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In the realm of physics, a variable force changes with respect to some variable, such as position or time. Unlike a constant force, which remains the same regardless of where you are, a variable force can increase or decrease, being dependent on certain conditions. In the context of our problem, the force acting on the particle is given by the formula:\[ F(x) = -\frac{k}{x^2} \]Here, the force depends specifically on the position \(x\). This kind of force is common in fields where the influence of the force diminishes as you move away from the source, like gravitational or electrostatic forces. For example, a gravitational force decreases as the distance from the mass increases. Managing a variable force requires special consideration when calculating work done, as it involves integration over the distance covered.

Integration is a powerful mathematical tool used extensively in physics to calculate quantities like work, especially when dealing with variable forces. When a force varies over a distance, we cannot simply multiply the force by the distance as we can with constant forces. Instead, we need to calculate the integral of the force with respect to the distance.For our exercise, the integral to find the work done is:\[ W = \int_{a}^{2a} F(x) \, dx = \int_{a}^{2a} \left( -\frac{k}{x^2} \right) \, dx \]This integral measures the cumulative effect of the force over the path from \(x = a\) to \(x = 2a\). The integral of \( -\frac{1}{x^2} \) is \( \frac{1}{x} \), evaluated at the boundaries, it simplifies to the work done. Integration thus allows us to properly quantify work in scenarios involving variable forces, providing an accurate assessment of energy transfer.

The work done by a force is an important concept that tells us how much energy is transferred when an object is moved by a force. With a constant force, calculating this is straightforward: it's simply the product of the force and the distance over which it acts. However, when the force varies, as with our example:\[ F(x) = -\frac{k}{x^2} \]we must account for how the force changes along the path. This is done through the work-energy theorem, which connects the concept of work done with energy changes. The formula to calculate the work, using integration, is:\[ W = \int_{a}^{2a} \left( -\frac{k}{x^2} \right) \, dx \]Evaluating this integral from \(x = a\) to \(x = 2a\), we found:\[ W = \frac{k}{2a} \]This tells us that the total work done in displacing the particle involves an energy transfer expressed in terms of the constant \(k\) and the boundary positions \(a\) and \(2a\). Understanding the work done by variable forces is crucial for problems in mechanics, especially those involving potential energy fields.