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Step-by-Step Solution
Step 1: Identify the given quantities
The amplitude of the piston’s simple harmonic motion (SHM) is given as 7 cm, which can be converted to meters as
$ A = 0.07 \, \text{m} $.
Step 2: Write down the condition for the washer losing contact
When the washer just loses contact with the piston, the normal force on it becomes zero. In SHM, this happens when the maximum acceleration of the washer (which is the same as that of the piston at the extreme position) equals the acceleration due to gravity $g$.
Step 3: Express the maximum acceleration in SHM
For simple harmonic motion, the maximum acceleration is given by
$ a_{\text{max}} = \omega^2 A $,
where $ \omega $ is the angular frequency of the motion.
Step 4: Equate maximum acceleration to $g$
Set $ \omega^2 A = g $, which gives
$ \omega^2 = \frac{g}{A} \quad \Rightarrow \quad \omega = \sqrt{\frac{g}{A}}. $
Step 5: Calculate the angular frequency $ \omega $
Taking $ g \approx 10 \, \text{m/s}^2 $ and $ A = 0.07 \, \text{m} $, we get
$ \omega = \sqrt{\frac{10}{0.07}} = \sqrt{\frac{1000}{7}}. $
Step 6: Convert angular frequency to linear frequency $f$
Recall that $ f = \frac{\omega}{2\pi} $. Hence,
$ f = \frac{1}{2\pi} \sqrt{\frac{1000}{7}}. $
Step 7: Simplify to find the numerical value of $f$
Approximating gives $ f \approx 1.9 \, \text{Hz}. $
Step 8: Conclude the result
Therefore, the frequency at which the washer just loses contact with the piston is about 1.9 Hz.
