Max Tension and Height

Question

The (massless) string of a pendulum has a length of 0.80 m with a 0.20-kg ball attached to one end. If the maximum speed of the ball at the lowest point of its swing equals 1.5 m/s, what are the maximum tension, T, in the string, and the maximum height, h, of the ball above the lowest point?

Answer 1

Let

r = 0.8 m be the length of the string,

m = 0.2 kg be the mass of the ball,

v = 1.5 m/s be the speed at the lowest point,

g = 9.81 be gravitational acceleration.

The tension, T, has two contributing components:

the weight of the ball, and the centrifugal force.

Maximum tension occurs at the bottom for two reasons:

the centrifugal force is at its maximum,

at the bottom both forces are aligned.

T = mg + mv²/r = m(g + v²/r) = 0.2(9.81 + 1.5²/0.8) = 2.5 N

We get the maximum height, h, from conservation of energy:

The kinetic energy at the lowest point gets converted to the potential energy at the highest point.

mgh = mv²/2

h = v²/(2g) = 1.5²/(2×9.81) = 0.11 m