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MCAT Content / Work / Work Done By A Constant Force W Fd Cos0

### Work done by a constant force: W = Fd cosθ

Topic: Work

The work done on a system by a constant force is the product of the component of the force in the direction of motion times the distance through which the force acts.

W is work, F is the magnitude of the force on the system, d is the magnitude of the displacement of the system, and θ is the angle between the force vector F and the displacement vector d.

Examples of work:

1. The work done by the force F on this lawnmower is Fdcosθ.

2. A person holding a briefcase does no work on it because there is no motion. No energy is transferred to or from the briefcase. The person moving the briefcase horizontally at a constant speed does no work on it and transfers no energy to it.

3. Work is done on the briefcase by carrying it upstairs at a constant speed because there is necessarily a component of force F in the direction of the motion. Energy is transferred to the briefcase and could, in turn, be used to do work.

The work can be calculated as the area under the force-displacement graph. This is true for work done by a constant force (like the examples shown here), as well as work done by a variable force. Work done by an elastic force, for example, follows the equation W = ksx2/2, where ks is the spring constant of the force and x is the displacement. Work can also be calculated in these systems from force-displacement graphs.

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Key Points

• Work is the transfer of energy by a force acting on an object as it is displaced.

• The work done by a force is zero if the displacement is either zero or perpendicular to the force.

• The work done is positive if the force and displacement have the same direction, and the work done is negative if they have opposite direction.

Key Terms

Energy: A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent

Force-displacement graph: A graph of the force applied on the y-axis, and distance moved on the x-axis. The area under the curve between two points is the work done.

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