In the case of a simple pendulum swinging back and forth, the tension in the string does not do any work on the bob. This is because the tension force acts perpendicular to the direction of the bob's motion.
To elaborate, the work done by a force is defined as the dot product of the force and the displacement vector. When the pendulum bob swings, the tension force from the string directs towards the pivot point, while the displacement of the bob is tangential to its path of motion. Since the angle between the tension force and the direction of the displacement is 90 degrees at all points in the swing, the work done is:
\[ W = \vec{F} \cdot \vec{d} = F d \cos(\theta) \]
Where:
- \( \vec{F} \) is the tension force,
- \( \vec{d} \) is the displacement,
- \( \theta \) is the angle between the force and displacement vectors.
Because \( \cos(90^\circ) = 0 \), the work done by the tension in the pendulum's string is:
\[ W = F d \cdot 0 = 0 \]
Thus, the work done by the string of a pendulum on its bob as it swings back and forth is zero. The energy of the system is transformed between potential energy and kinetic energy as the pendulum swings, but the tension force in the string does not perform any work on the bob.
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