Find the differential equation of motion for a gravitational pendulum of mass m and length a by using the fact that the torque of its weight relative to the point of suspension is equal to the time derivative its angular momentum. Solve this equation to obtain the equation of motion in the small oscillation approximation, by considering that initially the pendulum at rest makes an angle è0 with the Oy axis (vertical).
To find the differential equation of motion for a gravitational pendulum, we can start by considering the torque exerted by the weight relative to the point of suspension. The torque can be given by the product of the weight force (mg) and the perpendicular distance from the suspension point to the line of action of the weight force.
Let's assume that the pendulum makes an angle θ with respect to the vertical (Oy axis). The perpendicular distance from the suspension point to the line of action of the weight force is a sin θ, where 'a' is the length of the pendulum.
The torque τ is then given by τ = (mg)(a sin θ).
Now, let's use the fact that torque is equal to the time derivative of angular momentum to obtain the equation of motion. The angular momentum L is defined as L = Iω, where I is the moment of inertia and ω is the angular velocity. For a simple pendulum, the moment of inertia about the point of suspension is simply m * a^2.
Differentiating angular momentum with respect to time, we have dL/dt = d(Iω)/dt.
Since I is a constant for a simple pendulum, the above equation simplifies to dL/dt = I * dω/dt.
Using the chain rule and substituting dL/dt = τ (from torque equation), we can write:
τ = I * dω/dt
(mg)(a sin θ) = m * a^2 * dω/dt
Now, divide both sides of the equation by m * a^2:
g * sin θ = a * dω/dt
This is the differential equation of motion for a gravitational pendulum of mass m and length a.
To solve this equation in the small oscillation approximation, we assume that the angle θ is small (sin θ ≈ θ) and the angular acceleration dω/dt is small (dω/dt ≈ 0).
Under these approximations, the equation simplifies to:
g * θ = 0
This equation represents the equation of motion in the small oscillation approximation for a gravitational pendulum. It states that the restoring force acting on the pendulum is proportional to the angle θ, with the proportionality constant being g (acceleration due to gravity).
This signifies that a small oscillation around the equilibrium position (θ = 0) will result in simple harmonic motion.
Please note that this solution assumes small angles and does not take into account damping or other factors that may affect the motion of the pendulum.