Work Done By A Constant Force: Definition, Formula, Examples, Questions

Q: What is work in physics, and how is it related to force?

In physics, work is the energy transferred when a force moves an object. It's calculated as the product of force and displacement in the direction of the force. Work is done only when the force causes the object to move in the direction of the force.

Q: What's the difference between work and energy?

Work is the process of energy transfer, while energy is the capacity to do work. When work is done on an object, energy is transferred to or from that object. Work is measured in the same units as energy (joules) because it represents a change in energy.

Q: If a force is applied but an object doesn't move, is work done?

No work is done if there's no displacement, even if a force is applied. Work requires both force and displacement in the direction of that force. For example, pushing against a wall doesn't do work if the wall doesn't move.

Q: What's the significance of the joule as the unit of work?

The joule (J) is the SI unit of work and energy. One joule is the work done when a force of one newton causes a displacement of one meter in the direction of the force. It provides a standardized measure for comparing energy transfers across different systems.

Q: How is work related to power?

Power is the rate at which work is done, or energy is transferred. Mathematically, power is work divided by time (P = W/t). While work focuses on the total energy transfer, power considers how quickly that transfer occurs.

Q: How does the angle between force and displacement affect work done?

The angle between force and displacement affects work through the cosine component. Work is calculated as W = F * d * cos(θ), where θ is the angle between force and displacement. Maximum work is done when the force is parallel to displacement (cos 0° = 1), and no work is done when they're perpendicular (cos 90° = 0).

Q: How does the work done by a constant force relate to the area under a force-displacement graph?

For a constant force, the work done is equal to the area under the force-displacement graph. This is because work is the product of force and displacement, which geometrically represents a rectangle's area on this graph.

Q: Can a constant force do zero work?

Yes, a constant force can do zero work if it's perpendicular to the object's displacement. For example, when an object moves in a circular path, the centripetal force does no work because it's always perpendicular to the motion.

Q: Why is work a scalar quantity despite involving force (a vector)?

Work is a scalar because it represents energy transfer, which is a scalar quantity. Although force is a vector, work is calculated using only the component of force in the direction of displacement, resulting in a scalar product.

Q: How does friction affect the work done on an object?

Friction typically does negative work on a moving object, reducing its kinetic energy. The work done by friction is calculated as the product of the frictional force and the distance traveled, always opposing the motion and thus having a negative sign.

Work Done By A Constant Force

Edited By Vishal kumar | Updated on Jul 02, 2025 05:42 PM IST

Work is considered to be done when a force operating on a body can really move it a certain distance in the direction of the force. The watchman at the office gate is not causing any load to move and is simply standing there, thus both force and displacement are zero. Similarly, no work is done by the watchman. When the coolie bears a load on his head, he applies vertical effort to support it. With the load on his head, the coolie moves horizontally along the platform. Since no distance is covered along the vertical, that is, in the direction of the force exerted by him, the work performed by the coolie is also zero.

This Story also Contains

Define Work
Dependence Of Work Done By A Constant Force
Types of Work
Solved Examples Based on Work Done By a Constant Force
Summary

Work Done By A Constant Force

In this article, we will cover the concept of work and its types, including work done by a constant force. This concept falls under the broader category of work, energy, and power, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of six questions have been asked on this concept, including two in 2019, one in 2020, two in 2021, and one in 2023. And for NEET two questions were asked from this concept.

Let's read this entire article to gain an in-depth understanding of the work done by a constant force.

Define Work

Work is said to be done when a force applied on the body displaces the body through a certain distance along the direction of the force.

