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Work done is defined in such a manner that it includes both forces applied to the body and the total displacement of the body.
- Terms like Work, Energy, and Power are some of the commonly known words that we have been using in our lives.
- Like a coolie pulling the luggage, you do your homework, the driver driving the car, the chef cooking, etc.
- Usually, in our daily life, we call all the above-mentioned activities as work.
- But in the language of science, it is not the same.
Let’s see how we define work done in physics.
- Imagine if we push a block with a force ‘F’, the block starts moving with an acceleration ‘a’.
- Consequently, its speed changes by increasing or decreasing depending on the direction of the force applied.
- The kinetic energy of the system changes with the increase or decrease in speed.
- It means that this energy is definitely getting transferred to another form.
- So now, we can say that work has been done.
- The energy decreases when negative work gets done whereas it increases when positive work is done.
| Table of Content |
Key Terms: Power, Work, Energy, Force, Kinetic energy, Speed, Motion, Newton, Acceleration, Watt, Potential energy, Work done formula, Work-Energy Theorem
What is Work Done?
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The work done definition can be given as
| Work is said to be done if and only if a force is applied to a body and the body is moved to a certain displacement as a result of the exerted force. |
There are many conditions and situations in which we will say that work is done because of our real-life experience.
- But in the world of physics, certain activities may not be called work done.
- This is because they will not satisfy the conditions of getting work done like applying force and displacement.
- It is important to note that if there is no displacement then consequently there will be no work done.
- If a very large force has been applied to an object, and the object does not move or does not have any displacement.
- In such a case, according to physics, there is no work done.
- The SI unit of work is Joule, and its dimensional formula is [M L2 T-2].
Work Done
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| Related Topics | ||
|---|---|---|
| Unit Of Energy | SI Unit of Length | Unit of Mass (M) |
| Unit of Luminous intensity (J) | Kelvin | Unit of Electric current (I) |
Work Done Formula
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The formula for work done for a constant force acting on a body is given by
\(W =\vec F.\vec d = F\:d \:cos \theta\)
Where
- W is the work done
- F is the force applied to the body
- d is the displacement of the body
- θ is the angle between force and displacement.
Work done is equal to the dot product of the force applied and the displacement.
When the displacement of the body is in the direction of applied force, then θ = 0, and the work done is given as
\(W = F \: d\)
Work-Energy Theorem
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The work-energy theorem states that the work by all forces acting on a body is equal to the change in kinetic energy of the body.
Mathematically,
\(W = \Delta K.E = \frac {1}{2}mv_f^2 - \frac {1}{2}mv_i^2\)
Where
- m is the mass of the body
- vf is the final velocity
- vi is the initial velocity.
The work-energy principle can be derived from the law of conservation of energy.
Derivation for the Work Done Formula
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As we already know the work done is equal to the change in kinetic energy.
\(W = \Delta K.E = \frac {1}{2}mv_f^2 - \frac {1}{2}mv_i^2\)
or
\(W = \frac {1}{2}mv^2 - \frac {1}{2}mu^2\)
Where
- m is the mass of the body
- vf = v, is the final velocity
- vi = u, is the initial velocity.
The above equation can be written as
\(W = \frac {1}{2}m(v^2 - u^2)\)
From equations of motion, we have
\(v^2 - u^2 = 2as\)
Where
- s is the displacement
- a is the acceleration
Therefore,
\(W = \frac {1}{2}m \times 2as\)
\(\Rightarrow W = ma \times s\)
However, according to Newton’s second law of motion, we have
\(F = ma\)
Therefore, we can write
\(W = F \times s\)
The above equation is the mathematical formula for a work done.
Work Done by the System
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We have to assume that the system in which the work is being done is not affecting the surroundings. Thus, we define work as positive when the system exerts force on its surroundings. When work is done on the system (i.e., energy is added to it), the work is negative.
Types of Work Done
Work done can be positive, negative, or zero.
Positive Work
If a force causes an object to move in a certain direction, the work done is positive.
- The motion of a ball going down towards the earth is an example of positive work done.
- Here the ball is being displaced in the direction of the force of gravity.
