https://file.aiquickdraw.com/mathbot/file/1729739667765fj4e0iy0.jpg

To calculate the work done, we can apply the following principle: 

The work done by a force over a distance is given by the area under the force vs. distance graph. In this case, the graph consists of two sections: a triangle and a rectangle.

1. **Triangle**: 
   - Base = 4 meters (from 0 to 4 m)
   - Height = 30 N (the force reaches a maximum of 30 N)

   The area of the triangle is given by:
   \[
   \text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4 \, \text{m} \times 30 \, \text{N} = 60 \, \text{J}
   \]

2. **Rectangle**: 
   - Length = 2.5 meters (from 4 m to 6.5 m)
   - Height = 30 N (force is constant at 30 N)

   The area of the rectangle is:
   \[
   \text{Area of rectangle} = \text{Length} \times \text{Height} = 2.5 \, \text{m} \times 30 \, \text{N} = 75 \, \text{J}
   \]

Finally, the total work done is the sum of the areas of the triangle and rectangle:
\[
\text{Total work} = 60 \, \text{J} + 75 \, \text{J} = 135 \, \text{J}
\]

Thus, the work done in moving the object over a distance of 6.5 meters is **135 joules**.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How is work calculated in cases where force varies non-linearly?
2. What is the physical significance of the area under a force-distance graph?
3. How would the work change if the object were moved an additional 1 meter?
4. How is work related to energy in the context of Newtonian mechanics?
5. How would you calculate the work done if the force decreases instead of remaining constant?

**Tip:** The work done by a constant force over a distance is simply the product of the force and the distance, but when the force varies, integrating or calculating the area under the graph is necessary.

Shown below is the graph of a force function (in newtons) that increases to its maximum value and then remains constant. How much work (in joules) is done by the force in moving the object a distance of 6.5 meters?

Calculate the total work done by a varying force in moving an object over a distance of 6.5 meters. This problem covers key physics concepts, including calculating work as the area under the force vs. distance graph, combining the areas of a triangle and a rectangle. Suitable for students in Grades 10-12.

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