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Work-energy theorem states that the net work performed by external forces on an item equals the change in kinetic energy of the object. Work-energy theorem considers net work done by all forces rather than just work done by one force. Thus, kinetic energy may be defined as the effort necessary to accelerate an item from rest to the requisite velocity and work done is defined as the energy transferred from or to an item by applying a force in the direction of the object's displacement. The relationship between work and kinetic energy is known as the Work-Energy Theorem.
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Keyterms: Work, Energy, Kinetic energy, Acceleration, Velocity, Friction force, Initial kinetic energy, Final Kinetic Energy
What is Kinetic Energy?
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The kinetic energy of an object is the energy it contains as a result of its motion. It is defined as the amount of work required to accelerate a body of a given mass from rest to a certain velocity. The body retains its kinetic energy after gaining it during acceleration unless its speed changes. The body does the same amount of work while slowing down from its current pace to rest.
Formula of Work Energy Theorem
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According to the work-energy theorem, if K represents the change in kinetic energy of the body and W represents the work done on it by external forces,
then K = W or
Kf–Ki=W.
Where, Kf denotes the object's Final Kinetic Energy.
Ki denotes the object's initial kinetic energy.
This theorem is based on the law of energy conservation, which asserts that energy cannot be created and only moved from one form to another. Consider the following illustration of a box put on the surface of an inclined plane:
\(W = \int ^{xf}_{x_i} F(x) dx\)
The work-energy theorem states that
Work done by external forces = Change in Kinetic Energy.
If,
- Wg equals gravity's work,
- WN denotes the amount of work performed by a normal reaction.
- Wf denotes the amount of work done by friction force.
- Kf denotes the ultimate kinetic energy.
- Ki denotes the initial kinetic energy.
Then,
Wg + WN + Wf = Kf - Ki
Derivation of Work Energy Theorem
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We can derive the equation of work-energy theorem when the resulting force F is constant in magnitude and direction and parallel to the particle's movement. The particle is travelling in a straight path with constant acceleration.
The equation F = ma expresses the relationship between net force and acceleration, and the particle's displacement d may be calculated using the equation of Newton's second law of motion:
vf2 = vi2 + 2ad
Where,
vf = final velocity of the object;
vi = initial velocity of the object;
a = constant acceleration; and
d = displacement of the object.
Obtaining,
d = (vf2 - vi2) / 2a
The work of the net force is determined as the product of its magnitude (F=ma) and the displacement of the particle. When the preceding equations are substituted, the following results are obtained:
W= Fd
W= ma . (vf2 - vi2) / 2a
= 1/2mvf2 - 1/2mvi2
= KEf - KEi
= ΔK
This is the derivation of the Work-Energy Theorem. Thus, we can say that the work done on an object is equal to the change in the kinetic energy of the object.
Work Energy Theorem For Variable Force
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Plot a graph between an object's force and displacement by graphing the force applied on it along one axis and its subsequent displacement along the other.
To demonstrate the work-energy theorem, consider the force F along the y axis and the displacement x along the x axis as illustrated in the graph below:
Work Energy Theorem For Variable Force
Δx = displacement of the object
dK/dt = 1/2[d(mv2)/dt]
dK/dt = mvdv/dt
By comparing the equations, we get:
ΔK = W
This is the equation of work energy theorem.
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Work Energy Theorem for Constant Force
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To derive a formula for work done under a constant force, consider that an item with final velocity v and initial velocity u travels a distance s while accelerating at a constant rate a. The object's equation of motion is as follows:
v2 = u2 + 2as
v2 - u2 = 2as … (i)
The equation can be written as,
Kf -Ki = W
ΔK = W
This is the equation of work energy theorem.
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Example of Work Energy Theorem
Consider a skier moving at a constant speed across a flat, frictionless surface. If someone comes up behind him and pushes him for a distance - effectively exerting labour on the skier then he will accelerate as there will be an increase in the kinetic energy. Because the force was exerted in the skier's motion, this form of work is known as positive work. However, if the person pushed in the opposite direction of the skier's motion, the skier would slow down as there will be a decrease in the kinetic energy. This is referred to as negative work.
Example of Work Energy Theorem
Features and Applications of Work Energy Theorem
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The features and applications of work energy theorem are as follows:
- When the particle's speed remains constant, there is no change in kinetic energy and the work done by the resultant force is zero.
- Work and energy are two terms that can be used interchangeably. When work is completed, it manifests as energy. The energy can be reduced by allowing particles to undertake work on other particles.
- The work energy theorem is useful in understanding situations in which a rigid body moves under the influence of numerous forces. Because of its rigid lattice structure, a rigid body cannot store potential energy and can only contain kinetic energy. As a result, the work done by any force on a rigid body is equal to the body's change in kinetic energy.
