Torque Calculation Steps for Steel Shaft

This problem requires calculating the internal torque within a steel shaft based on the power it transmits and its rotational speed. We are given:

• Power transmitted, $ P = 40 $ kW
• Rotational speed frequency, $ f = \frac{75}{\pi} $ Hz

Power Unit Conversion

First, we convert the power from kilowatts (kW) to watts (W) for consistency in calculations:

$ P = 40 \text{ kW} = 40 \times 1000 \text{ W} = 40000 \text{ W} $

Angular Velocity Calculation

The rotational speed is given in Hertz (Hz), which represents cycles per second. To calculate torque using the power formula, we need the angular velocity ($\omega$) in radians per second (rad/s). The relationship between frequency ($f$) and angular velocity ($\omega$) is:

$ \omega = 2 \pi f $

Substituting the given frequency:

$ \omega = 2 \pi \times \frac{75}{\pi} \text{ rad/s} $

$ \omega = 2 \times 75 \text{ rad/s} $

$ \omega = 150 \text{ rad/s} $

Torque Calculation using Power Formula

The relationship between power ($P$), torque ($T$), and angular velocity ($\omega$) is given by the formula:

$ P = T \times \omega $

To find the torque ($T$), we rearrange the formula:

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

Now, substitute the values of power ($P$) and angular velocity ($\omega$):

$ T = \frac{40000 \text{ W}}{150 \text{ rad/s}} $

Simplify the fraction:

$ T = \frac{4000}{15} \text{ Nm} $

Divide both the numerator and the denominator by 5:

$ T = \frac{800}{3} \text{ Nm} $

Comparing Result with Options

The calculated internal torque is $ \frac{800}{3} $ Nm. Comparing this value with the given options:

• Option 1: $ \frac{812}{3} $ Nm
• Option 2: $ \frac{800}{3} $ Nm
• Option 3: $ \frac{541}{2} $ Nm
• Option 4: $ \frac{400}{3} $ Nm

Our calculated value matches Option 2.