What force must be applied to roll a 120-pound barrel up an inclined plane 9 feet long and 3 feet high?

To determine the force required to roll a 120-pound barrel up an inclined plane, it's essential to consider the angle of inclination and the gravitational force acting on the barrel.

First, calculate the angle of the inclined plane using the rise (height) and run (length). The rise is 3 feet, and the run corresponds to the length of the inclined plane, which is 9 feet. The tangent of the angle (( \theta )) can be determined as the ratio of rise to run:

[ \tan(\theta) = \frac{3}{9} = \frac{1}{3} ]

Using trigonometric relationships in a right triangle, you can find the sine of the angle:

[ \sin(\theta) = \frac{3}{\sqrt{3^2 + 9^2}} = \frac{3}{\sqrt{90}} = \frac{3}{9.49} \approx 0.316 ]

The force needed to roll the barrel up this incline can be approximated by considering the component of the weight of the barrel acting down the plane, which is equal to the weight multiplied by the sine of the angle of incline:

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