Work Done By a Constant Force

1. The scalar product of the force vector $(\vec{F})$ and the displacement vector $(\vec{S})$
$W = \vec{F} \cdot \vec{S}$
2. The product of the magnitude of force $(F)$ magnitude of displacement $(S)$ and cosine of the angle between them $(Θ)$
$W = F S \cos Θ$
3. If the number of forces $\vec{F_{1}}, \vec{F_{2}}, \vec{F_{3}} \dots \dots, {\vec{F}}_{n}$ , are acting on a body and it shifts from position vector $\vec{r_{1}}$ to position vector $\vec{r_{2}}$
$Then W = ({\vec{F}}_{1} + {\vec{F}}_{2} + {\vec{F}}_{3} \dots \dots \dots + {\vec{F}}_{n}) \cdot ({\vec{r}}_{2} - {\vec{r}}_{1}) = {\vec{F}}_{n e t} \cdot {\vec{r}}_{n e t}$
4. Units-
SI Unit-Joule
CGS Unit- Erg
1 Joule $= 10^{7} Erg$
5. Dimension- $M L^{2} T^{- 2}$

1. The scalar product of the force vector $(\vec{F}$ ) and the displacement vector ( $\vec{S}$ )
$W = \vec{F} \cdot \vec{S}$
2. The product of the magnitude of force $(F)$ magnitude of displacement $(S)$ and cosine of the angle between them $(Θ)$
$W = F S \cos Θ$
3. If the number of forces ${\vec{F}}_{1}, {\vec{F}}_{2}, {\vec{F}}_{3} \dots \dots, {\vec{F}}_{n}$ , are acting on a body and it shifts from position vector $\vec{r_{1}}$ to position vector $\vec{r_{2}}$
Then
$W = ({\vec{F}}_{1} + {\vec{F}}_{2} + {\vec{F}}_{3} \dots \dots . + {\vec{F}}_{n}) \cdot ({\vec{r}}_{2} - {\vec{r}}_{1}) = {\vec{F}}_{n e t} \cdot {\vec{r}}_{n e t}$

4. Units-
- SI Unit-Joule
- CGS Unit- Erg
- 1 Joule $= 10^{7} E r g$
5. Dimension- $M L^{2} T^{- 2}$

Dependence Of Work Done By A Constant Force

Till now we have studied the work and work done by a constant force now we are going to study the dependence of work done by a constant force, in which mainly covers the frame of reference.

- Frame of reference: With a change of frame of reference (inertial), force does not change while displacement may change. So the work done by a force will be different in different frames. i.e. A person is pushing a box inside a moving train with a force $\vec{F}$ , displacement inside train $\vec{S}$ and displacement of the train in the ground frame is ${\vec{S}}_{0}$

Then work done by $W = \vec{F} \cdot (\vec{S} + {\vec{S}}_{0})$

Types of Work

There are mainly three types of work which are mentioned below:

Positive Work-

Positive work means that force (or its component) is parallel to displacement.
Means $0 \leq Θ < \frac{π}{2}$

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Where $Θ$ is the angle between force vectors and displacement vector

Maximum work $= W_{max} = F S$ , When $θ = 0^{0}$
E.g When you move a block by pulling it then work done by you on the block is positive

Negative Work

Negative work means that force (or its component) is opposite to displacement.
Means $\frac{π}{2} < Θ \leq π$

Where $\Theta$ is the angle between force vectors and displacement vector

Minimum work= $W_{min} = - F S$ , When $θ = 180^{\circ}$
E.g When a body is made to slide over a rough surface, the work done by the frictional force is negative

Zero work

Under three conditions Work can be zero

a. If the force is perpendicular to the displacement

Means $Θ = \frac{π}{2}$

E.g-When a body moves in a circle the work done by the centripetal force is always zero.

b. If there is no displacement (means s = 0)

E.g- When a person tries to displace a wall by applying a force and can't able to move the wall

So the work done by the person on the wall is zero.

c. If there is no force acting on the body (means F=0)

E.g- Motion of an isolated body in free space.

Solved Examples Based on Work Done By a Constant Force

Example 1: A force $\vec{F} = 5 \hat{i} - 4 \hat{j} + \hat{k}$ moves a body of mass 10 Kg from position (3,4,5) to position (6,7,8). Find the work (Joule) done :

Solution:

The scalar product of the force vector $(\vec{F})$ and the displacement vector $(\vec{S})$
$\begin{aligned} w = \vec{F} \cdot \vec{S} \\ \vec{s_{1}} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \\ \vec{s_{2}} = 6 \hat{i} + 7 \hat{j} + 8 \hat{k} \\ \vec{S} = \vec{s_{2}} - \vec{s_{1}} = 3 \hat{i} + 3 \hat{j} + 3 \hat{k} \\ w = \vec{F} \cdot \vec{s} = (5 \hat{i} - 4 \hat{j} + \hat{k}) \cdot (3 \hat{i} + 3 \hat{j} + 3 \hat{k}) = 15 - 12 + 3 = 6 J \end{aligned}$

Hence, the answer is the option (4).

