What force is required to roll a 120-pound barrel up a 9-foot incline plane to a height of 3 feet?

To determine the force required to roll a 120-pound barrel up a 9-foot incline to a height of 3 feet, we can utilize the principles of physics related to inclined planes and gravitational force components.

First, we can find the angle of the incline using the relationship of the incline's rise and run. The height gained is 3 feet, and the length of the incline is 9 feet. The sine of the angle of incline (θ) can be found as follows:

[ \sin(\theta) = \frac{\text{height}}{\text{length}} = \frac{3}{9} = \frac{1}{3} ]

Next, the gravitational force acting down the incline can be calculated based on the weight of the barrel and the sine of the angle:

[ F_{\text{gravity}} = \text{weight} \times \sin(\theta) = 120 , \text{pounds} \times \sin(\theta) ]

Substituting the value we found earlier:

[ F_{\text{gravity}} = 120 , \text{pounds} \times \frac{1}{3} = 40 , \text{pounds