A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo witha radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

I looked at this problem and got that the vector field was conservative. I've asked my tutor if this was correct and she said it "seemed" like it. My question is how does the path taken not matter? The man isn't just cutting straight up the helix to reach the top right?

Wouldn't the stairs make the distance to the top longer for the man?

please answer.

never mind. i thought about it and it is the work done by gravity. not the man. which leads me to another question. how exactly would you find the work done by the man. would he be represented as another vector or or in the vector field?

To assess whether the path taken matters in this scenario, let's analyze the nature of the force involved - gravity.

Gravity is a conservative force, which means the work done against it only depends on the initial and final positions, not the path taken. This principle applies when the only force acting is gravity, and there is no external force acting on the can of paint.

In this case, as the man is carrying the can of paint up the helical staircase, the weight of the can acts vertically downward due to gravity. Since the helical staircase follows the shape of a helix or spiral, the vertical displacement between the starting and ending points (the bottom and top of the silo) is unaffected by the specific path followed.

Therefore, the work done by the man against gravity in climbing to the top depends solely on the vertical displacement and the weights involved, regardless of the specific route taken on the helical staircase.

To calculate the work done against gravity, we can use the formula:

Work = Force × Displacement × cos(θ)

In this case, the weight of the can (force) is 25 lb, and the vertical displacement is the height of the silo, which is 90 ft. As the path taken is a helical staircase that wraps around the silo, the angle between the force vector and the displacement vector (θ) is 0 degrees or 180 degrees. Since the cosine of 0 degrees or 180 degrees is 1, the cosine term becomes irrelevant in this case.

Thus, the work done by the man against gravity can be expressed as:

Work = Force × Displacement
= 25 lb × 90 ft
= 2250 lb-ft

Therefore, the man does 2250 foot-pounds of work against gravity in climbing to the top of the silo, regardless of the specific path taken on the helical staircase.