Two forces act on the bob of a pendulum, gravity and the tension in the string. Of these, the tension is always at right angles to the direction the bob travels.
The bob moves on a circle at the length of the string from the pivot, so its direction of travel is the tangent. The tension lies along the string, which is the radius, and on a circle the tangent and the radius meet at right angles everywhere.
A force at right angles does no work. Work is the force multiplied by the distance travelled along that force, and at right angles no distance is travelled along it at all. The tension only turns the bob and leaves its speed alone.
That leaves gravity as the only thing doing work. Gravity is a conservative force, so the mechanical energy is conserved. The bob speeds up or slows down only by as much as its height has changed.
So we need to know how far the bob is raised above the lowest point. Look at the vertical line in the figure. From the pivot down to the level of the bob is , and from there to the lowest point is .
The two add to the length of the string , so , that is, . At zero angle is , and the larger the angle the higher the bob is raised.
A appearing tends to put people on their guard, but all that is being done is one right-angled triangle. The hypotenuse is and its vertical side is . Energy problems about pendulums usually come unstuck right here.
With the height known, the potential energy follows. Taking the lowest point as the reference, .
As the bob moves, the kinetic energy and the potential energy trade places. Only the dividing line inside the bar moves; the full length does not. At the ends the speed is , so all of it is ; at the lowest point the height is , so all of it is .
The sum always equals its value at the point of release: , where is the height of release, that is, .
Solve for . From the cancels on both sides and .
The mass has gone. Heavy bob or light, released from the same height they reach the same speed. The time does not appear either. Without knowing what moment it is, knowing where the bob is gives its speed. This is the most useful tool of the lot in examinations.
Written with angles instead of heights, . At the lowest point gives , the greatest speed; at the ends gives . This is the exact reverse of the last lecture, where the force along the tangent was strongest at the ends and vanished at the lowest point.