Example 2: A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle, the motion of the particle takes place in a plane. It follows that :

its velocity is constant

its acceleration is constant

its kinetic energy is constant (correct)

it moves in a straight line.

Solution:

Net work done by all the forces gives the change in kinetic energy -

$\begin{aligned} W = \frac{1}{2} m v^{2} - \frac{1}{2} m v_{0}^{2} \\ W = k_{f} - k_{i} \\ - wherein \\ m = mass of the body \\ v_{0} = initial velocity \\ v = final velocity \end{aligned}$
- wherein
$m =$ mass of the body
$v_{0} =$ initial velocity
$v =$ final velocity
As $w = \vec{F} \cdot \vec{S}$ or
$W = \int \vec{F} \cdot \vec{v} d t$

In this case, since force is constant and it is always perpendicular to velocity, the work done by this force is zero. Hence kinetic energy remains constant.

Hence, the correct answer is option(3)

Example 3: A block of mass m is kept on a platform which starts from rest with constant acceleration g/2 upward, as shown in the figure. Work done by the normal reaction on the block in time t is:

1. $- \frac{m g^{2} t^{2}}{8}$
2. $\frac{m g^{2} t^{2}}{8}$
3. 0
4) $\frac{3 m g^{2} t^{2}}{8}$

Solution:

The scalar product of the force vector $(\vec{F})$ and the displacement vector $(\vec{S})$
By balancing the forces in the vertical direction, we get:
$\Rightarrow N - M g = \frac{M g}{2} \Rightarrow N = \frac{3 m g}{2}$
and displacement is given by the equation:
$s = u t + \frac{1}{2} a t^{2}$
where $u = 0$ and $a = g / 2$
$\Rightarrow \vec{S} = \frac{1}{2} \frac{\vec{g}}{2} t^{2}$
and we know,
$\begin{aligned} W_{N} = \vec{N} \cdot \vec{S} \\ \Rightarrow W = \frac{3 m g^{2} t^{2}}{8} \end{aligned}$

Example 4: A particle which is experiencing a force, given by $\vec{F} = 3 \vec{i} - 12 \vec{j},$ undergoes a displacement of $\vec{d} = 4 \vec{i}$ . If the particle had a kinetic energy of 3J at the beginning of the displacement, What is its kinetic energy (in Joule) at the end of the displacement?

Solution:

Net work done by all the forces gives the change in kinetic energy -

$\begin{aligned} W = \frac{1}{2} m v^{2} - \frac{1}{2} m v_{0}^{2} \\ W = k_{f} - k_{i} \\ - wherein \\ m = mass of the body \\ v_{0} = initial velocity \\ v = final velocity \\ W = F . d = 3 \times 4 = 12 J \\ W_{net} = Δ K E \\ 12 = Kf - 3 \\ Kf = 15 J \end{aligned}$

Hence, the answer is the option (3).

Example 5: A lift of mass m is moving in an upward direction with acceleration g/4 , its displacement is 'h'. Find out work (in mgh) done by the tension force

1) 0.25

2) 1.25

3) 2

4) 0.75

Solution:

1722492902676

$\begin{aligned} T - mg = m (g / 4) \\ T = 5 / 4 mg \\ W = F S \cos θ \\ W = [5 / 4 mg] h \cos θ \\ W = 5 / 4 mgh \end{aligned}$

Hence, the answer is option (2).

Summary

If nothing is actually moving, no work is done- no matter how great the force is involved. Work is a scalar quantity but you can have positive and negative work. positive work is where the force pulls in the same direction as the movement. Negative work is where the force is in the opposite direction. This is the very first concept of work energy and power after this we generally study the Work Done By Variable Force.

Frequently Asked Questions (FAQs)

1. What is work in physics, and how is it related to force?

In physics, work is the energy transferred when a force moves an object. It's calculated as the product of force and displacement in the direction of the force. Work is done only when the force causes the object to move in the direction of the force.