Negative Work
When force and displacement are in opposite directions, the work is considered to be negative.
An example of negative work done is when a ball is tossed high, the displacement is upward; and, the force due to gravity is downward.
Zero Work
If the force and displacement directions are perpendicular, the total work done by the force on the object is zero.
An example of zero work done is when we push strongly against a wall, the force we apply to the wall is useless since the displacement of the wall is zero, i.e. d = 0.
Conditions for Zero Work Done
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The several conditions in which the work done is zero are given below.
As we already know no displacement means no work done.
- If the body does not move due to the applied force, then regardless of how large the force is, the work done will be zero.
- Not only when the displacement is zero, but also, you may come across some situations in which the force applied is zero.
- For example, if an ice cube is floating on another slippery surface without any external force being applied.
- Here, the force is zero, so the work done will also be zero.
In a condition, where the direction of the force applied is perpendicular to the direction of the displacement, the work done will come out as zero. In this case, the angle between the direction of the force and displacement is 90 degrees, and as we all know W = F . d = F d cos θ, where the angle will be 90 degrees, consequently making W = 0 as the value of cos 90 = 0.
What is Force?
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Force is defined as any push or pull that causes acceleration in an object, whether with or without contact.
- The SI unit of force is Newton.
- Force can change (increase or decrease) the speed, and the direction of the motion of the body.
- It can also change the shape or size of the object.
The formula of Force is
\(F=ma\)
Where
- F is the force
- m is the mass
- a is the acceleration.
What is Power?
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In the normal human world, the word ‘Power’ can have a lot of meanings, but in world physics, Power is defined as the work done per unit of time.
- The rate at which work is done can be considered as Power.
- Its SI unit is Watt (W).
- We can say that 1 watt is equal to 1 Joule per second.
The formula of the Power is
\(P = \frac {W}{T}\)
Where
- P is the power,
- W is the work done,
- T is the time.
What is Energy?
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The capacity of doing work is considered as Energy.
- It also has a lot of forms like kinetic energy, potential energy, thermal energy, electrical energy, and more.
- All kinds of energy are linked with motion.
- The SI unit of energy is Joules.
The formula for energy is
\(Kinetic \: Energy = \frac {1}{2}mv^2\)
Where
- m is the mass
- v is the velocity
\(Potential\: Energy = mgh\)
Where
- m is the mass
- g is the acceleration due to gravity
- h is the height
Things To Remember
- Work is said to be done if a force produces a displacement on the body.
- The SI unit of work is Joule.
- The dimensional formula of work is [M L2 T-2].
- Work done is equal to the dot product of the force applied and the displacement.
- The formula for work done is W = F x d.
- According to the work-energy theorem work done by all forces acting on a body is equal to the change in kinetic energy of the body.
- Work done can be positive, negative, or zero.
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Sample Questions
Ques. What is the standard unit of measurement for work? (1 Mark)
Ans. Joule is the standard unit of measurement for work.
Ques. Consider a wooden box being pulled along the floor using a wire making an angle of 30° with the floor which is horizontal. The wooden box is dragged by 20 m, & the force applied by the rope is 220 N. What is the final Work Done by the force? (2 Marks)
Ans. We have been given-
- Force applied = 220 N
- The angle between the direction of force and the floor = 60 degrees
- Displacement = 20m
We know that- W = F d cos
Putting all the required values,
W = 220 x 20 x cos60
W = 4400 x 1/2
W = 2200 J
Ques. A force of magnitude 24 N acts at a distance of 10 m in the force direction. Calculate the Work Done. (3 Marks)
Ans. Here, we have been provided with-
- Force applied, F = 24 N
- Displacement made, d = 10
We have a formula-
W = F . d
Both the direction of force and the displacement are the same
W = F d
W = 24 x 10
W = 240
Ques. Discuss the relationship between Work and Energy. (4 Marks)
Ans. When we are moving an object, it means that we are transferring energies with the help of applying force. The transfer of energy by the force while moving an object is known as work done. Consequently, we have come across a direct relation between Work and Energy.
