The work-energy theorem connects the work done on a particle to its change in kinetic energy. It's a powerful tool for analyzing motion, allowing us to calculate velocities and displacements without needing to know the entire path of an object.

This theorem bridges the concepts of force, work, and energy. By understanding how work relates to changes in kinetic energy, we can solve complex problems involving particle motion and energy transfer in various physical scenarios.

Work-Energy Theorem

Work-energy theorem for particle motion

• States net work done on a particle equals change in its kinetic energy $W_{net} = \Delta KE$
  • $W_{net}$ represents net work done on particle by all forces
  • $\Delta KE$ represents change in particle's kinetic energy
• Calculate net work by summing work done by each force acting on particle $W_{net} = W_1 + W_2 + ... + W_n$
  • Calculate work done by a force using product of force and displacement in direction of force $W = \vec{F} \cdot \vec{d}$
    • For constant force $W = Fd \cos \theta$, $\theta$ represents angle between force and displacement vectors
    • For force varying with position $W = \int_{x_1}^{x_2} F(x) dx$
• Calculate change in kinetic energy by subtracting initial from final kinetic energy $\Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
  • $m$ represents particle's mass
  • $v_i$ and $v_f$ represent particle's initial and final velocities
• Determine particle's final velocity or displacement by applying work-energy theorem given initial conditions and acting forces (sliding block, roller coaster)
• Work-energy theorem relates to energy transfer between different forms (e.g., kinetic to potential energy)

Forces from motion using work-energy

• Determine net work done on particle using work-energy theorem if initial and final velocities (or kinetic energies) and displacement are known
• Rearrange work-energy theorem to solve for net work $W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
• Calculate work done by unknown force using net work and known forces $W_{unknown} = W_{net} - (W_1 + W_2 + ... + W_n)$
• Determine average force exerted on particle using work-displacement relationship $F_{avg} = \frac{W_{unknown}}{d \cos \theta}$ once work done by unknown force is calculated (pulling a sled, pushing a cart)

Kinetic energy changes from net work

• Calculate change in particle's kinetic energy directly from net work done on it $\Delta KE = W_{net}$
• To find change in kinetic energy:
  1. Calculate net work done by all forces acting on particle
  2. Equate net work to change in kinetic energy
• Alternatively, calculate change in kinetic energy using particle's initial and final velocities $\Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
  • Equation derived from work-energy theorem and definition of kinetic energy $KE = \frac{1}{2}mv^2$
• Understanding relationship between work and changes in kinetic energy is crucial for analyzing motion of particles under influence of forces (collision, projectile motion)

Energy Conservation and Mechanical Energy

• Conservation of energy principle states that total energy in an isolated system remains constant
• Mechanical energy is the sum of kinetic and potential energy in a system
• In conservative systems, mechanical energy is conserved when no non-conservative forces do work
• Power is the rate at which work is done or energy is transferred, measured in watts (W)