2. What's the difference between work and energy?

Work is the process of energy transfer, while energy is the capacity to do work. When work is done on an object, energy is transferred to or from that object. Work is measured in the same units as energy (joules) because it represents a change in energy.

3. If a force is applied but an object doesn't move, is work done?

No work is done if there's no displacement, even if a force is applied. Work requires both force and displacement in the direction of that force. For example, pushing against a wall doesn't do work if the wall doesn't move.

4. What's the significance of the joule as the unit of work?

The joule (J) is the SI unit of work and energy. One joule is the work done when a force of one newton causes a displacement of one meter in the direction of the force. It provides a standardized measure for comparing energy transfers across different systems.

5. How is work related to power?

Power is the rate at which work is done, or energy is transferred. Mathematically, power is work divided by time (P = W/t). While work focuses on the total energy transfer, power considers how quickly that transfer occurs.

6. How does the angle between force and displacement affect work done?

The angle between force and displacement affects work through the cosine component. Work is calculated as W = F * d * cos(θ), where θ is the angle between force and displacement. Maximum work is done when the force is parallel to displacement (cos 0° = 1), and no work is done when they're perpendicular (cos 90° = 0).

7. How does the work done by a constant force relate to the area under a force-displacement graph?

For a constant force, the work done is equal to the area under the force-displacement graph. This is because work is the product of force and displacement, which geometrically represents a rectangle's area on this graph.

8. Can a constant force do zero work?

Yes, a constant force can do zero work if it's perpendicular to the object's displacement. For example, when an object moves in a circular path, the centripetal force does no work because it's always perpendicular to the motion.

9. Why is work a scalar quantity despite involving force (a vector)?

Work is a scalar because it represents energy transfer, which is a scalar quantity. Although force is a vector, work is calculated using only the component of force in the direction of displacement, resulting in a scalar product.

10. How does friction affect the work done on an object?

Friction typically does negative work on a moving object, reducing its kinetic energy. The work done by friction is calculated as the product of the frictional force and the distance traveled, always opposing the motion and thus having a negative sign.

11. Can work be negative? If so, what does it mean?

Yes, work can be negative. Negative work occurs when the force applied is in the opposite direction of the object's motion. This means the force is removing energy from the system, like when friction slows down a moving object.

12. If a person carries a heavy box while walking on a level floor, is work being done?

No work is done in this scenario, despite the person's effort. Work requires displacement in the direction of the force. While walking on a level floor, the upward force to hold the box doesn't cause vertical displacement, so no work is done against gravity.

13. How does the concept of work apply to lifting objects?

When lifting an object, work is done against gravity. The work done is equal to the object's weight multiplied by the height it's lifted (W = mgh, where m is mass, g is gravitational acceleration, and h is height). This work increases the object's gravitational potential energy.

14. Can work be done by non-contact forces like gravity?

Yes, non-contact forces like gravity can do work. When an object falls, gravity does positive work on it, increasing its kinetic energy. Conversely, when an object is thrown upward, gravity does negative work, decreasing its kinetic energy.

15. How does the work-energy theorem relate work to kinetic energy?

The work-energy theorem states that the net work done on an object equals its change in kinetic energy. Mathematically, W_net = ΔKE = 1/2 * m * (v_f^2 - v_i^2), where m is mass, and v_f and v_i are final and initial velocities.

16. How does the work-energy principle apply to systems with multiple objects?

The work-energy principle applies to systems as a whole: the total work done on a system equals its change in mechanical energy (kinetic + potential). For multiple objects, you consider the work done on each object and the energy transfers between them.

17. How does the work-energy theorem apply to systems with varying mass?

For systems with varying mass, like rockets, the work-energy theorem must be modified. The change in kinetic energy isn't just due to external work, but also includes the effect of mass changes. This leads to more complex equations of motion and energy analysis.

18. How does the concept of work apply to quantum systems?

In quantum systems, work takes on a more abstract meaning. It's often defined in terms of changes in the system's Hamiltonian (energy operator). Understanding quantum work is crucial for developing quantum thermodynamics and explaining phenomena like quantum heat engines.

