Elastic Potential Energy
Elastic potential energy is the energy reserved in the configuration of a body or physical system. The object generates elastic energy when it is stretched or compressed in any manner. It is potential energy because the body converts it into other forms of energy like kinetic energy.
When you compress a spring, you feel the exact amount of force applied by you is used by the spring to regain its former position. The amount of energy used in regaining the former shape is nothing but spring potential energy.
In the notes, Vedantu has explained both potential energies in a detailed manner; students can refer to them to revise the topics in a short duration or during exam time. Faculties of physics at Vedantu have prepared them to give students clear and helpful knowledge in their preparation.
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Learn Uses of Elastic Energy and Hooke's Law
It is always necessary to understand the application of the concept; if you do not know why you are reading or learning about a topic, then it is of no use. For example, knowing the term thermometer is useless if you do not know what to do to measure fever when you are sick.
At Vedantu, you learn more than just definitions of the terms in this topic; potential energy experts have explained the uses of elastic energy and the application of Hooke's Law. The mathematical expression of Elastic Potential Energy, Spring Potential Energy and Hooke's Law help you solve the numerical questions in the exam. Also, teachers have pointed out how Hooke's Law explains the restoring force in the spring that helped it to regain its former position.
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More About Elastic Potential Energy and Spring Potential Energy
Elastic energy can be defined as the mechanical potential energy reserved in the configuration of a material or physical system. It is exposed to elastic deformation by work performed upon it. Elastic energy generated when objects are temporarily stretched, compressed, or generally deformed in any manner. Elasticity theory mainly develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used for calculating positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as sound energy and kinetic energy, when the object is allowed to reform by its elasticity.
\[ U = \frac{1}{2} k \Delta x^{2} \]
What Causes Elastic Energy?
A force acting on an object temporarily changes its shape, such as when you stretch an elastic band or squish a squishy ball with your hand.
Spring Potential Energy
Since the potential energy's change of an object between two positions is equal to the work that must be done to move the object from one point to another, the calculation of potential energy is identical to calculating the work. Since the force requires stretching a spring changes with distance, the calculation of the work involves an integral.
W =\[ \int_{0}^{x} k x dx = k \frac{x^{2}}{2} \]
The Potential Energy of a Spring
When we compress or extend a stretched spring, we feel a force equal to that applied by us in the opposite direction. So the reason for this happening is when a spring deviates from its mean position, it tends to restore its equilibrium by exerting a force equal and opposite to the external force. But the question remains in which way can this force be helpful to us? We all must have seen the uses of spring force in bicycle carriers and launching devices. The energy gained by disturbing the equilibrium of the spring is used as its potential energy and converted to other forms.
Hooke’s Law
The force that requires stretching an elastic object like a metal spring is always directly proportional to the spring extension for small-scale distances. The force applied back by the spring is known as Hooke's Law.
\[\overline{f_{s}} = - k\overrightarrow{ x } \]
Where \[ f_{s}\] is the force exerted by the spring, x is the displacement relative to the unstretched length of the spring, and k is the spring constant.
The spring force can be called a restoring force because the force exerted by the spring is always in the opposite direction to the displacement, this is the reason behind a negative sign in the Hooke's law equation. Pulling down on a spring stretches the spring downward, which results in the spring exerting an upward force.
Uses of Elastic Energy
We Have Listed a few Uses of Elastic Energy Below:
A spring is used to reserve elastic potential energy in many mechanical devices like the shock absorbers present in cars. Elastic energy can be used in many ways since the spring can remain in its compressed or stretched state for extended periods without dissipating energy. Balloons, rubber bands, bungees, and trampolines use elastic energy for the stretch. We can find uses of elastic energy in squishy balls, a bow and arrow, and coiled springs. Catapults and slingshots are also uses of elastic energy.
Solved Examples
Question 1: What happens when a spring is stretched too far?
Answer: If a force is applied to spring to exceed its elastic limit, then it will no longer return to its original shape.
Question 2: How to analyse a spring force versus displacement graph?
Answer: The area under the force in the spring versus displacement curve is done in the spring. The diagram below shows a plot of force on the spring versus displacement where displacement is 0 when the spring is unstretched. The work is done on a spring store elastic potential energy Us in the spring until the spring returns to its original length. Therefore, the Us is equal to the work done and also to the area under the curve.
