91Ó°ÊÓ

In physics, potential energy is the energy stored within a system due to the position or arrangement of its components. Here, it is expressed as a function of position along the x-axis:

• \(U(x) = 20 + (x-2)^2\)

This equation tells us how the potential energy varies with the position, \(x\).
This specific form shows that as you move away from \(x = 2\), the potential energy increases due to the squared term \((x-2)^2\), which describes a parabolic shape, centered at \(x=2\) where potential energy is minimized.
Conservative forces, such as the one we're dealing with here, are forces where the work done only depends on the initial and final positions, not the path taken. The potential energy can help us better understand how this force might act on the particle.

Kinetic energy is the energy of motion. The formula for kinetic energy in one dimension is:

• \(KE = \frac{1}{2}mv^2\)

where \(m\) is mass and \(v\) is velocity.
In our problem, a 1.0-kg particle has a kinetic energy of 20 Joules at \(x = 5.0\, m\).
This gives insight into the particle's speed at this point: as kinetic energy and potential energy convert into one another in conservative systems, the particle moves faster or slower depending on its energy configuration.

The derivative of potential energy with respect to position helps us find the force acting on the particle. This is derived from the principle that the force exerted by a conservative field is the negative gradient (or derivative) of the potential energy.
We calculate it as follows:

• Find the derivative of \(U(x) = 20 + (x - 2)^2\).
• The derivative of the constant \(20\) is \(0\).
• The derivative of \((x - 2)^2\) is \(2(x - 2)\).

As per the relation, the force is:

• \(F(x) = -\left(2(x - 2)\right)\)

This negative sign is crucial as it indicates that the force opposes the change in potential energy as expected in conservative forces.

The force function describes how force varies with position, derived from the potential energy function. Using the derivative we discussed:

• \(F(x) = -2(x - 2)\)

This simplifies to:

• \(F(x) = -2x + 4\)

This force function shows that the force depends linearly on the position \(x\), with a negative slope.
It tells us that the force is strongest when \(x = 0\) and decreases towards \(x = 2\). Beyond this point, force acts in the opposite direction pulling the particle back towards \(x = 2\).
Among the given options, we verified that option (3) \(F = 2(2-x)\) simplifies to the same expression \(-2x + 4\), confirming it as the correct function.
This aligns with conservative forces where potential energy changes are directly linked to the forces countering or aiding motion.