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# work done

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The work done by a force F during the time interval from t=t1 to t=t2 is equal towhere v is the velocity of the point of application of F.

From the equation of motion ma=F for a particle of mass m moving with acceleration a, it follows that ma. v=F. v, and this then gives (d/dt)(½mv.v)= F.V. By integration, it follows that the change in kinetic energy is equal to the work done by the force.

Suppose that a particle, moving along the x-axis in the positive direction, with displacement x(t)i and velocity v(t)i at time t, is acted on by a constant force Fi in the same direction. Then the work done by this constant force during the time interval from t=t1 to t=t2 is equal towhich equals F(x(t2)−x(t1). This is usually interpreted as ‘work=force×distance’.

The work done against a force F should be interpreted as the work done by an applied force equal and opposite to F. When the force is conservative, this is equal to the change in potential energy. When a person lifts an object of mass m from ground level to a height z above the ground, work is done against the uniform gravitational force and the work done equals the increase mgz in potential energy.

Work has the dimensions ML2 T−2, the same as energy, and the SI unit of measurement is the joule.

Subjects: Mathematics.

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