Pendulum Oscillation: Q Calculation for 0.7m Length, 0.4kg Mass

[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?

 

[Jhero]

Thank you for your question. It seems like you are on the right track with using equation 1 to solve for the Q of the system. However, you are correct in that you need to use the information given about the pendulum dropping to find the coefficient of damping (b).

To do this, you can use the conservation of energy principle. The potential energy (PE) of the falling mass is equal to the kinetic energy (KE) of the pendulum. We can set this up as follows:

PE = mgh = KE = 1/2mv^2

where m is the mass of the falling object, g is the acceleration due to gravity, h is the height the object falls, and v is the velocity of the pendulum bob.

We know the values of m, g, and h from the problem, and we can solve for v. This velocity will be equal to the maximum velocity (vmax) of the pendulum bob, since it is at its maximum amplitude when the object falls.

Next, we can use the equation for Q and substitute in our values for m, v, and the pendulum's properties (length and mass) to solve for b.

I hope this helps. Let me know if you have any further questions.