To determine the change in the kinetic energy of the object, we use the work-energy theorem, which states:

$\Delta K = W = \int F(x) \, dx$

The change in kinetic energy ($\Delta K$) is equal to the work done on the object by the net force, which can be calculated as the area under the $F(x)$ vs. $x$-position graph.

Analyzing the graph:

1. From $x = 0$ to $x = d$:

   • The force $F(x)$ is constant (slope of zero).
   • Let the force be $F_0$.
   • Work done in this region is: $W_1 = F_0 \cdot d$

2. From $x = d$ to $x = 2d$:

   • The force $F(x)$ decreases linearly from $F_0$ to 0.
   • This forms a triangular area under the curve.
   • The work done is the area of the triangle: $W_2 = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot d \cdot F_0 = \frac{F_0 d}{2}$

3. From $x = 2d$ to $x = 3d$:

   • The force $F(x) = 0$ (slope is zero).
   • No work is done in this region: $W_3 = 0$

Total Work Done:

$W_{\text{total}} = W_1 + W_2 + W_3 = F_0 d + \frac{F_0 d}{2} + 0$ $W_{\text{total}} = \frac{3}{2} F_0 d$

Expression for the change in kinetic energy:

$\Delta K = \frac{3}{2} F_0 d$

Would you like further clarifications or additional examples?

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5 Related Questions:

1. What is the significance of the work-energy theorem in physics?
2. How is work calculated when the force varies with position?
3. How would the result change if $F(x)$ decreased non-linearly in the region $d \leq x \leq 2d$?
4. How can we determine the velocity of the object from the kinetic energy?
5. What happens to kinetic energy if the total work done is negative?

Tip:

When analyzing force-position graphs, always focus on the area under the curve to compute the work done.