Study Guide -- Mechanical Energy (Mr. Rogers AP Physics C )

Unit Plan Practice Test Study Guide

Mathematical models

Work   Potential Energy vs Conservative Force
W = ∫ F(s) ds The most general form.   F(x) = - dU / dx (the spring force at x) = (the slope at a point on the U vs x curve)
W = F d Work done by a constant force   U = - ∫ F(x) dx
(U at a given value of x) = (the area under the F vs x curve from 0 to x)
(U at a given value of x) = (the work required to compress the spring)
Wc = Ui - Uf Work against a conservative force  
    Fs = - kx Linear spring force
Forms of Mechanical Energy   Power
K  = 1/2 mv2 the work needed to accelerate an object from rest to its current velocity   P = W / Δt The rate of doing work.
Us = 1/2 kx2 the work required to compress a linear spring.   P = F (Δx / Δt) The rate of doing work with a constant force at an average velocity.
Ug = mgh the minimum work needed to move a mass from one position to another in a constant gravity field   P = dE / dt The rate of using energy

Key Principles

  • Positive work: a force doing positive work on an object will increase the object's kinetic energy.
  • Negative  work: a force doing negative work on an object will decrease the object's kinetic energy
  • Negative work--conservative force: kinetic energy transferred potential energy.
  • Negative work--sliding friction: kinetic energy transferred into heat.

Conservation of Energy: often called the first law of thermodynamics, conservation of energy says that energy can change forms but never be created or destroyed. This law is as close to absolute truth as anything in all of science.

All forms of energy are scalars (including power)


  • Energy: joule = (kg) (m2 / s2)
  • Power: watt = j /s

Vector Multiplication

Dot Product: one of 2 ways to multiply vectors.
  • Gives a maximum value when the two vectors are in the same dimension
  • Gives a value of zero when the two vectors are in different dimensions (ө = 90).
  • Dot products always yield a scalar.

For the constant force vector F and displacement vector d:

W = F d
     = F (d) cos ө
W = work
ө = the angle between the two vectors


  Cross Product: one of 2 ways to multiply vectors.
  • Gives a value of zero when the two vectors are in the same dimension.
  • Gives a maximum value when the two vectors are in different dimensions (ө = 90)
  • Cross products always yield a vector.

For the force vector F and displacement vector v:

t = F x d
     = F (d) sin ө
t = torque
ө = the angle between the two vectors


Problem Solving Tips: Mechanical Energy


Example Problems




mechanical energy kinetic energy conservative force
work potential energy simple harmonic motion
dot product (scalar product)   periodic motion

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