Given a simple pendulum of length r and mass m, determine the oscillation period T if the perturbation angle θ is very small (i.e. θ < 10degrees).
How can I derive an equation for period using just that info?
Period = 2 pi sqrt(r/g)
for small angle oscillations.
You need to solve the differential equation of motion for small angle theta to derive that result.
Many proofs can be found online and in textbooks, but they require calculus.
To derive an equation for the period T of a simple pendulum with a small perturbation angle θ, you can use the following steps:
1. Start with the equation of motion for a simple pendulum: τ = Iα, where τ is the torque acting on the pendulum, I is the moment of inertia, and α is the angular acceleration.
2. For a simple pendulum, the moment of inertia I is approximately equal to mr^2 because the mass is concentrated at a point.
3. The torque τ can be calculated as τ = -mgd⊥, where m is the mass of the pendulum, g is the acceleration due to gravity, and d⊥ is the perpendicular distance from the pivot point to the center of mass.
4. The angular acceleration α can be related to the angle θ by α = d^2θ / dt^2, where d^2θ / dt^2 represents the second derivative of θ with respect to time.
5. Substituting the expressions for torque, moment of inertia, and angular acceleration into the equation of motion, we have -mgd⊥ = mr^2(d^2θ / dt^2).
6. Rearrange the equation to obtain d^2θ / dt^2 = - (g / r)θ.
7. This differential equation represents simple harmonic motion, and its general solution is θ(t) = A cos(√(g / r)t + φ), where A is the amplitude of oscillation and φ is the phase constant.
8. To determine the oscillation period T, we need to find the value of t at which θ(t) completes a full oscillation (from one extreme to the other and back). We can find this by setting √(g / r)t = π/2, which gives t = (π/2)√(r / g).
9. Therefore, the period T of the simple pendulum is T = 2t = π√(r / g).
By following these steps, you can derive an equation for the period T of a simple pendulum with a small perturbation angle θ.