Scivra
Pro 🔒~25 min

Work-Energy Theorem

Net work equals change in kinetic energy on inclined surfaces

How it works

The work-energy theorem states that the net work done on an object equals its change in kinetic energy. On an incline, multiple forces act simultaneously: gravity component along the slope, normal force, friction, and any applied force. Only forces with a component along the displacement do work. Friction converts mechanical energy to thermal energy, reducing the net kinetic energy gain. Power measures how quickly work is performed — the same task done in less time requires more power. Understanding this theorem bridges Newton's force-based approach with the energy-based perspective essential for thermodynamics and modern physics.

If you push a box twice as far, do you do twice the work?

Only if the force stays constant — in the real world, friction, angles, and changing forces make the answer surprisingly nuanced.

What you'll learn

  • What Is Work?. Work is energy transferred by a force acting over a displacement. Only the component of force parallel to the displacement does work. A force perpendicular to motion (like the normal force on a flat surface) does zero work, no matter how large it is.
  • Kinetic Energy. Kinetic energy is the energy of motion: ½mv². It depends on both mass and the square of velocity — so doubling speed quadruples kinetic energy. This is why car crashes at high speed are so much more devastating than at low speed.
  • The Work-Energy Theorem. The net work done on an object equals its change in kinetic energy. This powerful theorem connects force-based analysis (Newton's laws) with energy-based analysis. It works even when multiple forces act simultaneously — just sum all the work contributions.
  • Conservative vs Non-Conservative Forces. Gravity is conservative — work depends only on height change, not the path taken. Friction is non-conservative — it always removes mechanical energy as heat. On an incline with friction, the block gains less kinetic energy than gravity alone would provide because friction steals some as thermal energy.

Step-by-step

  1. Set the block mass and incline angle.
  2. Press Play to release the block and observe its acceleration down the slope.
  3. Watch the energy bar chart showing KE, gravitational PE, and thermal energy.
  4. Enable friction via the μ slider (Pro) to see energy dissipation.
  5. Add an applied force pushing the block up the slope.
  6. The work readout shows W_gravity, W_friction, W_applied, and W_net — verify that W_net = ΔKE at every snapshot.

Key formulas

  • Wnet=ΔKE=12mv2−12mv02W_{net} = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2Work-Energy Theorem: net work equals change in kinetic energy
  • W=F⋅d⋅cos⁡θW = F \cdot d \cdot \cos\thetaWork done by a constant force over displacement
  • P=Wt=F⋅vP = \frac{W}{t} = F \cdot vPower: rate of doing work or force times velocity
  • ΔEmech=Wfriction=−μmgcos⁡θ⋅d\Delta E_{mech} = W_{friction} = -\mu mg\cos\theta \cdot dMechanical energy lost to friction on the incline

Frequently asked questions

A 5 kg block slides from rest at h = 2 m on a frictionless incline. What is its speed at the bottom?
The correct answer is: v ≈ 6.26 m/s. W_net = W_gravity = mgh. Then ½mv² = mgh → v = √(2gh).
With μ = 0.3 on a 30° incline, length d = 4 m, how much energy is lost to friction?
The correct answer is: W_f ≈ 50.9 J. W_friction = μ·mg·cos30°·d. Normal force on incline = mg·cosθ.
A 5 kg block reaches 6 m/s after 10 seconds on the incline. What is the average power delivered by net force?
The correct answer is: P = 9 W. W_net = ΔKE = ½mv². P_avg = W_net / t.