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Work Done By Variable Force

Work Done By Variable Force

Edited By Vishal kumar | Updated on Jul 02, 2025 07:29 PM IST

When a force acts on an object and causes displacement, work is done. In the real world, forces often vary rather than remain constant, leading to what is known as variable force. Understanding the work done by a variable force is crucial in fields such as physics and engineering, where it helps to explain how energy is transferred in various situations. For example, when a car accelerates, the engine exerts a variable force on the car, changing its speed and kinetic energy. Similarly, when you stretch a spring or lift an object using a pulley system, the force exerted changes with distance, making the work done dependent on how the force varies with displacement. This concept is vital for designing efficient machines, optimizing energy usage, and predicting system behaviour in complex environments

This Story also Contains
  1. Work Done By Variable Force
  2. Solved Examples Based on Work Done By Variable Force
  3. Summary
Work Done By Variable Force
Work Done By Variable Force

Work Done By Variable Force

Force is a vector quantity. So it has a magnitude as well as direction. A variable force means when its magnitude its direction or both varies with position.

And work done by the variable force is given by

W=Fds

Where F is a variable force and ds is a small displacement

When Force is Time-Dependent

And we can write ds=vdt

So,

W=Fvdt
Where F and v are force and velocity vectors at any instant.

Work Done Calculation by Force Displacement Graph

The area under the force-displacement curve with the proper algebraic sign represents work done by the force.

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Solved Examples Based on Work Done By Variable Force

Background wave

Example 1: When a rubber band is stretched by a distance x, it exerts a restoring force of magnitude F=ax+bx2 where a and b are constants. The work done in stretching the unstretched rubber band by L is :

1) aL2+bL3
2) 12(aL2+bL3)
3) aL22+bL33
4) 12(aL22+bL33)

Solution:

Definition of work done by a variable force
W=Fds

wherein
F is variable force and ds is a small displacement
At x=x

F=ax+bx2
Work done in displacing rubber through dx=Fdx

W=0L(ax+bx2)dx=a0Lxdx+b0Lx2dxw=aL22+bL33

Hence, the answer is the option (3).

Example 2: A force acts on a 2kg object so that its position is given as a function of time as x = 3t2 + 5. What is the work (in Joule) done by this force in the first 5 seconds?

1) 850

2) 875

3) 950

4) 900

Solution:

Definition of work done by a variable force
W=Fds

wherein
F is variable force and ds is a small displacement
Work done = change in kinetic energy

V=dxdt=6tWork done=12mv20=12(2)(6×5)20=900J

Hence, the answer is the option (4).

Example 3: A time-dependent force F=6t acts on a particle of mass 1 kg. If the particle starts from rest, the work (in Joule) is done by the force during the first 1 sec. will be :

1) 4.5

2) 22

3) 9

4) 18

Solution:

Force is given F=6t

F=ma=mdvdt=6tdvdt=6t(Sincem=1kg)dv=6tdt
On integrating, 0vdV=601tdt=3
Therefore v=3m/s
Therefore change in kinetic energy in one second =12×m×320=4.5J
Since the change in kinetic energy is equal to the work done.

W=ΔKE=4.5J

Hence, the answer is the option (1).

Example 4: A person pushes a box on a rough horizontal surface. He applies a force of 200N over a distance of 15m. Thereafter, he gets progressively tired and his applied force reduces linearly with distance to 100N. The total distance through which the box has been moved is 30m. What is the work done by the person during the total movement of the box?

1) 3280 J

2) 2780 J

3) 5960 J

4) 5250 J

Solution:

F=200Nfor0x15=20010015(x15)for15x<30W=Fdx=015200dx+1530(30010015x)dx=200×15+300×1510015×(302152)2=3000+45002250=5250J

Hence, the answer is the option (4).

