solve "A Cessna 152 propeller shaft 14 in. in diameter and 18 ft long is used to transmit 5000 hp at
189 rpm. If G = 12 × 106 psi, determine the maximum shearing stress."

To solve for the maximum shearing stress in the propeller shaft, we'll proceed with the following steps:

### Step 1: Calculate the torque \(T\)

The power \(P\) transmitted by the shaft is related to the torque \(T\) and the angular speed \(\omega\) by:

\[
P = T \cdot \omega
\]

Given:
- Power \(P = 5000\) hp
- Rotational speed \(N = 189\) rpm

First, convert the power \(P\) to consistent units (from horsepower to pound-inch per second):

\[
P = 5000 \, \text{hp} \times 63,025 \, \text{in-lb/s per hp} = 315,125,000 \, \text{in-lb/s}
\]

Next, convert the rotational speed to radians per second:

\[
\omega = \frac{2\pi \cdot N}{60} = \frac{2\pi \cdot 189}{60} \, \text{rad/s} \approx 19.8 \, \text{rad/s}
\]

Now, solve for the torque \(T\):

\[
T = \frac{P}{\omega} = \frac{315,125,000 \, \text{in-lb/s}}{19.8 \, \text{rad/s}} \approx 15,911,616 \, \text{in-lb}
\]

### Step 2: Calculate the polar moment of inertia \(J\)

For a solid circular shaft:

\[
J = \frac{\pi c^4}{2}
\]

where \(c\) is the radius of the shaft, half of the diameter. Given the diameter \(d = 14\) inches:

\[
c = \frac{d}{2} = \frac{14}{2} = 7 \, \text{inches}
\]

Now, compute \(J\):

\[
J = \frac{\pi \cdot (7)^4}{2} = \frac{\pi \cdot 2401}{2} \approx 3770.57 \, \text{in}^4
\]

### Step 3: Calculate the maximum shearing stress \(\tau_{\text{max}}\)

The maximum shearing stress is given by:

\[
\tau_{\text{max}} = \frac{T \cdot c}{J}
\]

Substitute \(T\), \(c\), and \(J\) into the formula:

\[
\tau_{\text{max}} = \frac{15,911,616 \cdot 7}{3770.57} \, \text{psi} \approx 29,548.5 \, \text{psi}
\]

Thus, the maximum shearing stress in the propeller shaft is approximately **29,548.5 psi**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the shearing stress change if the shaft length was doubled?
2. What is the impact on the maximum shearing stress if the diameter of the shaft is reduced?
3. How does the modulus of rigidity \(G\) influence the shaft's torsional behavior?
4. What happens to the shearing stress if the rotational speed is increased by 50%?
5. How would the stress analysis change if the shaft were hollow instead of solid?

**Tip:** When working with torsion problems, ensure you correctly convert all units to avoid discrepancies in results.

Learn how to calculate the maximum shearing stress in a Cessna 152 propeller shaft using torque, polar moment of inertia, and shearing stress formulas. This problem involves mechanics and solid mechanics concepts and is suitable for undergraduate-level engineering students.

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