Young's Modulus is the ratio of stress and strain. It is named after the famous British physicist Thomas Young. Young's Modulus provides a relation between stress and strain in any object. When a certain load is added to a rigid material, it deforms. When the weight is withdrawn from an elastic material, the body returns to its original form, this property is called Elasticity.
Elastic bodies have a steady linear Young's modulus. Young's modulus of Steel is 2×1011 Nm-2. Young Modulus is also called the Modulus of Elasticity. In this article, we will learn about Young’s Modulus, its Young's Modulus formula, unit, Stress, Strain, and how to calculate Young's Modulus.
What Is Young’s Modulus?
Young’s Modulus, is the measure of the deformation in the length of the solid such as rods, or wires when the stress is applied along the x-axis. Bulk modulus and Shearing modulus are also used to measure the deformation of the object according to the stress applied.
Young's Modulus Definition
Young Modulus is the property of the material which allows it to resist the change in its length according to stress applied to it. Young’s modulus is also called the modulus of elasticity.
It is represented using the letters E or Y.
Before proceeding any further first learn in brief about the stress and strain.
- Stress is defined as the force applied per unit length of the object.
- Strain is the change in shape or length of the object with respect to its original length.
Young’s modulus provides a relation between stress and strain. A solid object deforms when a particular load is applied to it. When the force is applied to an object it changes its shape and as soon as the force is removed from the object it regains its original position. This is called the elastic property of the object.
The more elastic the material is more it will resist the change in its shape.
Young's Modulus of Elasticity
Young's Modulus is a mathematical constant. It was named after Thomas Young, an 18th-century English physician and scientist. It defines the elastic characteristics of a solid that is subjected to tension or compression only in one direction. For Example, consider a metal rod that returns to its original length after being stretched or squeezed longitudinally.
It is a measurement of a material's capacity to endure changes in length when subjected to longitudinal tension or compression. It's also known as the Modulus of Elasticity. It is calculated as the longitudinal stress divided by the strain. In the instance of a tensioned metal bar, both stress and strain may be stated.
Young's Modulus, also known as Elastic Modulus or Tensile Modulus, is a mechanical property measurement of linear elastic solids such as rods, wires, and so on. Other numbers exist that give us a measure of a material's elastic characteristics. Bulk modulus and shear modulus are two examples. However, the value of Young's Modulus is most commonly utilized. This is because it provides information about a material's tensile elasticity.
When a material is compressed or stretched, it experiences elastic deformation and returns to its original shape when the load is released. When a flexible material deforms, it deforms more than when a rigid substance deforms. In other words, it can be interpreted as:
- A solid with a low Young's Modulus value is Elastic.
- A solid with a high Young's Modulus value is Inelastic or stiff.
Young's Modulus is described as a material's mechanical ability to tolerate compression or elongation with respect to its initial length.
Mathematically, Young's Modulus is defined as the ratio of the stress applied to the material and the strain corresponding to the applied stress in the material as shown below:
Young's Modulus = Stress / Strain
Y = σ / ϵ
where
Y is Young’s Modulus of the material
σ is the stress applied to the material
ϵ is the strain corresponding to the applied stress
Units of Young's Modulus
SI unit for Young's modulus is Pascal (Pa).
Dimensional formula for Young's Modulus is [ML-1T-2].
The values are most often expressed in terms of Megapascal (MPa), Newtons per square millimeter (N/mm2), Gigapascals (GPa), or kilonewtons per square millimeter (kN/mm2).
We know that,
Y = σ / ϵ...(1)
Also,
σ = F/A
ϵ = ΔL/L0
Putting these values in eq(1)
Y = σ / ϵ
= (F/A)×(L0/ΔL)
Y = FL0 / AΔL
- Y is Young’s modulus
- σ is Stress applied
- ε is Strain related to the applied stress
- F is Force exerted by the object
- A is Actual cross-sectional area
- ΔL is change in the length
- L0 is actual length
Young’s Modulus Factors
Young's Modulus of any material is used to explain the deformation in the length of the material when force is applied to it. As it is clear that the Young Modulus of steel is greater than rubber or plastic it is safe to say that steel is more elastic than both rubber and plastic.
