A `pi-meason` hydrogen atom is a bound state of negative charged pion (denoted by `pi^(bar), m_(pi) = 273 m_(e))` and a proton. Estimate the number of revolutions a `pi-meason` makes (averagely ) in the ground state on the atom before , it decays (mean life of a `pi-meason ~= 10^(-8) s`, mass of proton `= 1.67 xx 10^(-27) kg)`.

Explanation

To estimate the number of revolutions a pi meson makes in a hydrogen atom before it decays, we can follow these steps:

### Step 1: Understand the Problem
We have a pi meson (negative charged pion) with a mass of \( m_{\pi} = 273 m_e \) (where \( m_e \) is the mass of the electron) and a proton. The mean life of the pi meson is given as \( \tau = 10^{-8} \) seconds. We need to find out how many revolutions it makes before it decays.

### Step 2: Calculate the Reduced Mass
The reduced mass \( \mu \) of the system (pi meson and proton) is calculated using the formula:
\[ \mu = \frac{m_{\pi} m_p}{m_{\pi} + m_p} \]
Where:
- \( m_{\pi} = 273 m_e \)
- \( m_p = 1.67 \times 10^{-27} \) kg
- \( m_e = 9.1 \times 10^{-31} \) kg

Substituting the values:
\[ \mu = \frac{(273 \times 9.1 \times 10^{-31}) (1.67 \times 10^{-27})}{(273 \times 9.1 \times 10^{-31}) + (1.67 \times 10^{-27})} \]

### Step 3: Calculate the Frequency of Revolution
The frequency \( f \) of the pi meson in a hydrogen-like atom can be expressed as:
\[ f = \frac{4 \pi^2 k^2 e^4 \mu}{h^2} \]
Where:
- \( k = \frac{1}{4 \pi \epsilon_0} \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \) (charge of the electron)
- \( h = 6.63 \times 10^{-34} \, \text{Js} \) (Planck's constant)

Substituting the reduced mass \( \mu \) calculated in Step 2 into the frequency formula.

### Step 4: Calculate the Total Number of Revolutions
The total number of revolutions \( N \) before the pi meson decays can be calculated as:
\[ N = f \cdot \tau \]
Where \( \tau = 10^{-8} \) seconds is the mean life of the pi meson.

### Step 5: Final Calculation
After calculating \( f \) and substituting it into the equation for \( N \), we can find the total number of revolutions.

### Solution Summary
1. Calculate the reduced mass \( \mu \).
2. Calculate the frequency \( f \) using the reduced mass.
3. Calculate the total number of revolutions \( N \) using the mean life.