Question

Arrange the four graphs in descending order of total work done; where W₁, W₂, W₃ and W₄ are the work done corresponding to figure a, b, c and d respectively.

• A. W₃ > W₂ > W₁ > W₄
• B. W₃ > W₂ > W₄ > W₁
• C. W₂ > W₃ > W₄ > W₁
• D. W₂ > W₃ > W₁ > W₄

Answer 1

The Correct Option is A

To determine the total work done in each of the graphs, we need to calculate the area under the force (F) versus displacement (x) graph for each figure. The total work done is equal to this area.

1. Figure a:
   • Shape: Triangle
   • Area = \frac{1}{2} \times \text{base} \times \text{height}
   • Base = x_1 - x_0, Height = 2F
   • Work done, W_1 = \frac{1}{2} \times (x_1 - x_0) \times 2F = (x_1 - x_0) \times F
2. Figure b:
   • Shape: Trapezium
   • Area = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}
   • Bases = x_1 - x_0 and x_2 - x_1, Height = 2F
   • Work done, W_2 = \frac{1}{2} \times [(x_1 - x_0) + (x_2 - x_1)] \times 2F = (x_2 - x_0) \times F
3. Figure c:
   • Shape: Trapezium and Triangle
   • Trapezium Area = (x_2 - x_1) \times 2F
   • Triangle Area = \frac{1}{2} \times (x_1 - x_0) \times 2F
   • Work done, W_3 = \frac{1}{2} \times (x_1 - x_0) \times 2F + (x_2 - x_1) \times 2F
   • Simplified: W_3 = (x_2 - x_0) \times 2F
4. Figure d:
   • Shape: Three Triangles
   • Positive and negative areas cancel each other out partially.
   • Total work done, after summing up areas.

Now, evaluate the order of work done based on these calculations:

• Figure c has the largest area due to having a large trapezium and triangle.
• Figure b comes next as it is a full trapezium.
• Figure a is smaller as it is a single triangle.
• Figure d has the least as the areas of positive and negative triangles reduce net work done significantly.

Thus, the order is W_3 > W_2 > W_1 > W_4.