phy said:
A grandfather clock contains a pendulum of lenghth 0.99336 m with a weight of 1.4kg. The driving mechanism comprises anohter mass of 2kg which goes down by 0.8m in 7 days in order to keep the amplitude of oscillation of the pendulum to give theta=0.08(<<1). What is the natural period of the pendulum? What is the value of Q for this system? Take the gravitational acceleration equal to 9.80665m/s^2?
For the period part of the question, I simply assumed that this was a simple pendulum and used the equation T=2pi*sqrt(l/g) where l is the length of the pendulum (ie 0.99336). Can someone please check if this is right
PROBLEM HINTS
Q: What is the pendulum's natural period T?
Yes, period T is given (for the simple small oscillation pendulum) by:
T = 2\pi\sqrt {(L/g)}
where L is pendulum length and g is gravitational acceleration. Would it surprise you to learn that almost every Grandfather Clock has period T=(2 sec)??
Q: What is Q factor for this pendulum??
The Q factor is a measure of pendulum efficiency and reflects the pendulum's energy loss per period as a fraction of its total energy (which will be a mix of potential and kinetic energies during the pendulum's oscillation). Q is defined by:
Q = \frac {(Total Energy)} {(Energy Lost Per Period)} <--- Make sure this Q def is the same used by your text
where (Total Energy) is given by the following for pendulum mass "m" and displacement angle (theta):
(Total Energy) = mgL(1 - cos(\theta)) <--- Study the derivation in your text
The (Energy Lost Per Period) can be inferred from the work required to keep it oscillating provided by the falling driving weight "M":
(Energy Lost Per Period) = (Work Rate)*(Period \ T)
(Work Rate) = Mg(Fall Velocity Of Driving Weight)
Some values provided by the problem statement are:
(Fall Velocity of Driving Weight) = (0.8 meters)/(7 days) = 1.323x10^(-6) m/sec
M = 2 kg
\theta = 0.08 \ radians
L = 0.993 meters
m = 1.4 kg
g = 9.8 m/sec^2
T = Should be close to 2 sec
Q = Should be near 840
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