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Constant-force work

For a force $F$ acting through displacement $s$ ,

$W=Fs\cos\theta$

Only the component parallel to displacement transfers energy by work.

Area under a force–distance graph

For a variable force, the area under an $F$ -against- $s$ graph gives work. A negative area represents work against the chosen displacement direction.

Check the angle

Use the angle between force and displacement, not the angle between the force and an unrelated axis unless the component has first been resolved.

Worked example from local Question Bank row 39177

A kite pulls a ship with force $2.50\times10^5\,\mathrm{N}$ at $39^\circ$ to its $1.00\,\mathrm{km}$ displacement. Convert $1.00\,\mathrm{km}=1.00\times10^3\,\mathrm{m}$ , then

$W=Fs\cos\theta=(2.50\times10^5)(1.00\times10^3)\cos39^\circ=1.94\times10^8\,\mathrm{J}\approx1.9\times10^8\,\mathrm{J}$

Only the force component along the ship's displacement transfers energy.

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State what is represented by the area under the graph.

Account for energy

Define the system, identify energy stores and describe transfers. Work done by a force transfers energy; total energy is conserved even when mechanical energy is not.

Use the mechanical model

$E_k= rac12mv^2,\quad \Delta E_{p,g}=mg\Delta h,\quad E_{p,elastic}= rac12k(\Delta x)^2$

Conserve their sum only when resistive transfers are absent or included explicitly.

Use rates and ratios

$P= rac{\Delta E}{\Delta t}=Fv,\qquad \eta= rac{E_{useful}}{E_{input}}= rac{P_{useful}}{P_{input}}$

Fuel energy density connects available input energy to a chosen volume.

Final checks

Check the system boundary, signs of work and potential-energy changes, the reference height, extension from natural length, and whether the quantity is energy, power, efficiency or energy density.

A.3.4—Work by constant force

Calculate Work by a Constant Force

Retrieve the A.3 Work, Energy and Power Model

A.3.4 Exam Analysis