A) There Is No Force between The Molecules (B) There Is A Repulsive
Force Between The Molecules(C) There Is An Attractive Force Between
The Molecules (D) there is zero resultant force between the molecules
19. If a material is heated and annealed, then its elasticity is__________
(A) Increased (B) Decreased (C) Not Changed (D)
Becomes zero
20. When impurities are added to an elastic substance, its
elasticity_________
(A) Increases (B) Decreases (C) Becomes Zero (D) may
increase or decrease
21. Strain has_________
(A) No Units But Only Dimensions (B) Only Units But No Dimensions (C)
No Units, No Dimensions But A Constant Value (D) no units, no
dimensions but a variable value
22. If by applying a force, the shape of a body is changed, then the
corresponding stress is known as_________
(A) Tensile Stress (B) Bulk Stress (C) Shearing Stress (D)
Compressive Stress
23. The relative change is the dimensions or shape of a body when it is
subjected to external deforming forces is called________
 (A) Strain (B) Stress (C) Modulus Of Elasticity (D) Poisson's ratio
24. Out of the following materials, whose elasticity is independent of
temperature________
(A) Copper (B) Invar Steel (C) Brass (D) Silver
25. The ratio of the change in dimension at right angles to the applied
force to the initial dimension is known as________
(A) Young's Modulus (B) Poisson's Ratio (C) Lateral Strain
(D) Shearing strain

26. For Hooke's law to hold well, the intermolecular distance must
be__________ as compared to the equilibrium distance.
(A) Much More (B) Zero (C) Much Less (D) approximately
same

27. A wire is stretched through 2 mm by a certain load. The extension

produced in a wire of the same material with double the length and radius
with the same load will be________
(A) 2 mm (B) 4 mm (C) 1 mm (D) 0.5 mm
28. Which of the following formula is used to estimate tensile stress?
A. area / force B. force / area C. force + area D. force
× area

29. The unit of strain is

A. newton B. joule C. Pascal D. no unit

30. A comparison of such a change caused by the stress with the original
shape, volume or length is called
A. stress B. strain C. density D. elasticity

31. The property of a body to restore its original size and shape as the
deforming force ceases to act is called
A. energy B. floating C. elasticity D. density

32. If stress produces a change in the length of an object then the strain is
termed as
A. zero strain B. constant strain C. former strain D. tensile strain

33. A body of mass 11kg is suspended from a ceiling fan by an aluminum

wire of length 2m and diameter 2mm. Calculate the elastic energy stored
in the wire [Young's modulus of aluminum is 7.0 x 10 10Nm-2, g = 10ms-
, π = 3.142]`
2

A. 1.1 x 10-1J B. 5.5 x 10-2J C. 1.1 x 10-4J D. 5.5 x 10-8J

34. An elastic material has a length of 36cm when a load of 40N is hung
on it and a length of 45cm when a load of 60N is hung on it. The original
length of the string is
A. 12cm B. 20cm C. 18cm D. 15cm
35. The point beyond which a stretched spring does not return to its
original length is called the
A. Breaking point B. Elastic limit C. Spring constant D. Elasticity
point E. Release point

ASSIGNMENT
1(a). State Hooke’s law. Describe how you can verify it in the laboratory.
b. A force of 40N stretches a wire through 3.0 cm. What force will stretch
it through 5.0 cm and through what length will a force of 100 N stretch it?
What assumption have you made?

2(a) Define Young modulus of elasticity;

(b) A spiral spring extends from a length of 10.0 cm to 10.01 cm when a
force of 20 N is applied on it. Calculate the force constant of the spring.

3(a) Define the following terms: Stress, Strain, Young’s modulus.

(b) State the law that relates the stress and the strain in an elastic
material.
An elastic string of length l is stretched through a length e by a force F.
The area of cross-section of the string is A and its Young’s modulus is E.
Deduce the expression between F, E, e and l
4(a).What do you understand by the following set of bodies?
 A Rigid Body
 An Elastic Body
  An Isotropic Body
 A Plastic Body
b. Explain the following terms:
Elastic limit, Proportionality limit, Yield point and Breaking point
With the aid of a diagram, describe briefly the behaviour of a wire that is
gradually loaded to its breaking point
5 (a) Sketch the graph of relation between the extension of a spiral spring
and the load attached to it when it is gradually loaded up the elastic limit.
What is the physical significance of the area under the graph?
b. A spiral spring is 20cm long when at 10N.What is the length of the
spring at 2N provided Hooke’s law is obeyed.