91Ó°ÊÓ

When dealing with non-linear restoring forces, such as the one given by the spring in this problem, we express the force as a function of position: \( F_{x}(x) = -\alpha x - \beta x^{2} \). This equation signifies that the spring force is dependent on both linear and quadratic terms of displacement \( x \). To find the potential energy function \( U(x) \) related to this force, you integrate the force function with respect to \( x \), as potential energy is the energy stored in the system due to its position.

The process begins by setting up the integral of the negative restoration force: \( U(x) = -\int F_{x}(x) \, dx \). This yields the equation: \( U(x) = \alpha \frac{x^2}{2} + \beta \frac{x^3}{3} + C \). By applying the condition \( U(0) = 0 \), we determine that the constant \( C = 0 \), refining our potential energy function to:
- \( U(x) = \frac{\alpha x^2}{2} + \frac{\beta x^3}{3} \).
This formula allows us to calculate the potential energy for any given displacement, providing insights into the energy stored at that position.

This important principle states that the total energy of an isolated system remains constant over time. In physics, it implies that energy can neither be created nor destroyed; instead, it can only be transferred or transformed from one form to another. For example, in this spring exercise, the initial potential energy is converted into kinetic energy as the system evolves.

In the given scenario, start with the initial potential energy calculated for the spring stretched 1.00 m, denoted as \( U_{\text{initial}} = 36.0 \, \text{J} \). When the spring is compressed or stretched to 0.50 m, the potential energy changes to \( U_{\text{final}} = 8.25 \, \text{J} \).
By conservation of energy, the energy initially stored in the potential energy of the stretched spring must have been converted into kinetic energy if no other opposing forces (like friction) are present:
- \( U_{\text{initial}} = U_{\text{final}} + K \)
Solving for \( K \), we find it to be \( 27.75 \, \text{J} \). This is the energy that now accounts for the speed of the moving object attached to the spring.

Kinetic energy is the energy of an object due to its motion, and it’s quantified by the equation \( K = \frac{1}{2}mv^2 \). To find the velocity of the object attached to the spring, utilize the kinetic energy determined from conservation of energy: \( K = 27.75 \, \text{J} \).

Given that you know the kinetic energy and the mass of the object, 0.900 kg, solve for its velocity \( v \) using:
- \( K = \frac{1}{2}(0.900) v^2 \)
Plug the numbers into the equation:
- \( 27.75 = \frac{1}{2}(0.900) v^2 \)
Solving for \( v \) yields:
1. Multiply the kinetic energy by 2 and divide by the mass: \( v^2 = \frac{27.75 \times 2}{0.900} = 61.67 \)
2. Take the square root of the result: \( v = \sqrt{61.67} = 7.85 \, \text{m/s} \)
This calculation reveals that the object’s speed is 7.85 m/s as it moves through the position 0.50 m from its equilibrium point. This result illustrates how energy transformation affects motion in a non-linear force system.