91Ó°ÊÓ

Potential energy is the energy stored in a system due to its position or configuration. In the context of conservative forces, potential energy is particularly significant. Conservative forces, like gravitational and elastic forces, have the unique property that the work done by them on a particle moving between two points depends only on the initial and final positions, not on the path taken. For this exercise, we are dealing with a force that is a function of position, specifically given as \( \vec{F} = (6.0x - 12) \hat{\mathrm{i}} \). The potential energy \( U(x) \) associated with this conservative force can be found by integrating the force with respect to position, taking into account that any force field capable of doing work can alter the potential energy of the system.

Integration is a core mathematical technique used to find potential energy from a conservative force. It essentially involves finding the "area under the curve" of a function. For conservative forces, potential energy \( U(x) \) is obtained through integration. The integral is computed by taking \[ U(x) = -\int \vec{F} \, dx + C \] where \( C \) is a constant determined by boundary conditions. In our example, this means integrating the force expression \( 6.0x - 12 \) with respect to \( x \), resulting in an anti-derivative. This yields the function \[ U(x) = -3.0x^2 + 12x + C \]. The integration process is vital as it transforms a mathematical expression for force into a comprehensible expression for potential energy.

The quadratic formula is an essential tool used to solve equations of the form \( ax^2 + bx + c = 0 \). It takes the form: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. In our exercise, after integrating the force and determining the constant of integration, we have a quadratic expression for potential energy: \( -3.0x^2 + 12x + 27 = 0 \). To find where the potential energy is zero, we apply the quadratic formula. Plugging in our values, the solutions are calculated as \( x = \frac{-12 \pm 18}{-6} \), which simplifies to the results \( x = -1 \) and \( x = 9 \). This shows where the potential energy in the system equates to zero for given positions on the \( x \)-axis.

Boundary conditions are crucial in solving physics problems because they provide specific constraints that help in defining constants within general solutions. In a potential energy problem, understanding these conditions is essential to pinning down the exact function. In our problem, it's given that the potential energy \( U(0) = 27 \) Joules when \( x = 0 \). This information allows us to determine the constant \( C \) within our integrated potential energy function. Plugging \( x = 0 \) and \( U(0) = 27 \) into \[ U(x) = -3.0x^2 + 12x + C \] yields \( C = 27 \). Such boundary conditions ensure that our mathematical representations of physical phenomena align with real-world scenarios, providing accurate predictive power for the potential energy at any position \( x \).