Example 5: A particle experiences a variable force F=(4xi^+3y2j^) in a horizontal xy plane. Assume distance in meters and force is Newton. If the particle moves from point (1,2) to point (2,3) in the xy plane, then the Kinetic energy changes by :

1) 50.0 J
2) 12.5 J
3) 25.0 J
4) 0 J

Solution:

The force experienced by particles in a horizontal XY plane is,

F¯=4x(i^)+3y2(j^)
Comparing with,

F¯=Fxi^+Fyj^Fx=4x,Fy=3y2dW=F¯ds¯=(Fxi^+Fyj^)(dxi^+dyj^)dW=Fxdx+FydyW=dW=x=1x=24xdx+y=2y=33y2dy=ΔKE

(from Work-energy theorem)

W=4[x22]12+3[y33]23=4[212]+3[983]W=6+19=25J=ΔKE

Hence, the answer is the option (3)

Summary

The article discusses the concept of work done by variable forces, which are forces whose magnitude, direction, or both change with position. It explains how work done by such forces can be calculated using integration and how it can be visualized using force-displacement graphs. The article also provides several solved examples to illustrate the application of these concepts, such as calculating work done by stretching a rubber band, analyzing time-dependent forces, and determining the energy changes in particles experiencing variable forces. These examples help in understanding the practical implications of variable force in real-life situations.

Frequently Asked Questions (FAQs)

