Calculating Work Done (College Board AP® Physics 1: Algebra-Based): Revision Note

Work done equation

• When an applied force displaces an object over a distance, work is done

  • If a force is applied but the object is not displaced, then no work is done

• For a constant force applied parallel to the displacement of the object, work is calculated using the equation:

$W=Fd$

• Where:

  • $W$ = work, measured in $\mathrm{N}·\mathrm{m}$

  • $F$ = applied force parallel to displacement, measured in $\mathrm{N}$

  • $d$ = distance, measured in $\mathrm{m}$

• For example, if a bucket of water is raised from a well by pulling a rope with a force of $200\mathrm{N}$ through a height of $2\mathrm{m}$, then the work done is:

$W=200·20=4000\mathrm{N}·\mathrm{m}$

Work done at an angle

• Only the components of the applied force that are parallel to the displacement contribute to the work done

• When a constant force is applied at an angle to the displacement of the object, work can be calculated using:

$W=F_{\parallel }d=Fd\left(\cos \theta \right)$

• Where:

  • $W$ = work, measured in $\mathrm{N}·\mathrm{m}$

  • $F_{\parallel }$ = component of the force that is parallel to the displacement of the object, measured in $\mathrm{N}$

  • $d$ = distance, measured in $\mathrm{m}$

  • $\theta$ = angle at which the force is applied, measured in $°$ from the displacement vector

• The point of application of the force will determine the system's total energy

• When the angle of the applied force is $0\le \theta \le 90°$:

  • $\cos \theta$ is positive

  • work done is positive

  • the speed of the object increases

  • the kinetic energy of the object increases

• When the angle of the applied force is perpendicular to the displacement, $\theta =90°$

  • $\cos \theta$ is zero

  • work done is zero

  • the speed of the object does not change

  • the kinetic energy of the object does not change

• When the angle of the applied force is $90\le \theta \le 180°$:

  • $\cos \theta$ is negative

  • work done is negative

  • the speed of the object decreases

  • the kinetic energy of the object decreases

• When a force is applied perpendicularly to the direction of the displacement of the object, it is possible for the object to change direction without changing the kinetic energy of the object

• Examples include:

  • elastic collisions (detailed in the study guide on Elastic and inelastic collisions)

  • circular motion (detailed in the study guide on Uniform circular motion)

Force-versus-displacement graph

• If the force applied on an object is not constant, then a force-versus-displacement graph can be used to determine the work done

• The force applied to an object may not always be constant

  • This is more representative of real-world scenarios

• In this case, the parallel component of the variable force can be plotted as a function of the displacement of the object and the work done is then equal to the area under the graph

• The work done is equivalent to the area under the graph, whether there is:

  • a small force applied over a large displacement 

  • a large force applied over a small displacement