Question

You use your hand to stretch a spring to a displacement \(x\) from its equilibrium position and then slowly bring it back to that position. Which is true? a) The spring's \(\Delta U\) is positive. b) The spring's \(\Delta U\) is negative. c) The hand's \(\Delta U\) is positive. d) The hand's \(\Delta U\) is negative. e) None of the above statements is true.

Short answer

a) The spring's ΔU is positive b) The spring's ΔU is negative c) The hand's ΔU is positive d) The hand's ΔU is negative e) None of the above statements is true Answer: b) The spring's ΔU is negative and d) The hand's ΔU is negative

Step-by-step solution

Determine potential energy changes in the spring

In the given situation, the spring is first stretched to a displacement \(x\) and then brought back to its equilibrium position. Let's calculate the potential energy change during this process: Initial potential energy, \(U_i = \frac{1}{2} kx^2\) Final potential energy, \(U_f = 0\), because the spring is back to its equilibrium position. Now, calculate the change in potential energy, \(\Delta U = U_f - U_i\). \(\Delta U_{spring} = 0 - \frac{1}{2} kx^2 = - \frac{1}{2} kx^2\) For the spring, \(\Delta U_{spring} < 0\), which means the potential energy of the spring has decreased during the process.

Determine potential energy changes in the hand

The work done by the hand to stretch the spring and bring it back to its equilibrium position is equal to the change in potential energy in the spring. In this case, the work done by the hand is given by: \(W_{hand} = -\Delta U_{spring}\) \(W_{hand} = -(- \frac{1}{2} kx^2)\) \(W_{hand} = \frac{1}{2} kx^2\) Since work done by the hand is positive, it means that the potential energy of the hand has decreased during the process. Therefore, \(\Delta U_{hand} < 0\).

Analyze the options

Based on the analysis in the previous steps, let's go through each option: a) The spring's \(\Delta U\) is positive - False, because \(\Delta U_{spring} < 0\). b) The spring's \(\Delta U\) is negative - True, as \(\Delta U_{spring} = -\frac{1}{2} kx^2\). c) The hand's \(\Delta U\) is positive - False, because \(\Delta U_{hand} < 0\). d) The hand's \(\Delta U\) is negative - True, as the hand's potential energy has decreased during the process. e) None of the above statements is true - False, because options b) and d) are true. Therefore, the correct answer is that the spring's \(\Delta U\) is negative (option b) and the hand's \(\Delta U\) is negative (option d).

Key concepts

Spring Potential Energy

Understanding spring potential energy is key to grasping various physics concepts. Imagine a spring at rest. When you apply a force to stretch or compress it, you're storing energy in the form of spring potential energy. Mathematically, it's given by the equation \( U = \frac{1}{2} kx^2 \), where \( k \) represents the spring constant, indicating the stiffness of the spring, and \( x \) is the displacement from its equilibrium position.

In the context of the exercise, when you stretch the spring to displacement \( x \) and then allow it to return to the equilibrium, the stored potential energy is transformed back into other forms, usually kinetic energy or work done by the spring's force. The concept that the energy stored in the spring depends on the square of the displacement is essential, meaning even small stretches or compressions can store significant amounts of energy within stiff springs. This fundamental principle is widely used in mechanics, from watches to car suspensions.

Work-Energy Principle

The work-energy principle is a cornerstone in physics, providing a clear link between the forces acting on an object and its energy. In simplest terms, it states that work done by forces on an object results in a change in that object's kinetic energy. Mathematically, \( W = \Delta KE \), where \( W \) is the work and \( KE \) is the kinetic energy. When looking at springs, this principle can be extended; the work done in stretching or compressing a spring is manifested as potential energy within the spring.

For the exercise, when you use your hand to stretch and then release the spring, your hand is doing work against the spring force. This work is initially stored as potential energy in the spring and is then released as the spring returns to equilibrium. The work-energy principle reminds us that energy is neither created nor destroyed but simply transformed from one type to another, such as from potential to kinetic energy or vice-versa.

Mechanical Energy Conservation

The conservation of mechanical energy is a fundamental concept that states when only conservative forces (like gravity and spring forces) act on a system, the total mechanical energy remains constant. This total includes both kinetic energy (\( KE \) - energy of motion) and potential energy (\( PE \) - stored energy).

In the exercise we've been considering, even as the spring goes from stretched back to its equilibrium position, the total mechanical energy of the system remains the same (if we ignore non-conservative forces like friction). Energy transformations occur within the system: Potential energy (spring potential energy when stretched) converts into kinetic energy (motion) and potentially into work (done by the hand), but the total amount of mechanical energy stays constant throughout the process.

The mechanical energy conservation principle helps explain why the spring returns to its original position after being disturbed – it's an interplay of energy conversion within a conservative force field, with no net loss in the system's mechanical energy.