You see an optical illusion of an ever-upward spiral staircase. The climber trudges up and up and never gets anywhere, going in circles instead. Suppose the staircase is provided with a narrow ramp, allowing the tired stair-climber to push a wheelbarrow up the stairs. The loaded wheelbarrow weighs 300.0 N, and the ramp makes an angle of 15.0° with the horizontal, all along its length. The ramp consists of four straight sections, with slant lengths 12.0 m, 8.0 m, 20.0 m, and 20.0 m. How much work does the climber do on the wheelbarrow when he pushes it up the ramp from the red marker, all the way around the loop, and (supposedly) back to the red marker again? An ordinary inclined-plane computation will give an accurate value for the work. (In the illusory illustration, the fact that he ends up where he started means that, impossibly, he does NO work.)
To calculate the work done on the wheelbarrow when pushed up the ramp, we can use the formula:
Work = Force * Distance * Cosine(theta)
First, let's break down the problem into sections:
Section 1: The first straight section of the ramp has a slant length of 12.0 m.
Section 2: The second straight section of the ramp has a slant length of 8.0 m.
Section 3: The third straight section of the ramp has a slant length of 20.0 m.
Section 4: The fourth straight section of the ramp has a slant length of 20.0 m.
Now, let's calculate the work done on the wheelbarrow for each section and sum them up:
Section 1: The component of the weight force acting parallel to the ramp is given by: Force_parallel = Weight * sin(theta)
= 300.0 N * sin(15°)
≈ 78.447 N
Distance traveled in section 1 = slant length of section 1 = 12.0 m
Work done in section 1 = Force_parallel * Distance * Cosine(theta)
= 78.447 N * 12.0 m * cos(15°)
Section 2: Similar to section 1, the component of the weight force parallel to the ramp is:
Force_parallel = Weight * sin(theta)
= 300.0 N * sin(15°)
≈ 78.447 N
Distance traveled in section 2 = slant length of section 2 = 8.0 m
Work done in section 2 = Force_parallel * Distance * Cosine(theta)
= 78.447 N * 8.0 m * cos(15°)
Section 3: Now, the ramp consists of a horizontal section, so the angle with the horizontal is 0°.
Force_parallel = Weight * sin(0°)
= 0 N
Distance traveled in section 3 = slant length of section 3 = 20.0 m
Work done in section 3 = Force_parallel * Distance * Cosine(0°)
= 0 N * 20.0 m * cos(0°)
= 0 J
Section 4: Similar to section 3, since the ramp is horizontal, the angle with the horizontal is 0°.
Force_parallel = Weight * sin(0°)
= 0 N
Distance traveled in section 4 = slant length of section 4 = 20.0 m
Work done in section 4 = Force_parallel * Distance * Cosine(0°)
= 0 N * 20.0 m * cos(0°)
= 0 J
Finally, we can sum up the work done in each section:
Total work done = Work section 1 + Work section 2 + Work section 3 + Work section 4
= Work section 1 + Work section 2 + 0 J + 0 J
Note: The total work will only include the sections where the wheelbarrow is being pushed against gravity.
Calculating the values, we get:
Total work done ≈ (78.447 N * 12.0 m * cos(15°)) + (78.447 N * 8.0 m * cos(15°))
With the respective calculations for section 1 and section 2, you can calculate the total work done by substituting the values into the equation.