Things to Remember
- Work-energy theorem states that the net work performed by external forces on an item equals the change in kinetic energy of the object.
- The equation of work-energy theorem: W = KEf - KEi = 1/2mvf2 - 1/2mvi2
- The kinetic energy of an object is defined as the amount of work required to accelerate a body of a given mass from rest to a certain velocity.
- Newton's second law can be used to construct the work-energy theorem.
- A constant force is rarely visible in our daily lives. In reality, a constant force is an ideal circumstance, and all the forces that occur around us are variable.
- Work can be either positive or negative.
- If the force is applied in the same direction as the displacement of the object then the force is doing positive work.
- If the force is applied in the direction opposite to the displacement, the force does negative work.
Important Questions Based on Work-Energy Theorem
Ques: What exactly is the work-energy theorem proof? (2 Marks)
Ans: The work energy theorem describes the relationship between the change in the kinetic energy and work done by a force. When the force exerted on a particle affects its location, the work is said to be done. Then, by integrating it, we can show that work done by a force equals kinetic energy change.
Ques: (i) What are the work-energy theorem's applications? (2 Marks)
(ii) Why is the work-energy theorem not applicable to all forces?
Ans: (i) The Work-Energy theorem is extremely useful in assessing scenarios in which a rigid body moves under the influence of many forces. Because of its rigid structure, a rigid body cannot store potential energy in its lattice and can only contain kinetic energy.
(ii) The work-energy concept holds true even in the presence of non-conservative factors. The work energy concept is valid as long as you use the work done by the resultant force in the equation or equivalently sum the work done by each force/moment.
Ques: (i) Is it possible for kinetic friction to provide positive work? (3 Marks)
(ii) Is it possible for work to be negative?
Ans: (i) Kinetic friction on the item will act backwards, whereas it will act forwards on the earth. Because there is no movement along this forward kinetic friction force, the work done is nil. Thus, the work done by Kinetic Friction on an Object may be both positive and negative.
(ii) The work done might be positive or negative depending on whether the force doing the work is directed opposing the motion of the object or in the same direction as the motion of the item. Positive work is done on an item if the force and displacement are both in the same direction.
Ques: (i) When can we use the work-energy theorem? (3 Marks)
(ii) What is the significance of the work-energy theorem?
(iii) Is the work-energy theorem always correct?
Ans: (i) The work-energy theorem may be able to provide some information about the forces in circumstances when the motion of an item is known but the values of one or more of the forces acting on it are unknown.
(ii) Work Energy Theorem provides a direct relationship between net-work and kinetic energy. Though the Work-Energy theorem's full relevance cannot be observed until we examine energy conservation, we can apply the theory to find the velocity of a particle given a known force at any place.
(iii) The work-energy theorem applies to any non-conservative force. As we can see, we are using the work energy theorem for the work done by the resulting force, and this theory holds true everywhere.
Ques: (i) What is the relationship between work and energy? (3 Marks)
(ii) Is the work-energy theorem applicable in a non-inertial frame?
(iii) Is the work-energy theorem valid in the presence of friction?
Ans: (i) Work or work done refers to the quantity of energy delivered by a force to move an item. As a result, the relationship between Work and Energy is straightforward. That is, the difference in an object's Kinetic energy is work done by the object.
(ii) Since the dynamics of a body cannot be explained in a non-inertial frame using just real forces, the work-energy theorem is invalid. However, if we incorporate pseudo forces and Newton's equations, the non-inertial frame can be considered as an inertial frame, and the work-energy theorem is valid.
(iii) When a non-conservative force (such as friction, air resistance, or applied force) works on an item, its total mechanical energy (Ek+Ep) varies. This is due to the item experiencing acceleration and hence a change in velocity. The work-energy theorem follows because of this.
Ques: A bullet of mass 10g moving with a velocity of 500 m/s, strikes a tree and goes out from the other side with a velocity of 400 m/s. Calculate the work done by the bullet (in joules) in passing through the tree. (5 Marks)
Ans: Given that-
The mass of the bullet,
m = 10g = 0.01kg
The initial velocity of the bullet,
u = 500m/s
The final velocity of the bullet,
v = 400m/s
The change in kinetic energy of the bullet will be equal to the work done on the bullet by the tree. Thus, by applying the work-energy theorem, we get-
Wb = Kf – Ki
Wb= 12×m×v2 – 12×m×u2
Wb= 12×m×[v2–u2]
Wb= 0.012×[160000–250000]
Wb = –450J
The work done by the bullet = 450J.
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