Elasticity is the property of the material which resists the change in its length as soon as the applied stress is removed.
Young’s Modulus of the material explains how a material behaved when stress is applied to it. The lower value of Young’s Modulus in materials tells us that this material is not fit for dealing with large stress and applying large stress will change the shape of the object completely.
How to Calculate Young’s Modulus
Young’s Modulus of any object is calculated using the formula,
Young’s Modulus = Stress / Strain = σ / ϵ
We can also plot a stress-strain curve to find Young's Modulus of the material.
-(1).png)
The figure discussed above it is the stress-strain curve and the initial slope of the first segment of the curve is Young's modulus.
If continuously increasing stress is applied to the material it reaches a point when its elasticity gets disappeared and any further stress can create a more significant strain. This point is called the elastic limit of the material.
Further increasing the stress make the material such that it start to deform without even applying stress the point where this started to happen is called the plastic limit.
Young's Modulus of Some Materials
Young's Modulus of some common materials are discussed in the table below:
Materials | Young's Modulus (Y) in Nm-2 |
|---|
| Rubber | 5 × 108 |
| Bone | 1.4 × 1010 |
| Lead | 1.6 × 1010 |
| Aluminum | 7.0 × 1010 |
| Brass | 9.0 × 1010 |
| Copper | 11.0 × 1010 |
| Iron | 19.0 × 1010 |
Mathematical Interpretation of Young's Modulus
Consider a wire of radius r and length L. Let a force F be applied on the wire along its length i.e., normal to the surface of the wire as shown in the figure. If △L is the change in length of the wire, then Tensile stress (σ = F/A), where A is the area of the cross-section of the wire and the Longitudinal strain (ϵ = △L/L).
Therefore, Young's Modulus for this case is given by:
Y = (F/A) / (△L/L)
= (F × L) / (A × △L)
If the extension is produced by the load of mass m, then Force, F is mg, where m is the mass and g is the gravitational acceleration.
And the area of the cross-section of the wire, A is πr2 where r is the radius of the wire.
Therefore, the above expression can be written as:
Y = (m × g × L) / (πr2 × △L)
Factors Affecting Young's Modulus
The Factors on which Young's Modulus of material depends are,
- Larger the value of Young's modulus of the material, the larger the value of the force required to change of length of the material.
- Young's modulus of an object depends upon the nature of the material of the object.
- Young's modulus of an object does not depend upon the dimensions (i.e., length, breadth, area, etc) of the object.
- Young's modulus of a substance decreases with an increase in temperature.
- Young's modulus of elasticity of a perfectly rigid body is infinite.
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Solved Examples on Young's Modulus
Example 1: A cable is cut to half of its length. Why this change has no effect on maximum load cable cab support?
Solution:
The maximum load a cable can support is given by:
F = (YA△L) / L
Here Y and A is constant, there is no change in the value of △L/L.
Hence, no effect on the maximum load.
Example 2: What is Young's modulus for a perfectly rigid body?
Solution:
The Young's modulus for a material is,
Y=(F/A) / (△L/L)
Here, △L = 0 for rigid body. Hence, Young's Modulus is infinite.
Example 3: Young's Modulus of steel is much more than that of rubber. If the longitudinal strain is the same which one will have greater tensile stress?
Solution:
Since the Tensile stress of material is equal to the product of Young's modulus (Y) and the longitudinal strain. As steel have larger Young's modulus therefore have more tensile strain.
Example 4: A force of 500 N causes an increase of 0.5% in the length of a wire of an area of cross-section 10-6 m2. Calculate Young's modulus of the wire.
Solution:
Given that,
The force acting, F = 1000 N,
The cross-sectional area of the wire, A = 10-6 m2
Therefore,
△L/L = 0.5 = 5/1000 = 0.005
Y = (F/A)/(△L/L)
= 1012 Nm-2
Example 5: What is the bulk modulus of a perfectly rigid body?