1. How do you calculate work done by a variable force in a rotating reference frame?
In a rotating reference frame, additional terms like centrifugal and Coriolis forces must be included. The work is calculated by integrating these forces, along with any applied forces, over the path in the rotating frame.
2. What is the role of Hamiltonian mechanics in analyzing systems with variable forces?
Hamiltonian mechanics provides a powerful framework for analyzing systems with variable forces. It uses generalized coordinates and momenta, making it particularly useful for complex systems where forces depend on position and velocity.
3. What is meant by variable force in physics?
Variable force refers to a force that changes in magnitude, direction, or both as it acts on an object over time or distance. Unlike constant forces, variable forces can produce complex motion and require calculus to analyze fully.
4. How is work done by a variable force calculated?
Work done by a variable force is calculated by integrating the force over the displacement. Mathematically, it's expressed as W = ∫F·dx, where F is the force vector and dx is the displacement vector. This integral sums up the infinitesimal amounts of work done over the entire path.
5. Why can't we use the simple formula W = F * d for variable forces?
The formula W = F * d only applies to constant forces. For variable forces, using this formula would give incorrect results because the force changes over the displacement. Instead, we need to use integration to account for the continuously changing force.
6. What's the difference between work done by a constant force and a variable force?
Work done by a constant force is simply the product of force and displacement (W = F * d). For variable forces, work is calculated using integration (W = ∫F·dx) to account for the changing force over the displacement.
7. Can work done by a variable force be negative?
Yes, work done by a variable force can be negative. This occurs when the force opposes the direction of motion for a portion or all of the displacement. The negative work indicates that energy is being removed from the system.
8. What is the significance of Green's theorem in analyzing work done by variable forces in a plane?
Green's theorem relates line integrals around a closed curve to double integrals over the enclosed region. It's useful in simplifying calculations of work done by variable forces in two dimensions, especially for conservative force fields.
9. How does a force-displacement graph help in calculating work done by a variable force?
A force-displacement graph visually represents how the force changes over displacement. The area under this curve equals the work done by the force. For variable forces, this area can be calculated using integration or approximated using methods like the trapezoidal rule.
10. What is the significance of the dot product in the work integral (∫F·dx)?
The dot product in the work integral (∫F·dx) ensures that only the component of force parallel to the displacement contributes to the work done. It accounts for situations where the force and displacement vectors are not parallel.
11. How does the concept of path independence apply to work done by variable forces?
Work done by variable forces is generally path-dependent, meaning the work done depends on the specific path taken between two points. This is in contrast to conservative forces, where the work is path-independent.
12. What is the relationship between work done by a variable force and potential energy?
For conservative variable forces, the work done is equal to the negative change in potential energy. This relationship is expressed as W = -ΔU, where W is the work done and ΔU is the change in potential energy.
13. How does spring force relate to variable forces?
Spring force is a common example of a variable force. As a spring is stretched or compressed, the force it exerts changes according to Hooke's Law (F = -kx). This varying force makes it a perfect example for studying work done by variable forces.
14. What is the work-energy theorem and how does it apply to variable forces?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy. For variable forces, this is expressed as W = ΔK = ½mv₂² - ½mv₁², where W is the work done, and v₁ and v₂ are initial and final velocities.
15. How do you determine the direction of work done by a variable force?
The direction of work is determined by the relative directions of the force and displacement vectors. If these vectors point in the same general direction (angle < 90°), work is positive. If they point in generally opposite directions (angle > 90°), work is negative.
16. What role does calculus play in analyzing work done by variable forces?
Calculus is essential for analyzing work done by variable forces. Integration is used to sum up infinitesimal amounts of work over a path. Differential calculus helps in finding instantaneous rates of change of force with respect to displacement.
17. How does gravity act as a variable force when considering large vertical displacements?
For large vertical displacements, the gravitational force varies with height above Earth's surface according to the inverse square law. This variation makes gravity a variable force in such scenarios, requiring integration to calculate work done.
18. What is the significance of the scalar product in work calculations for variable forces?
The scalar product (dot product) in work calculations ensures that only the component of force parallel to the displacement contributes to work. It's crucial for variable forces that may not always align with the direction of motion.
19. How does air resistance exemplify a variable force?
Air resistance is a variable force that typically increases with velocity. As an object moves faster, the air resistance force increases, making it a non-constant force that requires integration to calculate the work done against it.
20. What is the relationship between power and work done by a variable force?
Power is the rate at which work is done. For variable forces, instantaneous power is given by P = F·v, where F is the force vector and v is the velocity vector. The total work is the integral of power over time.
21. How do you graphically represent work done by a variable force?
Work done by a variable force can be represented graphically as the area under the force-displacement curve. For complex force functions, this area might need to be calculated using numerical integration methods.
22. What is the principle of virtual work and how does it relate to variable forces?
The principle of virtual work states that the total virtual work done by all forces acting on a system in equilibrium is zero for any virtual displacement. This principle is particularly useful in analyzing complex systems with variable forces.
23. How does the concept of work done by variable forces apply in thermodynamics?
In thermodynamics, work done by variable forces is often encountered in processes involving changing pressure, such as the expansion or compression of gases. The work done is calculated by integrating PdV, where P is pressure and V is volume.
24. What is the difference between conservative and non-conservative variable forces?
Conservative variable forces, like gravity, have work that is path-independent and can be associated with a potential energy. Non-conservative variable forces, like friction, have work that is path-dependent and cannot be associated with a potential energy function.
25. How do you calculate work done by a variable torque?
Work done by a variable torque is calculated by integrating the torque over the angular displacement: W = ∫τ·dθ, where τ is the torque vector and dθ is the angular displacement vector.
26. What is the significance of line integrals in calculating work done by variable forces?
Line integrals are used to calculate work done by variable forces along a specific path. They allow us to sum up the infinitesimal work contributions along a curved path in two or three dimensions.
27. How does the work done by a variable force relate to the concept of energy conservation?
