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 determine the maximum shearing stress in the propeller shaft, we'll use the formula for the shearing stress due to torsion:

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

where:
- \(\tau_{\text{max}}\) is the maximum shearing stress.
- \(T\) is the torque applied to the shaft.
- \(c\) is the radius of the shaft.
- \(J\) is the polar moment of inertia for a solid circular shaft.

### 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:

\[
P = T \cdot \omega
\]

First, convert the power \(P\) to consistent units. The power is given in horsepower (hp), so convert it to pound-foot per second:

\[
P = 5000 \, \text{hp} \times 550 \, \text{ft-lb/s per hp} = 2,750,000 \, \text{ft-lb/s}
\]

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 \(T\):

\[
T = \frac{P}{\omega} = \frac{2,750,000 \, \text{ft-lb/s}}{19.8 \, \text{rad/s}} \approx 138,889 \, \text{ft-lb}
\]

Convert the torque to inch-pounds (since the diameter is given in inches):

\[
T = 138,889 \, \text{ft-lb} \times 12 \, \text{in/ft} \approx 1,666,668 \, \text{in-lb}
\]

### 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 \(\tau_{\text{max}}\)

Now, substitute \(T\), \(c\), and \(J\) into the shearing stress formula:

\[
\tau_{\text{max}} = \frac{T \cdot c}{J} = \frac{1,666,668 \cdot 7}{3770.57} \, \text{psi} \approx 3094.1 \, \text{psi}
\]

Thus, the maximum shearing stress in the propeller shaft is approximately **3094.1 psi**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the maximum shearing stress change if the shaft diameter increases?
2. What would be the effect of using a hollow shaft instead of a solid one?
3. How would the shearing stress change if the power transmitted was increased to 6000 hp?
4. What is the significance of the modulus of rigidity \(G\) in calculating the shearing stress?
5. How does the length of the shaft affect the shearing stress?

**Tip:** Always ensure the units are consistent when applying formulas to avoid errors in calculations.

Learn how to calculate the maximum shearing stress in a propeller shaft transmitting 5000 hp at 189 rpm using mechanics and strength of materials principles. This problem involves torsion and requires understanding of torque, polar moment of inertia, and shearing stress formulas. Suitable for college-level students and engineers.

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