Solution:
Since, the Bulk modulus of a material is defined as,
K= P / (△V/V)
Since, △V = 0 for perfect rigid body.
Hence, the bulk modulus is infinite for perfect rigid body.
Practice Problems on Young's Modulus
Problem 1: A steel rod with a length of 2 meters and a cross-sectional area of 0.01 square meters experiences a uniform force that stretches it by 1 mm. If the applied force is 10,000 N, calculate the Young's Modulus of steel.
Problem 2: A rubber band with a cross-sectional area of 2 mm² and a Young's Modulus of 0.01 GPa is stretched from an original length of 10 cm to 12 cm. Determine the force required to stretch the rubber band.
Problem 3: A concrete column is 3 meters tall and has a cross-sectional area of 0.05 square meters. The Young's Modulus of concrete is 25 GPa. If a force of 500,000 N is applied to the top of the column, calculate the change in length of the column.
Problem 4: An aluminum bar with a Young's Modulus of 70 GPa and a length of 1 meter is subjected to a stress that results in a strain of 0.0005. Calculate the force applied to the bar and the change in length of the bar.
Problem 5: In an experiment, a linear elastic wire is stretched, and the following data are collected: when a 200 N force is applied, the wire stretches by 0.2 mm; when a 400 N force is applied, the wire stretches by 0.4 mm. Assuming the wire has a constant cross-sectional area, calculate the Young's Modulus of the material of the wire.
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What is Frequency?Frequency is the rate at which the repetitive event that occurs over a specific period. Frequency shows the oscillations of waves, operation of electrical circuits and the recognition of sound. The frequency is the basic concept for different fields from physics and engineering to music and many mor
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Amplitude, Time Period and Frequency of a VibrationSound is a form of energy generated by vibrating bodies. Its spread necessitates the use of a medium. As a result, sound cannot travel in a vacuum because there is no material to transfer sound waves. Sound vibration is the back and forth motion of an entity that causes the sound to be made. That is
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Energy of a Wave FormulaWave energy, often referred to as the energy carried by waves, encompasses both the kinetic energy of their motion and the potential energy stored within their amplitude or frequency. This energy is not only essential for natural processes like ocean currents and seismic waves but also holds signifi
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Simple Harmonic MotionSimple Harmonic Motion is a fundament concept in the study of motion, especially oscillatory motion; which helps us understand many physical phenomena around like how strings produce pleasing sounds in a musical instrument such as the sitar, guitar, violin, etc., and also, how vibrations in the memb
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Displacement in Simple Harmonic MotionThe Oscillatory Motion has a big part to play in the world of Physics. Oscillatory motions are said to be harmonic if the displacement of the oscillatory body can be expressed as a function of sine or cosine of an angle depending upon time. In Harmonic Oscillations, the limits of oscillations on eit
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Sound
Production and Propagation of SoundHave you ever wonder how are we able to hear different sounds produced around us. How are these sounds produced? Or how a single instrument can produce a wide variety of sounds? Also, why do astronauts communicate in sign languages in outer space? A sound is a form of energy that helps in hearing to
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What are the Characteristics of Sound Waves?Sound is nothing but the vibrations (a form of energy) that propagates in the form of waves through a certain medium. Different types of medium affect the properties of the wave differently. Does this mean that Sound will not travel if the medium does not exist? Correct. It will not, It is impossibl
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Speed of SoundSpeed of Sound as the name suggests is the speed of the sound in any medium. We know that sound is a form of energy that is caused due to the vibration of the particles and sound travels in the form of waves. A wave is a vibratory disturbance that transfers energy from one point to another point wit
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Reflection of SoundReflection of Sound is the phenomenon of striking of sound with a barrier and bouncing back in the same medium. It is the most common phenomenon observed by us in our daily life. Let's take an example, suppose we are sitting in an empty hall and talking to a person we hear an echo sound which is cre
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Refraction of SoundA sound is a vibration that travels as a mechanical wave across a medium. It can spread via a solid, a liquid, or a gas as the medium. In solids, sound travels the quickest, comparatively more slowly in liquids, and the slowest in gases. A sound wave is a pattern of disturbance caused by energy trav
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How do we hear?Sound is produced from a vibrating object or the organ in the form of vibrations which is called propagation of sound and these vibrations have to be recognized by the brain to interpret the meaning which is possible only in the presence of a multi-functioning organ that is the ear which plays a hug
