Waves+Mechanics=Wachanics

Classical Mechanics Level 5

The system in the diagram is in equilibrium with the pulley having a friction coefficient of $\mu.$

If the strings have tensions $T_{1}$ and $T_{2},$ respectively, then for the string with tension $T_{1}$ a standing wave of $8^\text{th}$ harmonic is produced, whereas for the wire with tension $T_{2}$ a standing wave of $7^\text{th}$ harmonic is produced.

Find the value of $\mu.$

Details and Assumptions:

The masses $m_{1}$ and $m_{2}$ are taken such that the system is at rest.
The pulley is not frictionless.
In the last step of calculating $\mu,$ you may refer to a scientific calculator.
The frequencies given in both the wires are the same.

The answer is 0.0850087.

1 solution

Steven Chase
Feb 3, 2018

Relationship between vibrational frequency and tension:

$\large{f^2 \propto T}$

Relationship between the two tensions:

$\large{\frac{T_1}{T_2} = \frac{64}{49}}$

Using the Capstan equation , with $T_1$ being the greater (load) tension, and $T_2$ being the lesser (hold) tension:

$\large{T_1 = T_2 \, e^{\mu \, \phi}}$

Here, $\mu$ is the pulley friction coefficient and $\phi$ is the contact angle, which we can see is $\pi$ .

$\large{\frac{64}{49} = e^{\mu \, \pi} \\ \mu = \frac{1}{\pi} \, ln \Big(\frac{64}{49}\Big) \approx 0.085}$

Hmm Thanks sir. How was the problem? And sir, it would have been better if you proved $T_{1}=T_{2} e^{\mu \phi}$

Md Zuhair - 3 years, 4 months ago

Sure, it was a fun problem. I'm actually very rusty on these wave mechanics, so I had to do some reading for this one.

Steven Chase - 3 years, 4 months ago

Actually its not your fault. Wave mechanics is not that used and also it isnt that interesting like Pure mechanics or classical mechanics

Md Zuhair - 3 years, 4 months ago

Waves+Mechanics=Wachanics

The answer is 0.0850087.

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