19. How does the concept of work apply to fields like electromagnetism?

In electromagnetism, work is done when charges move in electric fields or when currents flow in magnetic fields. The work done by electromagnetic forces can change the kinetic or potential energy of charged particles, and is fundamental to understanding phenomena like electromagnetic induction.

20. Can work be done by fictitious forces in non-inertial reference frames?

Fictitious forces, like the Coriolis force in rotating reference frames, can appear to do work in non-inertial frames. However, this "work" doesn't represent a real energy transfer when viewed from an inertial frame. Understanding this helps in correctly analyzing motion in rotating systems.

21. How does the concept of work relate to the virial theorem in physics?

The virial theorem relates the time-averaged kinetic and potential energies of a stable system of particles. It's derived from considerations of work and energy. For example, in gravitational systems, it states that the average kinetic energy is half the magnitude of the average potential energy, providing insights into the behavior of bound systems.

22. What's the relationship between work and adiabatic processes in thermodynamics?

In an adiabatic process, no heat is exchanged with the surroundings, so any change in the system's internal energy must be due to work. Understanding this helps in analyzing processes like the adiabatic compression in diesel engines or the adiabatic expansion of gases in the atmosphere.

23. How does the concept of work apply to relativistic systems?

In relativistic systems, the classical definition of work must be modified. The work-energy theorem still holds, but kinetic energy includes relativistic effects. Understanding relativistic work is crucial for analyzing high-energy particle physics and astrophysical phenomena.

24. How does the concept of work relate to the uncertainty principle in quantum mechanics?

The uncertainty principle limits our ability to simultaneously know an object's position and momentum precisely. This affects how we understand work at the quantum level, as the classical definition involving force and displacement becomes problematic. Quantum work often involves statistical averages and probability distributions rather than deterministic values.

25. Can a force do positive work in one frame of reference and negative work in another?

Yes, the work done by a force can be positive in one frame and negative in another. Work depends on the relative motion between the force and the object. For example, a person walking in a moving train does positive work relative to the train but might do negative work relative to the ground if the train moves faster in the opposite direction.

26. How does the work done on a spring relate to its elastic potential energy?

The work done on a spring is stored as elastic potential energy. For an ideal spring, W = 1/2 * k * x^2, where k is the spring constant and x is the displacement from equilibrium. This work increases the spring's potential energy by the same amount.

27. What's the relationship between work and conservative forces?

Work done by conservative forces, like gravity or ideal springs, is path-independent and reversible. The total work done by these forces in a closed path is zero. This property allows for the definition of potential energy for conservative forces.

28. How does the concept of work apply to rotational motion?

In rotational motion, work is done by torque causing angular displacement. The work is calculated as W = τ * θ, where τ is torque and θ is the angle of rotation. This work changes the rotational kinetic energy of the object.

29. Can work be done on a system without changing its kinetic energy?

Yes, work can be done without changing kinetic energy if it's converted to other forms of energy. For example, work done to compress a spring increases its elastic potential energy without necessarily changing its kinetic energy.

30. How does the work done by a varying force differ from that of a constant force?

For a varying force, the work done is calculated by integrating the force over the displacement. Graphically, it's represented by the area under the force-displacement curve, which may not be a simple rectangle as with constant force.

31. What's the significance of the dot product in calculating work?

The dot product in work calculation (W = F · d) accounts for the directional relationship between force and displacement. It ensures that only the component of force parallel to the displacement contributes to work, reflecting the physical reality of energy transfer.

32. Can a force do work without changing an object's speed?

Yes, a force can do work without changing speed if it changes the object's direction. For example, in uniform circular motion, the centripetal force constantly does work to change the velocity's direction without altering its magnitude (speed).

33. How does air resistance affect the work done on a falling object?

Air resistance does negative work on a falling object, opposing its motion. This reduces the object's kinetic energy gain compared to free fall. As a result, the object reaches a terminal velocity where the work done by gravity equals the negative work done by air resistance.

34. What's the relationship between work and potential energy in a gravitational field?

The work done against gravity increases an object's gravitational potential energy. When an object moves in a gravitational field, the change in its gravitational potential energy is equal to the negative of the work done by gravity: ΔPE = -W_gravity.

