To calculate the shear stress transmitted by the shaft, we can use the following formula:

$\tau = \frac{T}{J} \cdot r$

Where:

• $\tau$ = Shear stress (Pa or N/m²)
• $T$ = Torque (Nm)
• $J$ = Polar moment of inertia for a circular shaft = $\frac{\pi d^4}{32}$
• $r$ = Radius of the shaft = $\frac{d}{2}$
• $d$ = Diameter of the shaft (in meters)

Given data:

• Power $P = 100 \text{ kW} = 100,000 \text{ W}$
• Shaft speed $N = 500 \text{ rpm}$
• Diameter of shaft $d = 80 \text{ mm} = 0.08 \text{ m}$

Step 1: Calculate the torque $T$

The power is related to torque and rotational speed by the formula:

$P = \frac{T \cdot \omega}{1000}$

Where:

• $P$ = Power (in kW)
• $T$ = Torque (in Nm)
• $\omega$ = Angular velocity (in rad/s)

We know that:

$\omega = \frac{2\pi N}{60}$

Substitute the values:

$\omega = \frac{2\pi \times 500}{60} = \frac{1000\pi}{60} \approx 52.36 \text{ rad/s}$

Now, we rearrange the formula to find $T$:

$T = \frac{P \cdot 1000}{\omega} = \frac{100000 \cdot 1000}{52.36} \approx 1909.86 \text{ Nm}$

Step 2: Calculate the polar moment of inertia $J$

The polar moment of inertia $J$ for a solid circular shaft is:

$J = \frac{\pi d^4}{32}$

Substitute $d = 0.08 \text{ m}$:

$J = \frac{\pi (0.08)^4}{32} \approx 1.256 \times 10^{-5} \text{ m}^4$

Step 3: Calculate the shear stress

Now, we can calculate the shear stress $\tau$ using the formula:

$\tau = \frac{T}{J} \cdot r$

Where $r = \frac{d}{2} = 0.04 \text{ m}$.

Substitute the values:

$\tau = \frac{1909.86}{1.256 \times 10^{-5}} \cdot 0.04 \approx 60,743,567 \text{ Pa} = 60.74 \text{ MPa}$

Final Answer:

The shear stress transmitted by the shaft is approximately 60.74 MPa.

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Do you want a more detailed explanation of any of the steps? Or have any other questions?

Related Questions:

1. How does the speed of the shaft affect the shear stress?
2. What would happen to the shear stress if the shaft diameter increased?
3. How do you calculate the torque transmitted by a shaft if the power and speed are known?
4. How does the material of the shaft affect the shear stress?
5. What is the significance of polar moment of inertia in rotational mechanics?

Tip:

When calculating shear stress in shafts, it’s important to ensure units are consistent (e.g., power in watts, speed in rpm, diameter in meters).