Question

A pendulum has a length 1 m and a mass 1 kg. Assume Earth free fall acceleration equal to 10 m/s^2. When the pendulum oscillates, the maximal deflection angle is +/- 1 degree.

a) How long will it take before the energy drops to half of the initial value at t=0?

b) How long will it take before the max deflection angle drops to half of the initial value at t=0?

c)

If the damping was produced by a force given by F = -v*k, where v is the velocity and k is some friction coefficient, find k.

(Hint: derive the equation of motion for the pendulum including friction and compare to the textbook one describing the damped harmonic oscillations.)

d)

Suppose you are aiming to excite a resonance of this pendulum by kicking it with a periodically modulated force at a frequency f. What should be f in Hz and approximately how accurately should you be able to adjust f, i.e. f +/- how much?

(Hint: look up the expression for the amplitude of forced vibration of a harmonic oscillator and see how it depends on the frequency of the external force and the dambing constant.)

Answer 1

A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. For small amplitudes, the period of such a pendulum can be approximated by:

Angular frequency () can be written as

For simple pendulum, the displacement from the center can be written as

Similarly, the angular displacement will be

Potential energy (PE) of simple pendulum is

Kinetic energy (KE) of simple pendulum is

Total energy (E_total) is give by

(a) For a simple pendulum with small deflection from the centre position, the total energy of the pendulum remains same for indefinite time. Therefore, it will take infinite time before the energy drops to half of the initial value at t=0 as per above discussion.

(b) The time period of the simple pendulum will be

The angular displacement will be (if you consider the position of pendulum at one end at t = 0)

Therefore, the pendulum takes 1/3 s before the max deflection angle drops to half of the initial value at t=0

(c) A pendulum oscillates about its rest position and at time t the displacement is x. Therefore, the restoring force will be F = -(mg/L)x or -Cx (where C = mg/L). Now, if a(t) is the acceleration of mass m at time t, then by Newton’s Law of Motion along the direction of motion, we have m.a(t) = -Cx(t) – k.υ(t), Therefore, using first and second derivatives of s(t), v(t) and a(t), we have,

Solutions should be oscillations within some form of damping envelope. We can consider the expression for displacement

Where w in the second term represent the angular frequency of the pendulum and r = iw - a. Putting the first and second derivative of x(t) in the above equation, you can obtain

  

which is a quadratic equation in r with solutions:

Note that there are two roots of r to the quadratic as long as the imaginary part is nonzero, corresponding to the two general solutions: (B₁ and B₂ are constants)

These solutions in general describe oscillation at frequency within a decay envelope of time-dependent amplitude . From the above expression, if you put the values of C, you can obtain

If you use the expression for , (f is the frequency of the oscillation) you can obtain the value of k as

(d) The amplitude of damped oscillation will be

If you put the value of k in the above expression, you can get