# Physics Tutorial: Types of Forces III (Elastic Force and Tension)

In this Physics tutorial, you will learn:

• What are elastic objects?
• How to determine whether an object is elastic or not?
• What happens when an object is not elastic?
• What are the factors affecting the elasticity of objects?
• Where does the elastic force depend on?
• How to represent graphically a situation involving the elastic behaviour of matter?
• What is limit of elasticity and breaking point?
• What is tension?
• Where does the tension differ from elastic force?

## Introduction

Think about the physical features of rubber bands. In which of the following situations they are good to use and in which they are not?

1. To pull a heavy object along a horizontal surface
2. To keep a rolled sheet of paper in a cylindrical shape
3. To tie you hair up the head
4. To hang a precious lamp board

For each case, explain your opinion.

## Elastic and non-Elastic Objects

From experience, we know that some objects regain their original shape after a distorting force has been exerted on them. When this distorting force stops acting, the objects turn to their original dimensions. Below we have mentioned some examples in this regard.

1. When you press a balloon or a ball, they lose the original round shape but immediately after releasing them, both the balloon and the ball take again their original shape.
2. When you stretch a rubber band (as mentioned in the Introduction paragraph), it becomes longer but when you release it, the rubber band returns to its original length.
3. A spring is compressed when a force acts on it but if the force stops, the spring will turn again to its original length, etc.

All the above examples deal with elastic objects, i.e. objects that after being deformed due to the action of a distorting force, return to their original shape when this distorting force stops acting.

Examples of elastic materials include steel, rubber, sponge, bamboo tree, etc.

On the other hand, objects that remain deformed after a distorting force has acted on them, are known as non-elastic (plastic) objects. Some examples of non-elastic objects include:

1. Working with clay. It is known that clay takes the shape according to the direction of the distorting force used and it remains so when this distorting force stops acting on it.
2. Unlike steel rods, an iron rod remains in that position after being bent (steel is an elastic material while iron is not).
3. A wooden stick cannot regain its original shape after breaking it in pieces. This means wood is not an elastic material, and so on.

In fact there are three kind of materials in regard to their elastic behaviour. They are:

1. Perfectly (absolutely) elastic. These materials turn exactly at their previous shape after the restoring force stops acting on them, even if they have been used multiple times.
2. Absolutely non-elastic (plastic). These materials keep exactly the actual shape after the restoring force stops acting on them.
3. An intermediate category in which objects try to regain their original shape after the restoring force stops acting. However, they remain deformed at a certain extent. Certain alloys, in which an elastic and a non-elastic material are mixed, manifest such a behaviour.

## Elastic Force and the Factors Affecting It

Suppose you have a perfectly elastic spring hanged in the vertical position as shown in the figure. The original (unstretched) length of the spring is denoted by L0. If we hook an object of mass m on it, as shown in the figure below, the final length of the stretched spring becomes L.

We denote by x the difference between the actual position L and the original position L0 of the spring. It represents the linear deformation of the spring. Therefore, we have

x = L - L0

Look at the figure below. Figure 2. An elastic spring before and after hanging an object on it

Even during the compression, we use the same reasoning as well. The only difference is that in compression, the linear deformation x is negative because L < L0. Therefore, we say the linear deformation x is a vector quantity as it involves direction.

Since the object m is in equilibrium, it is clear that there are two equal and opposite forces acting on it. One is the gravitational force which acts downwards as it tries to send the object on the Earth surface, while the other is the resistive force of the spring which acts upwards as the spring tries to resist to any deformation caused by external factors (such as the hanged object in the specific case). This resistive force is known as Elastic (or Restoring) Force, Fe as it tries to restore the spring's original length (look at again the figure above).

Since there is equilibrium, the resultant force acting on the object m is zero. Therefore, we have

Fe + Fg = 0

or

Fg = -Fe

The restoring force, also known as the Elastic Force is in the opposite direction to the deformation x. Its equation is

### Equation 1 (Hooke's Law)

Fe = -k × x

where k is a constant known as "coefficient of elasticity" or "spring constant" which depends on the properties of material, thickness of spring etc. We can determine the unit of k by rearranging the formula (1). Thus, we have

k = - Fe/x ⟹ the unit of k = [N/m]

The equation (1) is known as the "Hooke's Law."

### Example 1

A 300 g object is hanged on a spring as in the figure (2). As a result, the spring stretches by 12 cm. What is the magnitude of the spring constant expressed to the nearest whole number? Take g = 9.81 m/s2.

### Solution 1

First, we convert the units into the SI ones. Thus, m = 300 g = 0.300kg and x = 12cm = 0.12m.

Since there is equilibrium, we have

Fg = - Fe

where Fg = m × g and Fe = -k × x.

Thus, substituting the known formulae of gravitational and elastic force, we obtain

m × g = -(-k × x)
m × g = k × x
k = m × g/x
= 0.300 × 9.81/0.12
= 24.525 N/m
≈ 25 N/m

## The Graphical Representation of Hooke's Law. Limit of Elasticity.

The equation (1) written in the scalar form, i.e. Fe = k × x shows a linear function whose graph is a slope that starts from the origin like the one shown in the figure below. As long as the graph is linear, the elastic object involved (spring, rubber band etc.) shows a perfectly elastic behaviour. Therefore, the spring obeys the Hooke's law. However, when the pulling force is too large, the spring may not return to its original position. We say the force used has exceeded the limit of elasticity. For example, some dynamometers don't show anymore the value zero when they are not in use because heavy objects that exceed the limit of elasticity have been hanged on them.

