## Introduction

- The knowledge of stretching materials when forces are applied is important particularly in the construction industry.
- It helps engineers to determine the strength of the materials to be used for specific work.
- This topic deals with study of how materials behave when stretched and the relationship between the extent of stretching and stretching force.
- The pioneer of the topic is the physicist Robert Hooke.

## Characteristics of Materials

### 1. Strength

- It is the ability of a material to resist breakage when under stretching, compressing or shearing force.
- A strong material is one which can withstand a large force without breaking.

### 2. Stiffness

- Refers to the resistance a material offers to forces which tend to change its shape or size or both.
- Stiff materials are not flexible and resist bending.

### 3. Ductility

- This is the quality of a material which leads to permanent change of shape and size.
- Ductile materials elongate considerably when under stretching forces and undergo plastic determination until they break e.g.lead, copper, plasticine.

### 4. Brittleness

- This is the quality of a material which leads to breakage just after elastic limit is exceeded.
- Brittle materials do not undergo extension and break without warning on stretching. E.g.blackboard chalk, bricks, castiron, glass, and dry biscuits.

### 5. Elasticity

- This is the ability of a material to recover its original shape and size after the force causing deformation is removed.
- The materials with this ability are called elastic e.g. rubber bands, spring, and somewires.
- A material which does not recover its shape but is deformed permanently is called plastic e.g.plasticine.

## Hooke’s Law

- Hooke’s law relates the stretching force and extension produced.
- It states that “
**for a helical spring or any other elastic material, extension is directly proportional to the stretching force,provided elastic limit is not exceeded**”

i.e. F∝e; F=ke,

Where k is the constant of proportionality called spring constant. - Sl unit of spring constant is the newton per meter (N/m).
- Spring constant is defined as the measure of stiffness of a spring.
- Graphically, Hooke’s law can be expressed as below.
- The graph of stretching force against extension, for material that obeys Hooke’s law, is a straight line through the origin. The gradient(slope) of such a graph gives the spring constant of the spring used.

Gradient (slope)=^{change in F}/_{change in e }= spring constant

S=^{ΔF}/_{Δe}= k - If the stretching force exceeds a certain value, permanent stretching occurs.
- The point beyond which the elastic material does not obey Hooke’s law is called elastic limit.
- A point beyond which a material loses its elasticity is called yield point.
- Along OE the spring(or elastic material) is said to undergo elastic deformation.
- Along EA the spring is said to undergo plastic deformation

### Factors Affecting Spring Constant

- Type of material making the wire
- Length of the spring
- The number of turns per unit length of the spring
- The diameter (thickness) of the spring
- The thickness of the wire

**Examples**

- A spring stretches by 1.2cm when a 600g mass is suspended on it. What is its spring constant?

Solution - The figure below shows a spring when unloaded, when supporting a mass of 80g and when supporting a stone.Study the diagrams and use them to determine the mass of the stone.

Solution**=0.048kg (this is the mass of the stone)** - A spiral spring produces an extension of 6mm when a force of 0.3N is applied to it. Calculate the spring constant for a system when two such springs are arranged in:
- Series
- Parallel

Since the two springs will share the weight, extension of the system is^{1}/_{2}x 6mm = 3mm

Spring constant of the system, k_{P}is

k_{p }=^{F}/_{e}=^{0.3N}/_{0.003m}=100Nm^{-1}

- Series
- The data below represents the total length of a spring as the load suspended on it is increased.

**Weight, W (N)**0.5 1.0 1.5 2.0 2.5 3.0 **Total length, L (x10**^{-2}m)7.5 8.0 8.5 9.0 9.5 10.0 - Plot a graph of total length (y-axis) against weight.
- Use the graph to determine
- The length of the spring
*The length of the spring is that when force acting on it is zero. From the graph it is 7.1x10*^{-2}m - The spring constant,k.

- The length of the spring

- Plot a graph of total length (y-axis) against weight.

### Compressing a Spring

- Compression refers to change in length that occurs when a spring is squeezed from its two ends.
- A sketch of length against compression for a spring which obeys Hooke’s law is as below.
- Beyond the point E, the turns of the spring are virtually pressing onto one another and further increase in force achieves no noticeable decrease in length.

**Exercise**

- The figure below shows a simple apparatus for studying the behavior of a spring when subjected to forces of compression.

Describe how the apparatus may be used to obtain readings of compression force and corresponding length of spring. - In a similar experiment the following readings were obtained.

Force of compression, F (N) 0.0 5.0 10.0 15.0 17.5 22.5 25.0 30.0 35.0 40.0 45.0 50.0 Length of spring, L(cm) compression 14.50 13.00 11.50 10.00 9.25 7.75 7.00 6.50 6.25 6.00 6.00 6.00

Plot a graph of:- Compression forces versus length of the spring and from the graph determine the minimum force that will make the spring coils to just come into contact.
- Compression forces versus compression of spring and from the graph determine the spring constant.

### Work Done in Stretching or Compressing a Spring

- The area under force versus extension graph represents work done in stretching the spring.

Area under the graph=^{1}/_{2}Fe,

where F is the force applied and e the extension attained.

From Hooke's law, F=ke

Workdone=^{1}/_{2}(ke)e=^{1}/_{2}ke^{2}

**Exercise**

Two springs of negligible weights and of constants k_{1}= 50Nm^{-1 }and k_{2}=100Nm^{-1} respectively are connected end to end and suspended from a fixed point. Determine

- The total extension when a mass of 2.0kg is hung from the one end
- The constant of the combination.
- Work done in stretching each spring (elastic potential energy of each)

## Revision Exercise

- State Hooke’s law
- Define the following terms
- Elasticity
- Elastic material
- Plastic deformation
- Spring constant
- Stiffness
- A stiff material
- Elastic material
- Yieldpoint

- A 60g mass is suspended from a spring. When 1.5g wire is added, the spring stretches by 1.2cm. Given that the spring obeys Hooke’s law, find:
- The spring constant
- The total extension of the spring

- A piece of wire of length 12m is stretched through 2.5cm by a mass of 5kg. Assuming that the wire obeys Hooke’s law
- Through what length will a mass of 12.5kg stretch it?
- What force will stretch it through 4.0cm?

- The following readings were obtained in an experiment to verify Hooke’s law using a spring.

Mass(g) 0 25 50 75 100 125 Reading(cm) 10 11.5 12.5 13.5 14.4 16.0 Force(N) Extension(mm) - For each reading, calculate:
- The value of the force applied
- The extension in mm

- Plot a graph of extension against force. Does the spring obey Hooke’s law?
- From the graph determine:
- The elastic limit(mark on graph)
- The spring constant
- The weight of a bottle of ink hung from the spring if the reading obtained is 12cm
- The extension in mm when a force of 0.3N is applied
- The scale reading in cm for a mass of 0.02kg

- For each reading, calculate:

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