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Elastic potential energy is the energy that an object has stored inside of it due to its position. Frequently, while considering potential energy, the first image that comes to mind is of an item that is high in the air and is just beginning to descend. Due to its height, it has potential energy stored within it. As it falls, this energy will be converted into kinetic energy. An object can have potential energy in a few different circumstances, though. A material that is stretchy is one such example.
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Key Terms: Energy, Potential energy, Kinetic energy, Objects, Displacement, Force
What is elastic potential energy?
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Elastic potential energy is the internal energy that a shape-deformed object possesses. Any item with the ability to deform and then reform can be said to possess elastic potential energy. Rubber bands, sponges, bungee cords, and many other items are examples of them.
When we deform these things, they automatically return to their previous shape. The only thing that makes it possible is accumulated elastic potential energy. A compressible or stretchable object's stored energy is hence known as elastic potential energy.
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Elastic Potential Energy Formula
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With the help of the following fundamental formula, we can calculate elastic potential energy:
Elastic potential energy = Force × Displacement
It is calculated as the work required to extend the spring, which is dependent on both the stretched displacement and the spring constant k.
Hooke's law states that a spring's stretch is precisely proportionate to the force used to stretch it. Otherwise put, The spring's displacement is inversely proportional to the force needed to extend it. It is provided as
Elastic Potential Energy = Magnitude of Force × Displacement
Elastic Potential Energy = 12 k x 2
The minus sign (- ve) denotes the opposite direction.
Where,
k = Spring Constant
x = Displacement stretched
Derivation of Elastic Potential Energy
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Graphical Method
Create a graph of force (F) vs. displacement in order to use the graphical approach to determine the work performed by a non-constant force (x). The area beneath this line, which forms a straight line, is where the work was completed. Calculating the area of a triangle with a base of x and a height of F will reveal the area.
W = area = 12 b h = 12 F x
According to Hooke's Law, the formula for the force's magnitude is F = k x.
W = 12 k x x= 12 k x 2
Take into account that the change in the spring's elastic potential energy equals the work completed. As a result, a spring's elastic potential energy is equal to:
Us = 12 k x x = 12 k x 2
Calculus Method
Calculus can also be used to determine the elastic potential energy contained in a spring. Since the area under a function's graph is equal to the integral of that function, the following is true for the spring force:
Us = 0x k x â x = 12 k x 2
The calculus-based approach proves, similarly to the graphical approach, that the elastic potential energy held within a spring is given by:
Us = 12 k x 2
Things to Remember
- The total energy gained as a result of the shape of an object under examination deforming is known as the elastic potential energy.
- Any item that can recover its previous shape and size after deformation will have the elastic potential energy.
- Only the potential energy that has been accumulated, also known as the elastic potential energy, allows for the deformation of the objects.
- Elastic potential energy as the energy contained within a compressible or stretchable object, or, alternatively, as the energy stored within the completely elastic materials.
- When it hits the brick wall, all of the elastic elements, including the spring, the twisted rubber, and the bouncing ball, are crushed. The energy will be stored in the form of elastic potential energy in all of these materials.
Sample Questions
Ques. Define Electrical potential energy. (1 mark)
Ans. Elastic potential energy is the internal energy that a shape-deformed object possesses.
Ques. Mention elastic potential energy formula. (2 marks)
Ans. Elastic Potential Energy = Magnitude of Force × Displacement
Elastic Potential Energy = 12 k x 2
The minus sign (- ve) denotes the opposite direction.
