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| Conservative Forces and Non-Conservative Forces |
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| Let us consider two situations |
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| Suppose a block of mass 'm' rests on a rough horizontal table. It is dragged horizontally towards right through a distance 'l', and back to its initial position. Let 'm' be the coefficient of friction between the block and the table. Let us calculate the work done by friction during the round trip. |
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| The normal force between the table and the block is N = mg |
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 The force of friction = mmg. When the block moves towards right, friction on it is towards left and the work by friction is (-mmgl). When the block moves towards left, friction on it is towards right and work is again, (-mmgl). Hence, the total work done by friction in the round trip = -2 mmgl. |
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| Suppose a block connected by a spring is kept on a rough table as shown in the figure. The block is pulled aside and then released. It moves towards the centre A and has a velocity v0, as it passes through the centre. It goes to the other side of A and then comes back. This time, it passes through the centre with lesser velocity, v1. Compare the two cases in which the block is at A, once going towards left and then towards right. In both the cases, the system (table + block + spring) has the same configuration. The spring has the same length and the block is at the same point on the table. But, the kinetic energy in the second case is less than the kinetic energy in the first case. This loss in the kinetic energy is the real loss. Everytime, the block passes through the mean position A, the kinetic energy of the system becomes smaller and in due course, the block stops on the table. Here, the work done by friction in a round trip is negative and not zero. |
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| We divide the forces into two categories |
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 Conservative force |
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 Non-conservative force |
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| If the work done by a force during a round trip of a system is always zero, the force is said to be conservative. Otherwise, it is non-conservative. |
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| If the work done by a force depends only on the initial and final states and not on the path taken, then it is a conservative force. |
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| Examples of conservative force |
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 force of gravity |
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 coulomb's force in electrostatics |
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 the elastic force in a spring |
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| (1) (2) (3) |
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| Example of non-conservative force - frictional force |
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| Let us consider a body of mass 'm' being raised vertically to a height 'h' as in the figure (1). The work done is mgh. Suppose we take the body along the path as in figure (3). The work done during the horizontal motion is zero. Adding up the work done in the two vertical parts of the paths, we get the result mgh, once again. Any arbitrary path like the one shown in figure (3), can be broken into elementary horizontal and vertical portions. Work done along the horizontal parts is zero since gravitational force does not have a component in the horizontal direction. The work done along the vertical parts add up to mgh. So, we can conclude that the work done in raising a body against gravity is independent of the path taken. It depends only on the initial and the final positions of the body. |
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