The work done by a variable force contributes to changes in the system's energy. For conservative forces, it relates directly to changes in potential and kinetic energy, adhering to the principle of energy conservation.
28. What is the role of vector calculus in analyzing work done by variable forces in 3D space?
Vector calculus is crucial for analyzing work done by variable forces in 3D space. It allows for the calculation of line integrals along complex paths and the analysis of force fields using concepts like curl and divergence.
29. How do you determine if a variable force field is conservative?
A variable force field is conservative if the work done is path-independent. Mathematically, this is equivalent to the curl of the force field being zero everywhere, or if the force can be expressed as the gradient of a scalar potential function.
30. What is the significance of the work-energy principle for variable forces in orbital mechanics?
In orbital mechanics, the work-energy principle for variable forces helps analyze the energy changes of celestial bodies under the influence of varying gravitational forces. It's crucial for understanding orbital transfers and escape velocities.
31. How does the concept of work done by variable forces apply to fluid dynamics?
In fluid dynamics, work done by variable forces is often encountered in analyzing fluid flow and pressure changes. It's crucial in understanding concepts like Bernoulli's principle and calculating the work done in pumping fluids.
32. What is the relationship between work done by variable forces and the concept of potential wells in physics?
Potential wells, often created by variable forces, represent regions where objects tend to be trapped due to energy considerations. The work done by these forces determines the depth and shape of the potential well, affecting the behavior of particles within it.
33. How do you calculate work done by a variable magnetic force on a moving charged particle?
Work done by a variable magnetic force on a moving charged particle is calculated using the line integral W = ∫qv×B·dr, where q is the charge, v is the velocity vector, B is the magnetic field vector, and dr is the displacement vector.
34. What is the importance of path selection in calculating work done by non-conservative variable forces?
For non-conservative variable forces, the path selection is crucial as the work done depends on the specific path taken. Different paths between the same start and end points can result in different amounts of work done.
35. How does the concept of work done by variable forces apply to quantum mechanics?
In quantum mechanics, work done by variable forces is often considered in terms of expectation values and operators. It's crucial in understanding phenomena like tunneling and in calculating transition probabilities between energy states.
36. How do you analyze work done by variable forces in non-inertial reference frames?
In non-inertial reference frames, additional fictitious forces (like Coriolis force) must be considered. The work done by these apparent forces, which vary with position and velocity, is calculated using the same integration techniques as for real variable forces.
37. What is the role of work done by variable forces in understanding phase transitions in materials?
Work done by variable forces is crucial in understanding phase transitions. It helps explain the energy changes involved as materials transition between states (e.g., solid to liquid), accounting for the varying intermolecular forces during the process.
38. How does the concept of virtual work apply to systems with variable forces?
Virtual work for systems with variable forces involves considering infinitesimal displacements and the corresponding work. It's particularly useful in analyzing equilibrium conditions in complex mechanical systems where forces vary with position.
39. What is the significance of work done by variable forces in the context of field theory?
In field theory, work done by variable forces is often analyzed in terms of energy density and flux. It's crucial for understanding how fields (like electromagnetic fields) interact with matter and transfer energy across space.
40. What is the relationship between work done by variable forces and the concept of ergodicity in statistical mechanics?
In statistical mechanics, the concept of ergodicity relates to the exploration of phase space over time. Work done by variable forces can affect how a system explores its phase space, influencing whether the system behaves ergodically or not.
41. How does the principle of least action relate to work done by variable forces?
The principle of least action states that the path taken by a system between two points is the one for which the action integral is stationary. This principle provides an alternative way to analyze the motion and work done in systems with variable forces.
42. What is the significance of Stokes' theorem in calculating work done by variable forces?
Stokes' theorem relates the surface integral of the curl of a vector field to the line integral around the boundary of the surface. It's useful in simplifying calculations of work done by variable forces in three-dimensional space, especially for conservative fields.
43. How do you analyze work done by variable forces in the context of relativistic mechanics?
In relativistic mechanics, work done by variable forces must account for the relativistic form of Newton's second law. The calculations involve the proper time and four-vectors, ensuring consistency with special relativity.
44. How does the concept of work done by variable forces apply to the study of plasma physics?
In plasma physics, work done by variable forces is crucial in understanding the behavior of charged particles in electromagnetic fields. It's essential for analyzing phenomena like plasma confinement and acceleration in varying field configurations.
45. What is the significance of the work-energy theorem in analyzing variable forces in astrophysics?
In astrophysics, the work-energy theorem helps analyze the behavior of celestial bodies under varying gravitational forces. It's crucial for understanding phenomena like tidal forces, gravitational collapse, and energy transfer in binary star systems.
46. How do you calculate work done by variable forces in the context of continuum mechanics?
In continuum mechanics, work done by variable forces is often expressed in terms of stress and strain tensors. The work is calculated by integrating these tensors over the volume of the material, accounting for varying forces throughout the continuum.
47. What is the importance of understanding work done by variable forces in the field of materials science?
In materials science, understanding work done by variable forces is crucial for analyzing material behavior under varying loads. It's essential for studying phenomena like plastic deformation, fatigue, and fracture mechanics in materials with complex stress-strain relationships.
48. How does the concept of work done by variable forces apply to the study of chemical reactions?
In chemical reactions, work done by variable forces is often considered in terms of changes in intermolecular forces as reactants transform into products. It's crucial for understanding reaction energetics, particularly in processes involving large molecular rearrangements.
49. What is the role of tensor analysis in calculating work done by variable forces in complex systems?
Tensor analysis is crucial for calculating work done by variable forces in complex systems, especially those involving anisotropic materials or multidimensional stress states. It allows for a compact representation of forces and displacements in multiple dimensions.
50. How does the understanding of work done by variable forces contribute to advancements in renewable energy technologies?
Understanding work done by variable forces is essential in renewable energy technologies. It's crucial for optimizing wind turbine designs (analyzing variable wind forces), improving solar panel efficiency (considering varying solar radiation), and enhancing energy harvesting from ocean waves (analyzing complex fluid forces).

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