Problem 5 - University Physics with Modern Physics

Question

Which of the following is not a valid potential energy function for the spring force \(F=-k x ?\) a) \(\left(\frac{1}{2}\right) k x^{2}\) b) \(\left(\frac{1}{2}\right) k x^{2}+10 \mathrm{~J}\) c) \(\left(\frac{1}{2}\right) k x^{2}-10 \mathrm{~J}\) d) \(-\left(\frac{1}{2}\right) k x^{2}\) e) None of the above is valid.

Step-by-Step Solution

Answer

(a) \((\frac{1}{2}) k x^{2}\) (b) \((\frac{1}{2}) k x^{2} + 10\) (c) \((\frac{1}{2}) k x^{2} - 10\) (d) \(-(\frac{1}{2}) k x^{2}\) Answer: (d) \(-(\frac{1}{2}) k x^{2}\)

Step 1: Identify the valid potential energy functions

We already know that the standard potential energy function for spring force is \(U(x)=\left(\frac{1}{2}\right) k x^{2}\). Adding or subtracting a constant value, in this case \(10 \mathrm{~J}\), does not change the fact that the potential energy function is still valid. So, we can say that options a), b), and c) are valid potential energy functions.

Step 2: Examine the remaining option

Now, we only have to check option d) to determine whether it is a valid potential energy function for the spring force or not. The given function is \(U(x)=-\left(\frac{1}{2}\right) k x^{2}\). Comparing it to the standard potential energy function, we can see that it has a negative sign in front of the expression. This means that the potential energy will be negative, which is not possible for a spring system, as the potential energy should always be non-negative.

Step 3: Determine the answer

Based on our analysis, we can conclude that option d) is not a valid potential energy function for the spring force. Therefore, the correct answer is: (d) \(-\left(\frac{1}{2}\right) k x^{2}\)

Key Concepts

Spring Force

The spring force is a fascinating and important concept in physics. It's the force exerted by a spring when it is compressed or stretched. This force is crucial in understanding how springs work, whether in a mattress, a car suspension system, or even in a simple toy. A spring follows Hooke's Law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position. This can be expressed as:\[ F = -kx \]- **F** is the force exerted by the spring (in newtons).- **k** is the spring constant, which measures the stiffness of the spring (in newtons per meter).- **x** is the displacement of the spring from its equilibrium position (in meters). The negative sign indicates that the spring force is a restoring force, meaning it acts in the opposite direction to the displacement to bring the spring back to its original position. Understanding spring force is critical for solving problems related to oscillations, like those involving pendulums or the natural frequency of objects.

Conservative Forces

Conservative forces are forces where the work done on an object is path-independent and only depends on the initial and final positions. In simpler terms, this means the total mechanical energy (kinetic + potential) does not change as an object moves between two points under the influence of a conservative force. Examples of conservative forces include:

• Gravitational force
• Electrostatic force
• Spring force

For springs, the work done by the spring force can be completely described by the change in potential energy. Potential energy functions, such as the standard spring potential energy \( U(x) = \left(\frac{1}{2}\right) k x^2 \), represent stored energy due to the position of an object in the field of a conservative force. Thus, springs are ideal systems for illustrating the principles of conservative forces.

Mechanical Energy

Mechanical energy is a pivotal concept that combines kinetic energy and potential energy. This combination helps us understand the behavior of objects in motion and under the influence of various forces. Mechanical energy can be expressed as:\[ E_{mech} = K + U \]- **\(E_{mech}\)** is the total mechanical energy.- **K** is the kinetic energy.- **U** is the potential energy.In a closed system without external forces (such as friction), the mechanical energy remains constant. This is known as the conservation of mechanical energy and is an essential principle in physics.For a spring system, the potential energy is stored as the spring is either compressed or stretched:\[ U(x) = \left(\frac{1}{2}\right) k x^2 \]While the kinetic energy is derived from the motion of the mass attached to the spring, the conservation of mechanical energy within such a system is a foundational aspect of solving physics problems related to oscillations and dynamics. By understanding how mechanical energy is conserved, students can predict the future motion of objects in these systems effectively.