Answer :
Answer:
[tex]\frac{E_{2}}{E_{1}} \approx 1 -\frac{3\theta}{1-\theta}[/tex] (for small oscillations)
Explanation:
The total energy of the pendulum is equal to:
[tex]E_{1} = m\cdot g \cdot (1-\cos \theta)\cdot L[/tex]
For small oscillations, the equation can be re-arranged into the following form:
[tex]E_{1} \approx m\cdot g \cdot (1-\theta) \cdot L[/tex]
Where:
[tex]\theta = \frac{A}{L^{2}}[/tex], measured in radians.
If the amplitude of pendulum oscillations is increase by a factor of 4, the angle of oscillation is [tex]4\theta[/tex] and the total energy of the pendulum is:
[tex]E_{2} \approx m\cdot g \cdot (1-4\theta)\cdot L[/tex]
The factor of change is:
[tex]\frac{E_{2}}{E_{1}} \approx \frac{1 - 4\theta}{1-\theta}[/tex]
[tex]\frac{E_{2}}{E_{1}} \approx 1 -\frac{3\theta}{1-\theta}[/tex]