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Audible and Inaudible SoundsWe hear sound whenever we talk, listen to some music, or play any musical instrument, etc. But did you ever wondered what is that sound and how is it produced? Or why do we hear to our own voice when we shout in a big empty room loudly? What are the ranges of sound that we can hear? In this article,
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Explain the Working and Application of SONARSound energy is the type of energy that allows our ears to sense something. When a body vibrates or moves in a ‘to-and-fro' motion, a sound is made. Sound needs a medium to flow through in order to propagate. This medium could be in the form of a gas, a liquid, or a solid. Sound propagates through a
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Noise PollutionNoise pollution is the pollution caused by sound which results in various problems for Humans. A sound is a form of energy that enables us to hear. We hear the sound from the frequency range of 20 to 20000 Hertz (20kHz). Humans have a fixed range for which comfortably hear a sound if we are exposed
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Doppler Effect - Definition, Formula, ExamplesDoppler Effect is an important phenomenon when it comes to waves. This phenomenon has applications in a lot of fields of science. From nature's physical process to planetary motion, this effect comes into play wherever there are waves and the objects are traveling with respect to the wave. In the re
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Doppler Shift FormulaWhen it comes to sound propagation, the Doppler Shift is the shift in pitch of a source as it travels. The frequency seems to grow as the source approaches the listener and decreases as the origin fades away from the ear. When the source is going toward the listener, its velocity is positive; when i
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Electrostatics
ElectrostaticsElectrostatics is the study of electric charges that are fixed. It includes an study of the forces that exist between charges as defined by Coulomb's Law. The following concepts are involved in electrostatics: Electric charge, electric field, and electrostatic force.Electrostatic forces are non cont
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Electric ChargeElectric Charge is the basic property of a matter that causes the matter to experience a force when placed in a electromagnetic field. It is the amount of electric energy that is used for various purposes. Electric charges are categorized into two types, that are, Positive ChargeNegative ChargePosit
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Coulomb's LawCoulomb’s Law is defined as a mathematical concept that defines the electric force between charged objects. Columb's Law states that the force between any two charged particles is directly proportional to the product of the charge but is inversely proportional to the square of the distance between t
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Electric DipoleAn electric dipole is defined as a pair of equal and opposite electric charges that are separated, by a small distance. An example of an electric dipole includes two atoms separated by small distances. The magnitude of the electric dipole is obtained by taking the product of either of the charge and
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Dipole MomentTwo small charges (equal and opposite in nature) when placed at small distances behave as a system and are called as Electric Dipole. Now, electric dipole movement is defined as the product of either charge with the distance between them. Electric dipole movement is helpful in determining the symmet
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Electrostatic PotentialElectrostatic potential refers to the amount of electrical potential energy present at a specific point in space due to the presence of electric charges. It represents how much work would be done to move a unit of positive charge from infinity to that point without causing any acceleration. The unit
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Electric Potential EnergyElectrical potential energy is the cumulative effect of the position and configuration of a charged object and its neighboring charges. The electric potential energy of a charged object governs its motion in the local electric field.Sometimes electrical potential energy is confused with electric pot
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Potential due to an Electric DipoleThe potential due to an electric dipole at a point in space is the electric potential energy per unit charge that a test charge would experience at that point due to the dipole. An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific
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Equipotential SurfacesWhen an external force acts to do work, moving a body from a point to another against a force like spring force or gravitational force, that work gets collected or stores as the potential energy of the body. When the external force is excluded, the body moves, gaining the kinetic energy and losing a
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Capacitor and CapacitanceCapacitor and Capacitance are related to each other as capacitance is nothing but the ability to store the charge of the capacitor. Capacitors are essential components in electronic circuits that store electrical energy in the form of an electric charge. They are widely used in various applications,
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