A simple pendulum has a mass of 0.620 kg and a string of length 1.25 m. It is displaced through an angle of 40.0 degrees to point A and then released from rest. When the mass reach... A simple pendulum has a mass of 0.620 kg and a string of length 1.25 m. It is displaced through an angle of 40.0 degrees to point A and then released from rest. When the mass reaches point B, calculate: (a) the drop in height of the mass, (b) the speed of the mass, and (c) the tension in the string. Read more

Steps to Solve

1. Calculate the vertical height at point A

We can calculate the vertical height at point A using trigonometry: $$h_A = L \cos(\theta)$$ where $L$ is the length of the string and $\theta$ is the angle. $L = 1.25 m$ and $\theta = 40.0^\circ$, therefore: $$h_A = 1.25 \times \cos(40.0^\circ) = 0.95755 \text{ m}$$

2. Calculate the drop in height

The drop in height is the difference between the length of the string and the vertical height at point A:

$$ \Delta h = L - h_A$$ $$ \Delta h = 1.25 - 0.95755 = 0.29245 \text{ m} $$

3. Calculate the speed at point B using conservation of energy

The potential energy at point A is converted to kinetic energy at point B. $$ mgh = \frac{1}{2}mv^2 $$ $$ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 0.29245} = \sqrt{5.732} = 2.394 \text{ m/s} $$

4. Calculate the tension in the string at point B

At point B, the tension in the string provides the centripetal force and balances the weight of the mass. $$T - mg = \frac{mv^2}{L}$$ Therefore, the tension $T$ is: $$T = mg + \frac{mv^2}{L}$$ $$T = (0.620 \times 9.8) + \frac{0.620 \times (2.394)^2}{1.25} = 6.076 + \frac{3.549}{1.25}= 6.076 + 2.839 = 8.915 \text{ N}$$