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| Expression for Work |
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Let us take a ball that can slide along a frictionless wire, stretched
along the horizontal X-axis. A constant force  directed at an angle. |
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| f to the wire, accelerates the bead along the wire. According to Newton's second law |
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| Fx = max ______(1) |
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As the ball moves through a displacement
'd', the force changes the ball's velocity from an initial value
to another value . As is constant, acceleration 'a' is also constant. |
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| According to equation of motion |
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| v2 = v02 + 2 axd ______(2) |
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| Substituting (1) in (2), we get |
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| Hence, the work done by the force on the ball is equal to the energy transfer. Work done due to the force is W = Fx d. |
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| Therefore, to calculate work, we have to use only the force component along the object's displacement. |
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| W = Fx d. |
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=  |
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| This is the work done by a constant force (i.e., the magnitude and the direction of the force do not change.). |
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| Work is the energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work and energy transferred from the object is negative work. |
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| Let us consider a situation where the force is acting along the X-axis and the magnitude of the force is varying with position 'x'. Thus, as the ball moves, the magnitude of the work done by the force, on the ball, changes. The adjacent graph shows the plot of a one dimensional variable force. |
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| Let us calculate the work done on the ball by this force. We divide the area under the graph into number of narrow strips of width 'Dx'. It is small enough to assume that F (x) is uniform in that range. |
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| Let Fja be the average value of F (x) within the jth interval. Therefore,
Dwj is the work done in the jth
interval time; Dwj = FjaDx |
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| Total work, W = Dwj = FjaDx |
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| Geometrically, the work is equal to the area between the F(x) curve and the X-axis between the limits xi and xj. |
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| Unit work is the amount of work done when a unit force displaces a body through unit distance in the direction of the force. |
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| In International System of units, the absolute unit of work is 'joule'. |
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| 1 joule = 1 newton x 1 metre |
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| Positive work |
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| When q is acute, cos q is positive. Hence, the work done is positive. |
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 When a body falls freely under the force of gravity, q = 0, cos q = 1. |
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| Hence, the work done by gravity is positive. |
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 When a gas in a cylinder, fitted with a movable piston, is allowed to expand, work done by the gas is positive, since the force due to the gas pressure and displacement of piston are in the same direction. |
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 When a spring is stretched, work done by the stretching force is positive. |
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| Negative work |
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| and when q is obtuse, cos q is negative. Hence, the work done is negative. |
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 When a ball is thrown up, its upward motion is opposed by gravity. The displacement is in the upward direction and the gravitational force acts in the downward direction. Here, q = 180o and cos q = -1. Therefore, the work done by gravity on a body moving upward, is negative. |
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 Let us push a box on a table (a rough horizontal surface). The motion of the box is opposed by the force of friction. The force of friction and the displacement of the object always make an angle 180o and hence, the work done by the frictional force is always negative. |
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 When a positive charge is moved closer to another positive charge, the work done by the electrostatic force of repulsion between the charges is negative. |
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| Zero work |
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| When the applied force and the displacement make an angle 90o to each other, the work done is zero. This is because cos 90 = 0. |
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 A person carrying a briefcase, moves on a horizontal road. Since, the gravitational force acts vertically downwards and the displacement of the person is in the horizontal direction, the work done by the man against gravity is zero. |
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 Take the case of an oscillating simple pendulum. The displacement of the bob at any point on the arc is along the tangent drawn at that point. The tension of the string is along the radius of the arc. Hence, the tension of the string is always perpendicular to the displacement and the work done by the tension is always zero. |
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