It is important to note that the work and the energy are directly proportional to each other.
Scientifically, work done is written as
W = 1/2 mvi - 1/2 mui
Where,
- m = the mass of the object measured using kilograms
- W = the work done by an object measured using Joules
- vi = the final velocity of an object measured using m/s
- ui = the initial velocity of an object measured using m/s
According to the work-energy principle, the total work done by all the forces acting on a particle is equal to the change in kinetic energy of a particle.
Ques. A cyclist comes to a skidding stop at 10 m. During this process, the force on the cycle due to the road is 200 N and is directly opposed to the motion.
(a) How much work does the road do on the cycle?
(b) How much work does the cycle do on the road? (3 Marks)
Ans. In this case, the work done on the cycle by the road will be equal to the work done by the stopping or the frictional force on the cycle due to the road.
- The angle between the stopping force and the displacement is 180 degrees
So, now the work done by the road is-
W = F d cosθ
W = 200 × 10 × cos180
W = – 2000 J
The work done will be negative.
- According to Newton's Third Law of Motion, every action has its equal and opposite reaction.
This implies that an equal and opposite force acts on the road due to the cycle whose magnitude is 200 N. But, as we all have noticed the road undergoes no displacement. Consequently, the work done by cycle on the road will be zero.
Ques. In a ballistics demonstration, a police officer fires a bullet of mass 50.0 g with a speed of 200 m s-1 on soft plywood of thickness 2.00 cm. The bullet emerges with only 10% of its initial kinetic energy. What is the emergent speed of the bullet? (3 Marks)
Ans. The initial kinetic energy of the bullet is-
mv2/2 = 1000 J
The final kinetic energy will be
10% of the initial kinetic energy = 1000 x 10/100 = 100 J
Let’s consider v to be the emergent speed of the bullet, then
1/2mv2 = 100 J
v = 2 x 100 J0.05 kg
V = 63.2 m/s
Ques. When is the work done considered to be zero? (2 Marks)
Ans. The work done will be zero in three conditions-
- When the displacement is zero regardless of the amount of force exerted.
- When the force exerted is zero despite the object moving.
- When the direction between the force applied and the movement of the object is 90 degrees.
Ques. A man carries a 200 N suitcase up three flights of stairs at a height of 10.0 m and then pushes it with a horizontal force of 50 N at a constant speed of 0.5 m/s for a horizontal distance of 35.0 meters. How much work does the man do on his suitcase? (4 Marks)
Ans. The entire motion consists of pulling vertically to displace the suitcase vertically and pushing horizontally to displace the suitcase horizontally
For the vertical part,
W = (200 N) x (10 m) x cos(0
W = 2000 J
For the horizontal part,
W = (50 N) x (35 m) x cos (0)
W = 1750 J
Here, the total work done is-
Total work done = Work done for the vertical part + Work done for the horizontal part
Total work done = ( 2000 + 1750 ) J
Total work done = 3750 J
Ques. State the differences between work and energy. (3 Marks)
Ans. Both the work and energy are scalar quantities, and also, their unit is Newton-metre or Joules. Despite these two important similarities, both of them are far away from each other in a lot of terms.
- Work is done when a force is applied to an object and consequently, it changes the object's distance, but the ability to do some work is said to be energy.
- Work is dependent on displacement whereas energy is free of any displacement.
Ques. A block of mass m = 1 kg, moving on a horizontal surface with speed vi = 2 ms–1 enters a rough patch ranging from x = 0.10 m to x = 2.01 m. The retarding force Fr on the block in this range is inversely proportional to x over this range,
Fr = -kx for 0.1 < x < 2.01m
= 0 for x < 0.1m and x > 2.01 m
where k = 0.5 J.
What is the final kinetic energy and speed vf of the block as it crosses this patch? (3 Marks)
Ans. kf = ki + 0.12.01(-k)x dx
= 12mvi2 - k ln(x)|0.12.01
= 12mvi2 - k ln (2.01 / 0.1)
= 2 - 0.5 ln (20.1)
= 2 - 1.5
= 0.5 J
vf = 2kf / m = 1m/s
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