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Fun Facts
The elastic potential energy of the bowstring gets transferred to the arrow as kinetic energy. Elastic energy can be produced by stretching a muscle in your body. Scottish scientist William Rankine first struck the term potential energy in the 19th century. The equation for calculating the potential energy of a spring is PE=1/2*k*x2, where k is the spring constant, and x is the amount of compression.
FAQs on Elastic Potential Energy and Spring Potential Energy
1. What is meant by elastic potential energy?
Elastic potential energy is the mechanical potential energy stored in the configuration of an elastic object when work is done to deform it. This energy is generated when materials are temporarily stretched, compressed, or twisted. The object has the potential to convert this stored energy into other forms, such as kinetic energy or sound energy, once the deforming force is removed and it returns to its original shape.
2. What is the formula for the elastic potential energy stored in a spring?
The formula to calculate the elastic potential energy (U) stored in a spring that obeys Hooke's Law is: U = ½kx². In this equation, 'k' represents the spring constant, which is a measure of the spring's stiffness, and 'x' is the displacement of the spring from its equilibrium (unstretched or uncompressed) position.
3. What is the relationship between the stretch of a spring and its stored potential energy?
The relationship is not linear; the elastic potential energy stored in a spring is directly proportional to the square of its displacement (stretch or compression). This means if you double the stretch of a spring, the potential energy it stores increases by a factor of four (2²). Similarly, tripling the stretch increases the stored energy by a factor of nine (3²).
4. Can you provide some real-world examples of elastic potential energy?
Elastic potential energy is used in many everyday objects. Some common examples include:
- A stretched rubber band before it is released.
- The coiled spring in a wind-up toy or a mechanical watch.
- A bungee cord as it stretches during a jump.
- The bow of an archer when it is drawn back.
- The shock absorbers in a vehicle, which compress to absorb bumps.
5. What happens if a spring is stretched beyond its elastic limit?
If a spring is stretched beyond its elastic limit, it undergoes permanent deformation. This means it will not return to its original length and shape after the stretching force is removed. At this point, Hooke's Law is no longer applicable, and the energy stored is not fully recoverable as elastic potential energy; some of it is dissipated in permanently changing the material's structure.
6. Is elastic potential energy the same as spring potential energy?
Not exactly. Spring potential energy is a specific type of elastic potential energy. The term 'elastic potential energy' is a broader category that refers to the energy stored in any deformed elastic object, such as a stretched wire, a bent ruler, or a rubber balloon. Spring potential energy specifically refers to the energy stored in a spring that follows Hooke's Law.
7. How is the formula for a spring's potential energy, U = ½kx², derived?
The formula is derived from the work done to stretch or compress the spring. According to Hooke's Law, the restoring force (F) exerted by the spring is F = -kx. To stretch the spring, an external force equal in magnitude but opposite in direction (F_ext = kx) must be applied. Since this force varies with displacement, the work done (W) is calculated by integrating the force over the displacement from 0 to x.
W = ∫ F_ext dx = ∫ (kx) dx = k [x²/2] from 0 to x.
This gives W = ½kx². By the work-energy theorem, this work done is stored as potential energy (U) in the spring, hence U = ½kx².
8. How can we determine the work done on a spring using a force-displacement graph?
For a spring obeying Hooke's Law, a graph of applied force (F) versus displacement (x) is a straight line passing through the origin. The work done on the spring, which is equal to the elastic potential energy stored, can be determined by calculating the area under this force-displacement graph. Since the area forms a triangle, its area is calculated as ½ × base × height, which corresponds to ½ × x × F. Substituting F = kx gives the area as ½ × x × (kx) = ½kx².
9. How does elastic potential energy convert into kinetic energy?
The conversion of elastic potential energy into kinetic energy is a prime example of the conservation of mechanical energy. When a deformed elastic object (like a compressed spring or a stretched bowstring) is released, the stored potential energy begins to decrease. As the object returns to its equilibrium position, this potential energy is transformed into kinetic energy (the energy of motion), causing the object or an attached mass (like an arrow) to accelerate and gain speed.
10. Is elastic potential energy only stored in ideal springs? What about a stretched wire?
No, elastic potential energy is not exclusive to ideal springs. It can be stored in any elastic material that is deformed. For a stretched wire, the potential energy is stored due to the work done against the inter-atomic forces to change its length. The energy stored per unit volume in a stretched wire is given by the formula: U/Volume = ½ × Stress × Strain. This demonstrates that the concept of storing energy through deformation applies to solids in general, not just springs.
