Question

Show that gravitational force is a conservative force.

Solution

A force is said to be conservative if work done by the force is independent of the path followed and depends upon the initial and final positions.
Suppose a body of mass m be taken from A to B along different paths as shown in the figure.

i) Work done if body is taken up straight along Ab,
W₁ = -mgh
ii) Work done by gravity if body is taken up along path ACB,
W₂ = W_AC + W_CB
$straight W subscript AC space equals space straight m space straight g with rightwards harpoon with barb upwards on top. space AC with rightwards harpoon with barb upwards on top space space space space space space space space space space space equals space minus space mg space left parenthesis AC right parenthesis space cos space left parenthesis space 90 space minus space straight theta right parenthesis space space space space space space space space space space space equals space minus mg space AC space sin space straight theta space space space space space space space space space space space space equals space space minus space mg space left parenthesis AB right parenthesis space equals space minus space mgh space straight W subscript BC space equals space straight m space straight g with rightwards harpoon with barb upwards on top. space CB with rightwards harpoon with barb upwards on top space space space space space space space space space equals space minus mg space left parenthesis CB right parenthesis space cos space left parenthesis 90 right parenthesis space equals space 0 Therefore comma space straight W subscript 2 space equals space minus space mgh space$
iii) Work done by the gravity along the path ADEFGHB,
W₃ = W_AD + W_DE + W_EF + W_FG + W_GH + W_HB
$Here comma space the space displacements space DE with rightwards harpoon with barb upwards on top comma space FG with rightwards harpoon with barb upwards on top space and space JB with rightwards harpoon with barb upwards on top space are space perpendicular space to space gravity. space Therefore comma space straight W subscript DE space equals space straight W subscript FG space equals space straight W subscript HB space equals space 0 space comma space and space straight W subscript AD space equals left parenthesis space straight m. straight g with bar on top right parenthesis space. space AD with rightwards harpoon with barb upwards on top space equals space minus space mg space left parenthesis AD right parenthesis space straight W subscript EF space equals space left parenthesis space straight m. straight g with bar on top right parenthesis space. space EF with rightwards harpoon with barb upwards on top space equals space minus space mg space left parenthesis EF right parenthesis space straight W subscript GH space equals space left parenthesis space straight m. straight g with bar on top right parenthesis space. stack GH space with rightwards harpoon with barb upwards on top equals space minus space mg space left parenthesis GH right parenthesis space Therefore comma space straight W subscript 3 space equals space minus space mg space left parenthesis AD space plus space EF space plus space GH right parenthesis space equals space minus space mgh space Since space straight W subscript 1 space equals space straight W subscript 2 space equals space straight W subscript 3 space Therefore comma space gravitational space force space is space conservative. space$

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Work, Energy And Power

Show that gravitational force is a conservative force.

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