Clock Oscillation: Q Calculation for 0.7 m Pendulum w/ 0.4 kg Bob

[CaptainEvil]

Homework Statement

A grandfather clock has a pendulum length of 0.7 m and a mass bob of 0.4 kg. A mass
of 2 kg falls 0.8 m in seven days, providing the energy necessary to keep the amplitude
(from equilibrium) of the pendulum oscillation steady at 0.03 rad. What is the Q of the
system?

Homework Equations

1) Q = $$\omega$$_R/2$$\beta$$

2) Q = $$\omega$$₀/$$\Delta$$$$\omega$$

The Attempt at a Solution

I figured only equation 1 would help me here, and I can re-arrange it as follows:

$$\beta$$ = b/2m (b = damping coefficient)

Then Q = m$$\omega$$_R/b

when amplitude D is a maximum, we can differenciate wrt $$\omega$$ to obtain maximum (i.e $$\omega$$_R)

$$\omega$$_R = sqrt($$\omega$$²₀ - 2$$\beta$$²)

re-arranging yields

Q = m sqrt($$\omega$$²₀ - b²/2m²)/b

I'm kind of stuck because I don't know how to find the coefficient of damping b. Did I go in the wrong direction here? I know I have to use the information given about the pendulum dropping to find the flaw in the system, any help please?

 

[Redbelly98]

While I'm not certain how to solve the problem, the 2 kg mass dropping tells us at what rate energy is added to the pendulum to overcome damping.

Also, the power dissipated due to damping is definitely related to b. If you can express that power in terms of b, you should be in good shape.

 

[tomcue]

Hello,

Based on the information given, you are correct in using equation 1 to calculate the Q of the system. However, in order to use this equation, you will need to find the damping coefficient b. This can be done by using the given information about the pendulum dropping.

First, we can calculate the potential energy gained by the pendulum bob when it falls 0.8 m:

PE = mgh = (0.4 kg)(9.8 m/s^2)(0.8 m) = 3.136 J

Since this energy is used to keep the amplitude of the pendulum oscillation steady at 0.03 rad, we can equate it to the kinetic energy of the pendulum bob at its maximum displacement:

KE = (1/2)mv^2 = (1/2)(0.4 kg)(v^2)

Where v is the velocity of the pendulum bob at its maximum displacement. We can solve for v:

v = sqrt(6.272/0.4) = 3.95 m/s

Now, we can use this velocity to calculate the damping coefficient b:

b = 2m\omega0 - mv = 2(0.4 kg)(0.03 rad/s) - (0.4 kg)(3.95 m/s) = -0.088 Ns/m

Finally, we can plug in the values of mass, length, and b into equation 1 to calculate the Q of the system:

Q = (0.4 kg)(0.03 rad/s)(0.7 m)/(-0.088 Ns/m) = 0.012

Therefore, the Q of the system is 0.012. This indicates a relatively high level of damping, meaning the pendulum will lose its energy and stop oscillating more quickly compared to a system with a higher Q value.