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.
why stiffness (of a spring) ?
It is an analogy.
The general equation of motion make an analogy with a mass attached to the end of a spring:
mx"+Bx'+kx=0
where k represents the restoring force per unit displacement.
oww...ok...
To obtain the differential equation of motion for the simple pendulum, we can use Newton's second law and consider the forces acting on the pendulum.
Let's assume the length of the pendulum as "L" and the small angle it oscillates through as "θ." The force acting on the pendulum is the sum of the gravitational force and the resistance force due to the medium.
The gravitational force is given by F_g = -mg sin(θ), where "m" is the mass of the bob and "g" is the acceleration due to gravity.
The resistance force is proportional to the velocity of the pendulum. Let's denote the resistance constant as "k." The velocity of the pendulum can be expressed as the derivative of the angle with respect to time: v = dθ/dt.
The resistance force is then given by F_r = -k v = -k (dθ/dt).
Using Newton's second law (F = ma), we can equate the sum of forces with the mass times acceleration. In this case, the mass of the bob (m) cancels out, so we have:
-mg sin(θ) - k (dθ/dt) = - m L (d^2θ/dt^2).
Since L is the length of the pendulum, and the angle θ is small, we can use the small angle approximation sin(θ) ≈ θ.
So, the equation becomes:
-mg θ - k (dθ/dt) = - m L (d^2θ/dt^2).
Simplifying further by dividing through by "-m" and "L," we get:
g/ L θ + k/m (dθ/dt) = (d^2θ/dt^2).
This is the differential equation of motion for the simple pendulum in a medium with resistance proportional to velocity.
This is not an initial value problem!
External forces are zero, so the governing equation is homogeneous:
mx"+Bx'+kx=0
x"=d²x/dt².
x'=dx/dt
x=displacement, positive to the right
m=mass
B=resistance proportional to the velocity
k=stiffness (of a spring), which is resistance proportional to the displacement.
For a pendulum of length L and mass m,
making an angle θ with the vertical, three forces act on the mass m.
The vertical force due to gravity, mg.
A horizontal restoring force (towards the equilibrium position) of mgsin(θ), and the tension, which is equal and opposite to the resultant of the two forces.
The restoring force mgsin(θ) is often approximated by
kx=mgLsin(θ)≅mgLθ for small angles (when θ is in radians).