When the object hanged is too heavy, the spring breaks into two pieces. The breaking point is represented through the letter B in the graph above.

### Example 2

The elastic force vs deformation graph for a spring is shown in the figure below. 1. What is the value of the spring constant in N/m?
2. What is the minimum force to be used so that the spring does not turn again to its original position?
3. What is the minimum force needed to break the spring in two pieces?

### Solution 2

1. The spring constant is calculated considering only the part OA of the graph as only in this part the Fe vs x graph is linear, i.e. it obeys the Hooke's Law.
Since the graph is linear, we can take two points on it and calculate the slope as discussed in the Kinematics chapter when we calculated the instantaneous velocity. For simplicity, we can take the origin O and the point A (at the origin the spring is unstretched). Thus, we have:
k = FeA - Fe0/xA - x0
= 70N - 0N/5cm - 0cm
= 70N/5cm = 70N/0.05m
= 1400 N/m
1. The minimum force to make the spring permanently loose its elastic properties is shown at the point A in the graph. The corresponding value of the elastic force in that point is 70 N. This value represents the elastic limit of the spring.
2. The spring breaks at the point B as there is no more graph after this point. The corresponding value of the force is 80 N. This means when a 80 N force is used in the spring, it breaks in two pieces.

## What is Tension?

Tension is another resistive force that appears when a string is pulling a load. In other words, tension is the resistive force of a string against deformation caused when it is pulling an object. It is obvious that tension is in the opposite direction of the pulling force as shown in the figures below. In the first case (figure a) the tension acts against the gravity, while in the second case (figure b), it acts against friction. Tension has an intermolecular nature, i.e. it appears when the molecules of the string (which is a solid and as such, its molecules have a strong connection between them) display resistance towards extension.

Tension is different from elastic force, as it appears when the string is non-elastic. Therefore, no visible extension appears in the string, but even in cases when this extension is noticeable, the string does not turn back to its original length, and as a result, it is considered as a non-elastic material. Therefore, the Hooke's Law cannot be used to calculate the value of tension.

### Example 3

What is the minimum tension present in a rope if it is used to lift a 30 kg object? Take g = 9.81 m/s2.

### Solution 3

The situation is described in the figure below. The minimum tension occurs when its value is equal to that of the gravitational force. Thus, we have

T + Fg = 0
Tmin = -Fg
= -m × g
= -30kg × 9.81N/kg
= -294.3N

The sign minus means that the tension acts in the opposite direction of gravity.

## Summary

Elastic objects, are made from materials which after being deformed due to the action of a distorting force, return to their original shape when this distorting force stops acting.

Examples of elastic materials include: steel, rubber, sponge, bamboo tree, etc.

On the other hand, objects that remain deformed after a distorting force has acted on them are known as non-elastic (plastic) objects.

Examples of non-elastic materials include: clay, iron, glass, etc.

There are three kind of materials considering their elastic behaviour. They are:

1. Perfectly (absolutely) elastic. These materials regain their previous shape after the restoring force stops acting on them, even if they have been used multiple times.
2. Absolutely non-elastic (plastic). These materials keep exactly the actual shape after the restoring force stops acting on them.
3. An intermediate category in which objects try to regain their original shape after the restoring force stops acting. However, they remain deformed at a certain extent. Certain alloys, in which an elastic and a non-elastic material are mixed, manifest such a behaviour.

The equation used to determine the magnitude of the elastic force (restoring force) is known as the Hooke's Law. Mathematically, it is written as

Fe = -k × x

where k is a constant known as "coefficient of elasticity" or "spring constant" which depends on the properties of material, thickness of spring etc., and x = L - L0 is the extension or compression of the spring, i.e. the difference between the actual length L and the initial length L0 of the spring.

The graphical representation of Hooke's Law shows a linear function whose graph is a slope that starts from the origin. When the Fe vs x graph is not linear, the force used in the spring has exceeded the limit of elasticity. As a result, the spring cannot turn anymore to its original length when the force that has caused the extension stops acting.

When the force is very large, the spring breaks in two pieces and therefore, no Fe vs x graph exist anymore.

Tension is another resistive force that appears when a string is pulling a load. In other words, tension is the resistive force of a string against deformation caused when it is pulling an object. It is obvious that tension is in the opposite direction of the pulling force.

Tension has an intermolecular nature, i.e. it appears when the molecules of the string (which is a solid and as such, its molecules have a strong connection between them) display resistance towards extension.

Tension is different from elastic force, as it appears when the string is non-elastic. Therefore, no visible extension appears in the string, but even in cases when this extension is considerable, the string does not turn back to its original length, and as a result, it is considered as a non-elastic material. Therefore, the Hooke's Law cannot be used to calculate the value of tension.

## Types of Forces III (Elastic Force and Tension) Revision Questions

1. An object is hooked on a spring of constant 4000 N/m. As a result, the spring extends by 8cm. What is the mass of the object? (For simplicity, take the gravitational field strength equal to 10 N/kg).

1. 320 kg
2. 32 kg
3. 5 kg
4. 500 kg

2. An object is pulled horizontally due right by using a rope while it is hooked at a spring on the left as shown in the figure. The spring has a constant of 10000 N/m and it is tied to a fixed vertical wall. If the friction between the object and the ground is negligible, calculate the tension in the rope if the spring extends by 4 cm due to the action of the pulling force.

1. 40 N
2. 25 N
3. 400 N
4. 250 N

3. The elastic force vs deformation graph of a spring is shown in the figure below. What is the maximum weight the spring can hold without compromising its elastic behaviour?

1. 80 N
2. 1280 N
3. 2140 N
4. 560 N