Work is given by force multiplied by displacement, for a pendulum only the parallel component of gravity is causing a displacement so why don't we use that ?
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$\begingroup$ But you are actually doing that $\endgroup$– user126422Commented Apr 17, 2017 at 15:01
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$\begingroup$ Who said the work done by the gravitational force is ignored? As the pendulum oscillates, its height keeps changing. If the height changes, the gravitational potential energy change; this implies that gravitational force did some work. $\endgroup$– YashasCommented Apr 17, 2017 at 15:34
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Work isn't the force muliplied by the displacement. It's the dot product of the two. That means the only part of the displacement that has any effect on the work done is the component of the displacement parallel to the force. Gravity only points downward, so the work done is only done in that direction.
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1$\begingroup$ The tension does no work; the gravitational force does work. $\endgroup$– YashasCommented Apr 17, 2017 at 15:33
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$\begingroup$ Then why would the pendulum move horizontally $\endgroup$– LM26Commented Apr 17, 2017 at 20:15
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$\begingroup$ You can split the gravitational force into components. You will have a non-zero component along the direction of motion. $\endgroup$– YashasCommented Apr 18, 2017 at 2:40