Work is the result of a force on a point that follows a curve X, with a velocity v, at each instant.Energy must be transferred to an object to help it move, and the energy can be transferred in the form of force. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application. This scalar product of force and velocity is known as instantaneous power. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. It can change the direction of motion but never change the speed.įor moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force. The dot product of two perpendicular vectors is always zero, so the work W = F ⋅ v = 0, and the magnetic force does not do work. The result of a cross product is always perpendicular to both of the original vectors, so F ⊥ v. The magnetic force on a charged particle is F = q v × B, where q is the charge, v is the velocity of the particle, and B is the magnetic field. This force does zero work because it is perpendicular to the velocity of the ball. Another example is the centripetal force exerted inwards by a string on a ball in uniform circular motion sideways constrains the ball to circular motion restricting its movement away from the centre of the circle. Therefore, work need only be computed for the gravitational forces acting on the bodies. įor example, in a pulley system like the Atwood machine, the internal forces on the rope and at the supporting pulley do no work on the system. Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping. įixed, frictionless constraint forces do not perform work on the system, as the angle between the motion and the constraint forces is always 90°. Thus the virtual work done by the forces of constraint is zero, a result which is only true if friction forces are excluded. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system.įor a mechanical system, constraint forces eliminate movement in directions that characterize the constraint. For example, in the case of a slope plus gravity, the object is stuck to the slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. The work/energy principles discussed here are identical to electric work/energy principles.Ĭonstraint forces determine the object's displacement in the system, limiting it within a range. These formulas show that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, and units, of energy. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by: The work done is given by the dot product of the two vectors. If the ball is thrown upwards, the work done by its weight is negative, and is equal to the weight multiplied by the displacement in the upwards direction.īoth force and displacement are vectors. įor example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force. A force is said to do positive work if when applied it has a component in the direction of the displacement of the point of application. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. In physics, work is the energy transferred to or from an object via the application of force along a displacement.
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