Where,
k = Spring Constant
x = Displacement stretched
Ques. An extended long spring has a potential energy of U and is stretched by 2 cm. The spring's potential energy will be equal to if it is stretched by 10 cm. (1 mark)
(a) U/25
(b) U/5
(c) 5 U
(d) 25 U
Ans. (d) A spring's elastic potential energy, U = 25 U
As: U 2U 1= x 2 2x 2 1 = > U 2 (10 cm)U 1 (2 cm) = > U 2=25U
Ques. The vertical spring is connected to a 5 kg mass load that is compressed by 10m. Determine the spring's force constant. (3 marks)
Ans. Given: mass (m) = 5 kg
Distance x = 10 meters
The Force equation is provided by
F = ma
= 5 kg × 9.8 m/s²
= 49 N
The spring's extended force is
F = k x
Force constant k,
= F / x
= 49 / 10
= 4.9 N/m
Ques. The spring constant of a truck spring is 5.104 N/m. The truck is 0.8 m above the road when it is empty. It descends to 0.7 m above the ground when loaded with cargo. How much potential energy are the four springs able to hold? (3 marks)
Ans. The truck's height has changed by 0.1 meters (0.8 m – 0.7 m). This reveals the springs' Δx compression. For the potential energy in a spring, we can substitute:
U = 12 k Δx 2
= 0.5 * 5.104 N/m * (0.1 m)2
= 250 J/spring
= 1000 J
Ques. Surfaces with equipotential
a) are closer in areas with strong electric fields than in areas with weaker electric fields
b) will be crammed closer to a conductor's jagged edges.
c) will be packed in close proximity to areas with high charge densities
d) always have equal spacing (3 marks)
Ans. The correct answer is
a) are closer in areas with strong electric fields than in areas with weaker electric fields
b) will be crammed closer to a conductor's jagged edges.
c) will be packed in close proximity to areas with high charge densities
Ques. Do free electrons move toward an area with higher or lower potential? (3 marks)
Ans. F = q E is the force acting on the charged particle inside the electric field.
The free electrons experience an electrostatic force that is in opposition to the direction of the electric field.
The electrons go from a location of lower potential to one of higher potential because the direction of an electric field is greater than the potential.
Ques. Show that an equipotential volume must be enclosed by a closed equipotential surface that has no charge within it. (3 marks)
Ans. A closed equipotential surface has a variable potential from one position to the next. The potential gradient induced by the surface's dV / dr is distinct from the potential inside the surface.
This also implies that the electric field, which is given as E = - dV / dr, is not equal to zero. It may be claimed that the field lines are pointing either toward or away from the surface.
So, contrary to the first hypothesis, it may be claimed that the field lines are generated by the charges inside. The volume inside the surface must therefore be equal in both directions.
Ques. Show that if there are no other conductors present and an insulated, uncharged conductor is placed close to a charged conductor, the uncharged body must have a potential that is somewhere between the one of the charged-body and that of infinity. (3 marks)
Ans. E = dV / dr is the formula for the electric potential, which decreases along the direction of the electric field. When the path from the charged conductor to the uncharged conductor is taken in the direction of the electric field, the electric potential decreases.
This keeps happening until there are no more uncharged conductors, which lowers the potential even more.
The charged body is at infinity potential, and thus demonstrates that the uncharged body is already at intermediate potential.
Ques. Answer carefully:
(a) Q1 and Q2 are carried by two sizable conducting spheres that are brought in close proximity to one another. Is Q1Q2/4r2, where r is the distance between their centres, a precise expression for the magnitude of the electrostatic force between them?
(b) Would Gauss's law still hold true if Coulomb's law involved 1/r3 dependency (instead of 1/r2)?
(c) At a location in an electrostatic field arrangement, a little test charge is released while at rest. Will it follow the field line that passes through that location as it travels?
(d) What does a nucleus' field accomplish during an electron's whole circular orbit? If the orbit is elliptical, what then?
(e) We are aware that the electric field on the surface of a charged conductor is discontinuous. Is the electric potential there similarly irregular?
(f) How would you interpret a single conductor's capacitance? (5 marks)
Ans. (a) No, because the spheres' charge distributions won't be even.
(b) No.
(c) Definitely not. (Only true if the field line is parallel to the ground.) The field line displays the acceleration direction rather than the velocity.
(d) No matter how the entire orbit is shaped, zero.
(e) No, there is always a possibility.
(f) A capacitor with a single "plate" at infinity is made out of a single conductor.
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