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The following graph shows how a variable force, in this case a spring force, acts over a distance:

The following graph shows how a variable force, in this case a spring force, acts over a distance:

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Please answer parts A,B andC using the data given in the question.
### Understanding Variable Force through a Spring Force Graph 3) The graph illustrates how a variable force, specifically a spring force, acts over a distance. The force \( F \) is given by the equation \( F = kx \), where \( k \) represents the spring constant and \( x \) the displacement from the equilibrium position. #### Diagram Description - **Axes**: The horizontal axis represents the displacement (x), and the vertical axis represents the force (F). - **Graph Line**: A diagonal line indicating the linear relationship between the force and displacement as given by \( F = kx \). - **Shaded Area**: A right triangle under the graph line, ranging from \( x = x_0 \) to \( x = x_f \). #### Questions **a.)** What value of stretch (x) has the force least? Which has the greatest? - The force is least at \( x = x_0 \) and greatest at \( x = x_f \). **b.)** Show that the shaded area under the force curve is just the potential energy stored in the spring by the force: \[ \Delta U = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_0^2 \] *(Hint: What is the area of a triangle?)* **c.)** What units does the "length" of the triangle have? What units does the "height" of the triangle have? What units does the area under the curve have? (You will use this result in Part II of the lab.) - The "length" of the triangle (horizontal axis) is in meters (m). - The "height" of the triangle (vertical axis) is in newtons (N). - The area under the curve, representing potential energy, is in joules (J).
Transcribed Image Text:### Understanding Variable Force through a Spring Force Graph 3) The graph illustrates how a variable force, specifically a spring force, acts over a distance. The force \( F \) is given by the equation \( F = kx \), where \( k \) represents the spring constant and \( x \) the displacement from the equilibrium position. #### Diagram Description - **Axes**: The horizontal axis represents the displacement (x), and the vertical axis represents the force (F). - **Graph Line**: A diagonal line indicating the linear relationship between the force and displacement as given by \( F = kx \). - **Shaded Area**: A right triangle under the graph line, ranging from \( x = x_0 \) to \( x = x_f \). #### Questions **a.)** What value of stretch (x) has the force least? Which has the greatest? - The force is least at \( x = x_0 \) and greatest at \( x = x_f \). **b.)** Show that the shaded area under the force curve is just the potential energy stored in the spring by the force: \[ \Delta U = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_0^2 \] *(Hint: What is the area of a triangle?)* **c.)** What units does the "length" of the triangle have? What units does the "height" of the triangle have? What units does the area under the curve have? (You will use this result in Part II of the lab.) - The "length" of the triangle (horizontal axis) is in meters (m). - The "height" of the triangle (vertical axis) is in newtons (N). - The area under the curve, representing potential energy, is in joules (J).
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