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Work-Energy Theorem

Introduction​

The Work-Energy Theorem is a fundamental principle in physics that relates the work done on an object to its change in kinetic energy. This theorem provides a powerful tool for analyzing the motion of objects under the influence of forces, especially when the motion is complex or the forces are variable.

Definition​

The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy.

Mathematically: W_net = ΔK

Where:

  • W_net is the net work done on the object
  • ΔK is the change in kinetic energy of the object

Key Concepts​

1. Work​

Work is defined as the product of force and displacement in the direction of the force.

W = F · d = F d cos(θ)

Where:

  • W is work
  • F is force
  • d is displacement
  • θ is the angle between force and displacement vectors

2. Kinetic Energy​

Kinetic energy is the energy an object possesses due to its motion.

K = 1/2 mv^2

Where:

  • K is kinetic energy
  • m is mass
  • v is velocity

Mathematical Derivation​

The Work-Energy Theorem can be derived from Newton's Second Law:

  1. Start with F = ma
  2. Multiply both sides by displacement: F · d = ma · d
  3. Recognize that work W = F · d
  4. Use calculus to show that a · d = (v^2 - v_0^2) / 2
  5. Substitute to get: W = m(v^2 - v_0^2) / 2 = ΔK

Applications​

  1. Simple Machines: Analyzing the efficiency of levers, pulleys, and inclined planes.
  2. Roller Coasters: Calculating speeds at different points based on height changes.
  3. Ballistics: Determining the range of projectiles.
  4. Celestial Mechanics: Studying the motion of planets and satellites.
  5. Particle Physics: Analyzing collisions in particle accelerators.

Limitations and Considerations​

  1. The theorem applies to the net work done by all forces, including conservative and non-conservative forces.
  2. It doesn't directly account for changes in potential energy, but can be extended to include them.
  3. For systems where energy is not conserved (e.g., due to friction), additional terms may be needed.

Example Problems​

Problem 1: Block on a Frictionless Incline​

A 2 kg block slides down a frictionless incline that is 5 m long and inclined at 30° to the horizontal. If the block starts from rest, what is its speed at the bottom of the incline?

TODO: Add solution.

Problem 2: Work Done by a Variable Force​

A force F(x) = (3x^2 + 2x) N acts on a 1 kg particle as it moves along the x-axis from x = 0 to x = 2 m. If the particle starts from rest, what is its final velocity?

TODO: Add solution.

Problem 3: Stopping Distance​

A 1500 kg car is traveling at 20 m/s when the driver applies the brakes. If the coefficient of kinetic friction between the tires and the road is 0.6, how far does the car travel before coming to a stop?

TODO: Add solution.

Problem 4: Compound System​

A 50 kg crate is pushed up a rough incline (coefficient of friction μ = 0.2) that is 3 m high and 4 m long. If a horizontal force of 500 N is applied to the crate, what is its speed at the top of the incline if it starts from rest at the bottom?

TODO: Add solution.

Conclusion​

The Work-Energy Theorem is a powerful tool in physics that allows us to analyze complex motions by focusing on the initial and final states of a system, rather than the details of its path. By relating work to changes in kinetic energy, it provides a method for solving problems that might be difficult to approach using force and acceleration directly.

This theorem forms a bridge between the concepts of force and energy, and it serves as a foundation for understanding more advanced topics in mechanics, such as conservation of energy and potential energy. Mastering the Work-Energy Theorem is crucial for students of physics and engineering, as it provides insights into a wide range of real-world phenomena and technological applications.