Question

A mass is oscillating with amplitude \(A\) at the end of a spring. How far (in terms of \(A\)) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

Short answer

The displacement is \(\frac{A}{\sqrt{2}}\) from equilibrium.

Step-by-step solution

Identify Initial Conditions

The problem involves a mass oscillating on a spring with amplitude \(A\). We need to determine when the elastic potential energy equals the kinetic energy.

Understand Energy Relation

In a spring-mass system, total mechanical energy is constant and is the sum of elastic potential energy \(U\) and kinetic energy \(K\). When \(U = K\), the total energy \(E = U + K\) gives us the equation \(2U = E = 2K\).

Set Up Energy Equation

The total energy in a simple harmonic oscillator with amplitude \(A\) is given by \(E = \frac{1}{2}kA^2\), where \(k\) is the spring constant. At some displacement \(x\), the elastic potential energy is \(U = \frac{1}{2}kx^2\).

Equate Energies

Set \(U = K\), then \(2U = E\), so \(2 \left(\frac{1}{2} k x^2\right) = \frac{1}{2} k A^2\). Simplify this to get \(k x^2 = \frac{1}{2} k A^2\).

Solve for Displacement

Cancel \(k\) from both sides: \(x^2 = \frac{1}{2}A^2\). Now, solve for \(x\): \(x = \pm\frac{A}{\sqrt{2}}\). Thus, the displacement is \(x = \pm\frac{A}{\sqrt{2}}\).

Conclusion

The mass is \(\frac{A}{\sqrt{2}}\) (or approximately 0.7071 \(A\)) away from the equilibrium position in both directions when the energies are equal.

Key concepts

Elastic Potential Energy

Elastic potential energy is an essential concept when studying harmonic oscillation and spring-mass systems. It represents the energy stored in a spring when it is compressed or stretched. In mathematical terms, the elastic potential energy \( U \) can be expressed as \( U = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.
In our exercise scenario, when the mass attached to the spring oscillates with an amplitude \( A \), the maximum elastic potential energy happens when the spring is either at its maximum or minimum stretch, meaning \( x = A \) or \( x = -A \). However, for any other position, it's crucial to calculate how the potential energy compares to the spring's kinetic energy to understand the system's energy distribution.

• Elastic potential energy increases as the spring stretches or compresses further from its equilibrium position.
• Maximum potential energy is at the maximum displacement, while at equilibrium, it is minimal.
• This concept is crucial for understanding how energy converts between potential and kinetic forms in a spring-mass system.

Kinetic Energy

Kinetic energy relates to the motion of the mass in a spring-mass system. It describes the energy an object possesses due to its motion and is given by the expression \( K = \frac{1}{2}mv^2 \), where \( m \) represents the mass, and \( v \) denotes its velocity. In our exercise, the focus is on the point where the kinetic energy equals the elastic potential energy.
As the mass moves through its oscillation path, its speed varies, and thus its kinetic energy does as well. At the equilibrium position, the speed is greatest, making the kinetic energy maximum, while at the maximum displacement, the speed, and therefore the kinetic energy, is zero.

• Kinetic energy is highest at the equilibrium point where the velocity is maximum.
• There is a point in the oscillation where kinetic and potential energies are equal; this reflects a balance between the speed and stretch of the spring.
• Understanding kinetic energy changes helps in analyzing how the spring-mass system behaves dynamically.

Spring-Mass System

The spring-mass system is a classic model that explains how harmonic oscillation works. It's composed of a mass attached to a spring, which can move back and forth along the spring's length.
When studying this system, the total mechanical energy combines both elastic potential energy and kinetic energy. The principle of conservation of energy dictates that, while these energy forms interconvert, the total remains constant throughout the oscillation.
In our specific exercise, the problem examines the scenario when a mass oscillates at amplitude \( A \), questioning when its elastic potential energy equals its kinetic energy.

• A spring-mass system experiences continuous energy conversion between potential and kinetic forms.
• Amplitude \( A \) determines the maximum stretch or compression from the equilibrium position.
• At a displacement of \( x = \pm\frac{A}{\sqrt{2}} \), the energies are equal, demonstrating the interplay and balance of energies in the system.

By understanding these energy dynamics, it's easier to predict how the system will evolve over time, whether for engineering applications or in theoretical physics.