Questions: How much horsepower is required to rotate a shaft load of 1,500 lb . ft at 10 rpm for ten seconds 28.6 hp 2.86 hp 0.286 hp None of the Above

Transcript text: Previous Activity How much horsepower is required to rotate a shaft load of $1,500 \mathrm{lb} . \mathrm{ft}$ at 10 rpm for ten seconds 28.6 hp 2.86 hp 0.286 hp None of the Above

Solution

Solution Steps

Step 1: Understanding the Problem

We need to determine the horsepower required to rotate a shaft with a torque of $1500 \, \text{lb} \cdot \text{ft}$ at a speed of 10 rpm for ten seconds.

Step 2: Convert RPM to Radians per Second

First, convert the rotational speed from revolutions per minute (rpm) to radians per second (rad/s): \[ \omega = 10 \, \text{rpm} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{10 \times 2\pi}{60} \, \text{rad/s} = \frac{\pi}{3} \, \text{rad/s} \]

Step 3: Calculate Power in Foot-Pounds per Second

Power ($P$) in foot-pounds per second can be calculated using the formula: \[ P = \tau \cdot \omega \] where $\tau$ is the torque and $\omega$ is the angular velocity. Substituting the given values: \[ P = 1500 \, \text{lb} \cdot \text{ft} \times \frac{\pi}{3} \, \text{rad/s} = 1500 \times \frac{\pi}{3} \, \text{lb} \cdot \text{ft/s} = 500\pi \, \text{lb} \cdot \text{ft/s} \]

Step 4: Convert Power to Horsepower

Convert the power from foot-pounds per second to horsepower (1 hp = 550 ft-lb/s): \[ P_{\text{hp}} = \frac{500\pi}{550} \, \text{hp} = \frac{500\pi}{550} \approx 2.8562 \, \text{hp} \]