How much work is done against gravity by a workman carrying an 8-kg sheet of plywood up a 3-m ramp inclined at an angle of 20° to the horizontal?
I get that the answer is 80.44 but can some explain the diagram portion of it. Why do we take cos 70º?
To understand why we use the cosine of 70°, let's first break down the problem and create a diagram that illustrates the situation.
First, we have an 8-kg sheet of plywood being carried up a 3-m ramp inclined at an angle of 20° to the horizontal. We want to find the work done against gravity.
Now, let's create the diagram.
1. Draw a horizontal line to represent the ground.
2. From the ground, draw a vertical line upwards to represent the ramp.
3. Now, draw a line along the ramp, inclined at an angle of 20° to the horizontal line.
4. Mark the length of the ramp as 3 m.
5. Indicate the weight of the plywood (8 kg), acting vertically downward from the center of the plywood.
Next, we need to resolve the weight vector (acting vertically) into two components - one parallel to the ramp's slope and one perpendicular to it.
1. Take the weight vector (acting vertically) and draw a perpendicular line that intersects the ramp's inclined line.
2. This perpendicular line represents the component of weight acting perpendicular to the ramp.
Now, we can see that we have two right-angled triangles formed in our diagram:
- One triangle with the 20° angle, the vertical line representing the ramp, and the inclined line representing the ramp's slope.
- The other triangle with the 90° angle, the perpendicular line we drew, and the inclined line representing the ramp's slope.
The cosine of an angle is defined as the adjacent side divided by the hypotenuse in a right-angled triangle. In our diagram, the adjacent side is the component of weight acting perpendicular to the ramp, and the hypotenuse is the weight vector itself.
Therefore, we take the cosine of 70° (which is the complement angle to 20°) to find the ratio of the component of weight acting perpendicular to the ramp to the total weight of the plywood.
By multiplying this ratio by the total weight, we can calculate the work done against gravity.
In summary, we use the cosine of 70° because it allows us to find the component of weight acting perpendicular to the ramp, which is the relevant force in determining the work done against gravity.