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.

1.Suppose the pendulum started losing energy at the rate 1% per period. As a result, the energy of the pendulum drops according to E(t)= E(t=0)*exp(-z*t). Let’s call z damping constant, it has units 1/sec.

a) Find Z

b) Sketch E(t) for the time span of a few hundred periods.

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

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

e) 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.

f) 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?

Answer 1

Given information are

Lengh of pendulum (l) = 1m; Mass of pendum (m) = 1 kg; accleration due to gravity (g) = 10 m/s^2

The maximumm deflection angel is +/-1degree;

The pendulum looses energy by 1percent in each period

The engergy of pendum drops accroding to the equation E(t) = E(t=0)exp(-zt), where z is the damping constant

(a) Calculate the damping constant (z) ?

Let's calculate the time period of the pendulum and using the given info we get the time period T=1.99sec approximated to 2sec.

It is given the pendulum losses energy py one per cent after each period.

So after t=2 sec, the energy will be E(t=2) = E(t=0)-E(t=0)/100 = 0.99E(t=0)

So using this value in the energy equation we can calcuate z as follows

exp(-zt) = E(t)/E(t=0)

for t = 2sec

exp(-2z) = 0.99E(t=0 )/E(t=0) = 0.99

solving -2z = -0.01 which gives z=0.00503

Hence the damping constant is z=0.00503

(b) Sketch E(t) for the time span of a few hundred periods.

At 0 period the energy is E(t=0).

After 100 period the energy will be 0.99^100E(t=0) = 0.37E(t=0)

After 200 period the energy will be 0.99^200E(t=0) = 0.13E(t=0)

Similarly the energy will decrease, which is shown in the attached plot

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

So now it is asked to calculate time when E(t)/E(t=0) = 0.5

So using the energy equation will be

exp(-zt) = 0.5 , Using the value f z, which we calculate in (a) and solving the equation we will get t = 137.8 sec approximated to 138 sec.

Hence after 138 sec, the pendulum will lose half of its energy.