The Work-Energy Theorem for a Variable Force

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The work-energy theorem is connects the change in an object’s kinetic energy to the work that forces exert on it.
The Work-Energy Theorem for a Variable Force-Energy
Energy

What is the Work-Energy Theorem?

According to the work-energy theorem, the change in an object’s kinetic energy equals the work that the net force acting on it. This can be stated mathematically as:  Wnet = ΔKE = KEfinal − KEinitial
Where Wnet​ is the total work done, and KE represents the kinetic energy of the object.  Kinetic energy is given by:  KE = 1 / 2mv2
where m is the mass of the object and v is its velocity.
Since the force F may vary in magnitude or direction during motion, the work done for a variable force is determined through integration.

Work Done by a Variable Force

We cannot just multiply a force by the displacement as we would for a constant force when the force fluctuates along the displacement. The work is computed by adding up the tiny quantities of work performed over very small displacements dx. The expression for work becomes:  W = F(x) dx
  • Here:  F(x) is the variable force as a function of position x.
  • dx is a small displacement along the direction of the force.
If the motion occurs in three dimensions, the work is calculated using the dot product of the force and displacement vectors:
W = F ⋅ dr  where F is the force vector and dr is the displacement vector.
The Work-Energy Theorem for a Variable Force-Force
Force

Deriving the Work-Energy Theorem for a Variable Force

To derive the theorem, we start with Newton’s second law: 
     Fnet = ma
The net force Fnet can also be written in terms of work: 
Wnet = Fnet ⋅ dr
Substituting Fnet = ma we have: Wnet = ma ⋅ dr
Acceleration a can be expressed as dv / dt. Substituting this, the work becomes:
 Wnet = mdv/ dt⋅dr.  Using the relationship dr = v dt, we rewrite the equation: Wnet = mv ⋅ dv
This simplifies to: Wnet = mv . dv
Performing the integration gives:  Wnet = 1 / 2 mv2∣v initial v final​​
Thus:  Wnet = 1 / 2mv2 final – 1 / 2mv2initial​
This proves that the net work done on the object equals the change in its kinetic energy.
The Work-Energy Theorem for a Variable Force-
Object

Applications of the Work-Energy Theorem

1.Variable Forces in Real Life:
    • Springs, where the restoring force depends on the displacement (F = −kx).
    • Air resistance, which varies with velocity.
      2. Roller Coasters: The theorem describes how frictional and gravitational forces combine to alter a roller coaster’s speed at various locations.
      3. Rocket Propulsion: This idea is essential to space travel since rockets are subject to variable thrust forces.

Why It Matters

A flexible structure for analysing motion without having to deal with forces and accelerations directly is offered by the work-energy theorem. By concentrating on energy changes than complex force computations, it streamlines difficulties, particularly when forces fluctuate.
According to the work-energy theorem, the change in an object’s kinetic energy is equal to the net work performed by all forces acting on it. Integration is used to determine the work for a variable force, when the force varies with position or time:  W = F(x) dx
This allows us to find the total work done by summing up the small contributions of force over the object’s path.
When the force varies, work cannot be calculated using the simple formula W=F⋅d. Instead, the work is determined by integrating the force over the displacement: W = F(x) dx
In three dimensions, the formula becomes: W = F ⋅ dr
This accounts for changes in both the magnitude and direction of the force.
The work-energy theorem directly connects the work done on an object to its change in kinetic energy. Mathematically: 
Wnet = ΔKE = KEfinal − KEinitial​
This means if work is positive, the object gains kinetic energy (speeds up), and if work is negative, the object loses kinetic energy (slows down).
Since a variable force does not stay constant across displacement, integration is used. It computes the work performed by the force in each of the tiny segments that make up the displacement. The overall amount of work completed throughout the entire journey is obtained by adding up these contributions.
Some common examples of variable forces include:
  • Spring force: The restoring force in a spring changes with displacement (F=−kx).
  • Gravitational force: When dealing with large distances from Earth, gravity changes with distance
    (F = Gm1m2 / r2​​).
  • Air resistance: It depends on velocity and varies as the object moves through the air.
By emphasising energy changes over complex force computations, the work-energy theorem streamlines issues. For example: 
When energy transformations are important, it eliminates calculating complicated equations of motion and is utilised in situations with varying forces, such as stretching a spring or objects under uneven gravitational fields.
Yes, there are drawbacks to labour. When the force acts against the displacement, this is known as negative work. For example, friction reduces kinetic energy by opposing motion, which results in negative work.
• When an object resists compression, a compressed spring exerts negative work on it.
The object’s kinetic energy is reduced by negative work, which slows or stops it.

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