35. How does the concept of work apply to machines and mechanical advantage?

In ideal machines, the work input equals the work output (conservation of energy). Mechanical advantage allows a smaller input force to do the same work over a larger distance, trading force for distance while keeping the total work constant.

36. Can work be done instantaneously?

Theoretically, work cannot be done instantaneously because it requires displacement, which takes time. In practice, we often approximate very quick energy transfers as instantaneous, but this is a simplification of the underlying continuous process.

37. How does the work done by internal forces affect a system's total energy?

Internal forces within a closed system do not change the system's total energy. While they may redistribute energy within the system (e.g., between kinetic and potential forms), the total work done by internal forces on the entire system is zero.

38. What's the significance of the work-energy theorem in collision analysis?

The work-energy theorem is crucial in collision analysis, especially for inelastic collisions. It helps calculate energy losses during collisions by comparing the system's kinetic energy before and after the collision, with the difference representing work done by non-conservative forces.

39. How does the concept of pseudo-work relate to constrained motion?

Pseudo-work refers to the apparent work done by constraint forces in systems with constrained motion. While these forces may seem to do work, they actually don't contribute to energy changes. Understanding pseudo-work helps in correctly analyzing energy in constrained systems.

40. Can work be done on a particle in equilibrium?

A particle in equilibrium experiences no net force and no acceleration. However, work can still be done on parts of the system or by individual forces, as long as the net work is zero. This concept is important in understanding static equilibrium situations.

41. How does the work done by a force field relate to its conservative nature?

In a conservative force field, the work done is path-independent and depends only on the start and end points. This allows for the definition of a potential energy function. For non-conservative fields, work depends on the path and cannot be expressed as a simple difference in potential energy.

42. What's the relationship between work and heat in thermodynamics?

In thermodynamics, work and heat are two ways of transferring energy between a system and its surroundings. While work involves a force causing displacement, heat involves energy transfer due to temperature differences. Both contribute to changes in the system's internal energy.

43. How does the concept of virtual work apply to equilibrium problems?

Virtual work is a powerful method for analyzing equilibrium in complex systems. It considers hypothetical, infinitesimal displacements to determine if a system is in equilibrium. If the virtual work for any such displacement is zero, the system is in equilibrium.

44. Can negative work increase an object's energy?

While negative work generally decreases an object's energy, it can indirectly increase energy in certain contexts. For example, negative work done by a restraining force on a spring can increase the spring's potential energy, even though it decreases the kinetic energy of the object stretching the spring.

45. What's the significance of work in understanding energy conservation?

Work is crucial in understanding energy conservation because it quantifies energy transfer between different forms or systems. The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another, which is often facilitated through work.

46. Can work be done in a vacuum?

Yes, work can be done in a vacuum. Forces like gravity and electromagnetic forces can act through a vacuum and do work. For example, the gravitational force between celestial bodies does work even in the vacuum of space, changing their kinetic and potential energies.

47. How does the work done by a constant force relate to impulse?

While work relates force to displacement, impulse relates force to time. For a constant force, impulse (F * t) changes momentum, while work (F * d) changes energy. They're connected through the object's velocity: in some cases, the work done equals the impulse multiplied by the average velocity.

48. What's the significance of understanding work in analyzing energy efficiency?

Understanding work is crucial for analyzing energy efficiency in systems. By comparing the useful work output to the total work input, we can calculate efficiency. This concept is vital in engineering for optimizing machines and processes to minimize energy waste.

49. Can work be done in reversible processes, and how does it differ from irreversible processes?

Work can be done in both reversible and irreversible processes. In reversible processes, the work done is maximum (for work extracted) or minimum (for work input). Irreversible processes always involve some energy dissipation, resulting in less useful work. This concept is fundamental to understanding the limits of efficiency in thermodynamic cycles.

50. What's the significance of understanding work in the context of energy harvesting and sustainable technologies?

Understanding work is crucial for developing energy harvesting and sustainable technologies. It helps in designing systems that efficiently convert various forms of energy (like solar, wind, or mechanical) into useful work. This knowledge is essential for improving the efficiency of renewable energy sources and reducing reliance on non-renewable resources.