The formula to calculate the Periodic Time of SHM for a Compound Pendulum given its radius of gyration is:
\[ \text{Periodic Time} = 2 \pi \sqrt{\frac{\text{Radius of Gyration}^2 + \text{Distance}^2}{\text{Acceleration due to Gravity} \times \text{Distance}}} \]
The Periodic Time for a Compound Pendulum is the time taken by a complete cycle of the wave to pass a point. The Radius of Gyration is the radial distance to a point that would have a moment of inertia the same as the body's actual distribution of mass. The Distance of the Point of Suspension from the Center of Gravity is the measured length between that point and the center of gravity of the body. The Acceleration due to Gravity is the acceleration gained by an object because of gravitational force.
Let's assume the following values:
Using the formula:
\[ \text{Periodic Time} = 2 \pi \sqrt{\frac{0.048^2 + 0.042^2}{9.8 \times 0.042}} \approx 0.624644121838072 \, \text{seconds} \]
The Periodic Time of the Compound Pendulum is approximately 0.624644121838072 seconds.
| Radius of Gyration (meters) | Distance (meters) | Periodic Time (seconds) |
|---|---|---|
| 0.04 | 0.042 | 0.568028568702013 |
| 0.045 | 0.042 | 0.602843368605961 |
| 0.05 | 0.042 | 0.639515070522284 |
| 0.055 | 0.042 | 0.677742318230517 |
| 0.06 | 0.042 | 0.717276446398628 |