For the problem of finding the period of oscillation of a simple pendulum, obtain a formulation in the case when air resistance is proportional to the square of the velocity. Find a solution for this formulation.
To find the period of oscillation of a simple pendulum when air resistance is proportional to the square of the velocity, you can create a mathematical model for the system and solve it. The governing equation for the motion of the pendulum can be described as follows:
1. Equation of Motion:
Let:
- θ(t) be the angle the pendulum makes with the vertical axis at time t.
- L be the length of the pendulum.
- g be the acceleration due to gravity.
- k be the air resistance constant.
- θ'(t) be the angular velocity of the pendulum at time t.
Then, the equation of motion is:
\(L \cdot \theta''(t) = -g \cdot \sin(\theta(t)) - k \cdot (L \cdot \theta'(t))^2\)
2. Linearize the Equation:
To simplify the equation,
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