To tackle the problem regarding the period of a simple pendulum, we first need to understand the series expansion that describes it. The period \( T \) of a simple pendulum can be expressed as a series, where the first term is the period for small angles, and subsequent terms account for the effects of larger amplitudes. The second term in this series is particularly important for our calculations.
Understanding the Series Expansion
The period \( T \) of a simple pendulum can be approximated by the following series:
- First term: \( T_0 = 2\pi \sqrt{\frac{L}{g}} \)
- Second term: \( T_1 = \frac{1}{4} \left( \frac{L}{g} \right)^{1/2} \theta^2 \)
- Third term: \( T_2 = \frac{1}{96} \left( \frac{L}{g} \right)^{1/2} \theta^4 \)
Here, \( L \) is the length of the pendulum, \( g \) is the acceleration due to gravity, and \( \theta \) is the maximum angular displacement (in radians).
Finding the Value of m
To find the value of \( m \) for which the second term equals 0.02, we can set up the equation:
Equation: \( T_1 = \frac{1}{4} \left( \frac{L}{g} \right)^{1/2} \theta^2 = 0.02 \)
Rearranging this gives us:
Rearranged: \( \theta^2 = 0.08 \left( \frac{g}{L} \right)^{1/2} \)
Now, if we assume \( g \approx 9.81 \, \text{m/s}^2 \) and substitute this value, we can solve for \( \theta \) and subsequently for \( m \) (where \( m \) is related to the amplitude of the pendulum). If we let \( L = 1 \, \text{m} \) for simplicity, we can calculate:
Calculating the Second Term
Substituting \( L = 1 \, \text{m} \) into the equation:
Calculation: \( \theta^2 = 0.08 \cdot \sqrt{9.81} \approx 0.08 \cdot 3.13 \approx 0.25 \)
This gives us \( \theta \approx 0.5 \) radians. The value of \( m \) can be derived from the relationship between \( \theta \) and the amplitude of the pendulum's swing.
Determining the Third Term
Now, to find the value of the third term in the series at this amplitude, we use:
Third Term: \( T_2 = \frac{1}{96} \left( \frac{L}{g} \right)^{1/2} \theta^4 \)
Substituting \( \theta = 0.5 \) radians and \( L = 1 \, \text{m} \):
Calculation: \( T_2 = \frac{1}{96} \cdot \sqrt{1/9.81} \cdot (0.5)^4 \)
Calculating \( (0.5)^4 = 0.0625 \) and \( \sqrt{1/9.81} \approx 0.32 \):
Final Calculation: \( T_2 \approx \frac{1}{96} \cdot 0.32 \cdot 0.0625 \approx 0.000208 \, \text{s} \)
Summary of Results
In summary, we found that:
- The value of \( m \) for which the second term equals 0.02 is approximately related to an angular displacement of \( 0.5 \) radians.
- The value of the third term in the series at this amplitude is approximately \( 0.000208 \, \text{s} \).
This analysis illustrates how the period of a pendulum changes with amplitude and how we can use series expansions to approximate these values effectively.
