Elasticity

Key Terms

Elasticity is the ability of a material to regain its original shape or size after deformation, removal of stress/force, or after it has been compressed.

Deformation occurs when a wire is stretched or compressed. It is elastic if the wire returns to its original position, while it is plastic if it does not return to its original position.

Elastic limit is the limit of a force beyond which a stretched wire does not return to its original length when the stretching force is removed.

Yield point is the point beyond the elastic limit where the material has lost all its elasticity permanently and has become plastic.

Maximum load occurs when a wire can no longer withstand any further increase in load.

Breaking point is the point at which a material breaks after being stretched beyond the yield point and can no longer endure further stretching.

Hooke’s Law

Hooke’s Law states that the extension (e) of an elastic body is directly proportional to the applied force (F), provided that the elastic limit is not exceeded.

Mathematically:

F ∝ e

$$ F = K \times e $$

• K is the force constant (stiffness constant) in N/m.
• F is the applied force or weight in Newtons (N).
• e is the extension or compression in meters (m).

Force can also be expressed as:

$$ F = mg $$

• m is the mass in kilograms (kg).
• g is the acceleration due to gravity (9.8 m/s²).

Young’s Modulus

Young’s Modulus (Y) is the ratio of tensile stress to tensile strain.

Formula:

$$ Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}}$$

Tensile Strain

Tensile strain is the ratio of the extension of a material to its original length.

Mathematically:

$$ Strain = \frac{Extension (e)}{\text{original Length(L)}} $$

Tensile strain has no unit.

Tensile Stress

Tensile stress is the ratio of the force acting on a material to its cross-sectional area.

Mathematically:

Stress = Force (F) / Cross-Sectional Area (A)

Work Done in Springs and Elastic Strings

When a force (F) is applied to an elastic spring of original length (L), causing it to undergo an extension or compression (e), the average force is given by:

$$ F = \frac{1}{2} \times F $$

Work Done by a Spring

Work done is calculated as:

Work Done = Force × Distance

Since the average force is applied over the extension:

\[ W = \frac{1}{2} F e \]

Using \[ F = K e \], where K is the force constant:

\[ W = \frac{1}{2} K e^2 \]

This represents the work done by a spring when compressed or extended by a force.

Energy Stored in a Spring

The energy stored in a stretched or compressed spring (elastic potential energy) is also given by:

\[ W = \frac{1}{2} F e \]

Or using Hooke’s Law:

\[ W = \frac{1}{2} K e^2 \]

The unit of work or energy is Joules (J).

Spring Balance

A spring balance is a type of mechanical force gauge or weighing scale. It consists of a spring fixed at one end with a hook to attach an object at the other. At the top of the spring balance, there is a fixed end, and at the bottom, a free end that moves when a force is applied to it.

The key component of a Newton spring balance is the spring. When a force is applied, the spring extends. The larger the force, the greater the extension. The relationship between the force applied and the extension of the spring is governed by Hooke's Law.

To establish this relationship, you can measure how the spring extends in response to different forces. This can be done by adding a series of masses, increasing the force on the spring, and then plotting the force against the extension.

A spring balance is used to measure the weight (force acting on an object). It is also known as a spring scale or Newton meter.