Work-Energy Theorem

Understanding the Work-Energy Theorem

What is the Work-Energy Theorem?

The Work-Energy Theorem is a fundamental concept in physics that links the work done on an object to its change in kinetic energy. It states that the work done by the forces acting on an object is equal to the change in its kinetic energy. This theorem is crucial for solving various problems in mechanics, especially in exams like JEE Mains and Advanced.

Mathematical Expression

The Work-Energy Theorem can be expressed mathematically as:

W = ΔK

where:

• W is the work done by the forces acting on the object.
• ΔK is the change in the kinetic energy of the object.

In formula terms, if K_i is the initial kinetic energy and K_f is the final kinetic energy, then:

W = K_f - K_i

How to Apply the Work-Energy Theorem

To apply the Work-Energy Theorem effectively, follow these steps:

1. Identify the Forces: Determine the forces acting on the object. This could include gravity, friction, applied forces, etc.
2. Calculate Work Done: Compute the work done by each force. Work is calculated as W = F × d × cos(θ), where F is the force, d is the displacement, and θ is the angle between the force and displacement vectors.
3. Determine Initial and Final Kinetic Energies: Find the initial and final kinetic energies of the object using K = ½ m v², where m is the mass and v is the velocity.
4. Apply the Theorem: Use the Work-Energy Theorem to solve for unknown quantities or verify results.

Example: Consider a block of mass 2 kg sliding down a frictionless incline. If it starts from rest and reaches a speed of 10 m/s at the bottom, calculate the work done by gravity.

Initial kinetic energy, K_i = 0 (since it starts from rest)

Final kinetic energy, K_f = ½ × 2 × 10² = 100 J

Work done by gravity, W = K_f - K_i = 100 J - 0 J = 100 J

Understanding Work Done by Non-Conservative Forces

When non-conservative forces like friction are present, the Work-Energy Theorem still applies but includes the work done by these forces. For instance, in the presence of friction, some of the work done may be converted into thermal energy, and the equation adjusts to:

W_nc + W_c = ΔK

where W_nc is the work done by non-conservative forces, and W_c is the work done by conservative forces (like gravity).

Key Points to Remember

• The Work-Energy Theorem is useful for solving problems involving forces and motion.
• Work done by conservative forces changes the kinetic energy of the system.
• Non-conservative forces like friction result in energy loss or conversion to other forms.
• Always check the direction of forces and displacements to correctly compute work done.

Conclusion

The Work-Energy Theorem is a powerful tool in mechanics that simplifies the analysis of forces and motion. By understanding how to apply this theorem, you can solve complex problems efficiently and gain a deeper insight into the principles governing physical systems. Practice applying the theorem to various problems to build your skills for JEE Mains and Advanced.