A simple pendulum of length is oscillating through a small angle θ in a medium for which
the resistance is proportional to the velocity. Obtain the differential equation of its motion
and discuss the motion.
ML d^2è/dt^2 + k*L*dè/dt + Mg è = 0
or
d^2è/dt^2 + (k/M)*dè/dt + (g/L) è = 0
k is the damping constant of proportionality (force/velocity)
The solution is damped harmonic motion.
is that true answer?
d^2theta/dt^2 +(g/l)sin(theta)=0
Yes
To obtain the differential equation of motion for a simple pendulum in a medium where the resistance is proportional to the velocity, we can start with Newton's second law of motion.
Let's denote the mass of the pendulum bob as m, the displacement angle as θ, and the length of the pendulum as L.
The gravitational force acting on the pendulum is given by Fg = mg sin(θ), where g is the acceleration due to gravity. Since the pendulum is moving in a circular path, the net force acting on the pendulum is the sum of this gravitational force and the resistive force.
The resistive force is proportional to the velocity, which is given by Fd = -bθ', where b is a constant and θ' is the angular velocity (the derivative of θ with respect to time).
Newton's second law of motion states that the net force acting on an object is equal to its mass times its acceleration. For the pendulum, the acceleration is given by Lθ'' (since the radius of the circular path is L and the angular acceleration is the second derivative of θ).
Therefore, we can write the equation of motion as:
mg sin(θ) - bθ' = mLθ''
Dividing through by mL, we get:
g/L sin(θ) - (b/m)Lθ' = θ''
This is the differential equation of motion for the simple pendulum in the medium with resistance proportional to velocity.
To discuss the motion of the pendulum, let's focus on the small angle approximation, where sin(θ) approximately equals θ for small values of θ. In this approximation, the differential equation becomes:
g/L θ - (b/m)Lθ' = θ''
This equation is a second-order linear ordinary differential equation. The solution to this equation will depend on the initial conditions (initial displacement and initial angular velocity).
In general, for small angle oscillations, the motion of the pendulum will be periodic, with a period given by T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
The presence of the resistive force will cause the amplitude of the oscillations to decrease over time. The exact behavior of the pendulum will depend on the values of b, m, and L.
Overall, the differential equation of motion for the simple pendulum in a medium with resistance proportional to velocity allows us to study the damped oscillations of the pendulum and analyze its behavior under different initial conditions.