91Ó°ÊÓ

The exercise begins with converting the power from horsepower (hp) to watts (W). This is necessary because standard scientific formulas typically require power to be in watts. Power in horsepower is commonly used in automotive and mechanical fields, but we need it in watts to compute torque correctly. Given that 1 hp equals 746 watts, we multiply the power in hp by 746:
\[ P = 100 \text{ hp} \times 746 \text{ W/hp} = 74600 \text{ W} \]
This conversion helps us to align the power units with the SI system, making further calculations more straightforward.

The next step involves converting the rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). This is crucial because the relationship between power and torque requires angular speed in radians per second. The conversion process includes two key conversions:

• 1 revolution = 2Ï€ radians
• 1 minute = 60 seconds

Using these conversions, we calculate rotational speed as:
\[ \text{Rotational Speed} = 1800 \text{ rev/min} \times \frac{2Ï€ \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 188.4 \text{ rad/s} \]
This calculation is important because using radians per second keeps our formulas consistent with angular measurements in the SI unit system.

The most critical part of this exercise is applying the torque formula, which involves the relationship between power (P), torque (τ), and angular velocity (ω). This relationship is captured by the formula:
\[ P = τ \times ω \]
To isolate and solve for torque (Ï„), we rearrange the formula as:
\[ τ = \frac{P}{ω} \]
We can substitute the known values of power (74600 W) and angular velocity (188.4 rad/s) into the equation:
\[ Ï„ = \frac{74600 \text{ W}}{188.4 \text{ rad/s}} = 396 \text{ Nm} \]
The torque calculated here indicates the twisting force the crankshaft exerts, measured in newton-meters (Nm). Understanding this relationship helps to analyze and design components in mechanical systems, ensuring they function efficiently under specific power and speed conditions.