Question

A harmonic oscillator has angular frequency \(\omega\) and amplitude \(A\). (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that \(U =\) 0 at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to \(A\)/2, what fraction of the total energy of the system is kinetic and what fraction is potential?

Short answer

(a) Displacement: ±A/√2, Velocity: ±ωA/√2. (b) Twice per cycle, time is T/4. (c) Kinetic: 3/4E, Potential: 1/4E.

Step-by-step solution

Understanding Energy Equilibrium

In a harmonic oscillator, the total energy is the sum of kinetic and potential energy. When these are equal, the potential energy U = (1/2)kx^2 is equal to the kinetic energy K = (1/2)mv^2. Setting these equal and knowing the total energy is E = (1/2)kA^2, we have U = K = (1/2)E.

Finding Displacement in Energy Equilibrium

From the condition that U = K, each is (1/4)E; if U = (1/2)kx^2, then (1/4)E = (1/2)k(A^2)/2, solving for displacement gives x^2 = A^2/2, so x = ±A/√2.

Linking Velocity in Energy Equilibrium

Using velocity energy relation (1/2)mv^2 = K = (1/4)E = (1/4)(1/2)kA^2, solve for v^2: v = ±ωA/√2 by using v^2 = k/m*A^2/2 and k/m = ω^2.

Frequency of Equal Energy Occurrences

The energies are equal when the system is at x = ±A/√2. This displacement occurs twice per each half of a cycle: one on the positive side of oscillation, and one on the negative.

Calculating Time Between Occurrences

In a harmonic oscillator with period T, the time between zero crossings (half-cycle) is T/2. Energy equality (x = ±A/√2) occurs evenly in between, so time between these points is T/4.

Energy Fractions at Displacement A/2

Potential energy when displacement is A/2, U = (1/2)k(A/2)^2 = (1/8)kA^2, which is (1/4) of total energy E. Therefore, kinetic energy is (3/4)E by subtraction K = E-U.

Key concepts

Angular Frequency

Angular frequency, often represented by the symbol \( \omega \), is an essential concept in harmonic oscillators. It describes how quickly an object oscillates back and forth through its equilibrium point. Think of it as the rate of rotation in radians per second in circular motion, relating closely to the cycle of oscillation.

In a harmonic oscillator, angular frequency is calculated using the formula:

• \( \omega = \sqrt{\frac{k}{m}} \)

where \( k \) is the spring constant and \( m \) is the mass of the object. This equation highlights how both the stiffness of the spring and the mass influence the system's frequency.

Angular frequency helps determine the system's period, \( T \), through:

• \( T = \frac{2\pi}{\omega} \)

This period represents the time taken for one complete oscillation. Understanding \( \omega \) provides insight into the behavior and timing of the oscillating system, crucial in problem-solving scenarios such as determining how often potential and kinetic energy equalize.

Elastic Potential Energy

Elastic potential energy is the stored energy in a system due to deformation, like stretching or compressing a spring. In the context of a harmonic oscillator, it is a critical component.

The formula for elastic potential energy \( U \) is:

• \( U = \frac{1}{2} k x^2 \)

where \( k \) stands for the spring constant, and \( x \) is the displacement from the equilibrium position. This formula shows how potential energy is directly proportional to the square of the displacement.

When a harmonic oscillator moves, it shifts between kinetic and potential energy. At maximum displacement (amplitude), all energy is potential. As the oscillator moves back through equilibrium, potential energy converts entirely into kinetic energy.

Recognizing when potential energy equals kinetic energy, such as when \( x = \frac{A}{\sqrt{2}} \), is vital in understanding the energy distribution in systems like springs or pendulums.

Kinetic Energy

Kinetic energy in a harmonic oscillator is the energy associated with its motion. It is influenced by both the mass of the object and its velocity. When discussing oscillators, kinetic energy is particularly crucial because it fluctuates with displacement.

The formula for kinetic energy \( K \) is:

• \( K = \frac{1}{2} mv^2 \)

where \( m \) represents mass, and \( v \) denotes velocity. As displacement increases from zero, velocity decreases, converting kinetic energy into potential energy.

By understanding when kinetic energy equals potential energy (\( U = K \)), we grasp how energy shifts in oscillators. This typically occurs at points when the object's velocity is enough to carry it through the equilibrium point but is not at its fastest.

These energy transitions between potential and kinetic are fundamental to problems where finding the velocity at specific displacements, such as \( \pm \frac{\omega A}{\sqrt{2}} \), defines behavior.

Energy Equilibrium

Energy equilibrium in harmonic oscillators is the state where potential and kinetic energies are equal. This balance is a crucial characteristic of harmonic motion.

In any harmonic oscillator, total energy \( E \) is conserved and can be expressed as:

• \( E = U + K \)
• With \( U = K \rightarrow U = \frac{1}{4}E \text{,} \ K = \frac{1}{4}E \)

This balance means at some point during oscillation, both energy forms share the total energy equally, creating a special equilibrium state.

For the harmonic oscillator exercise, equilibrium is reached when displacement equals \( \pm \frac{A}{\sqrt{2}} \). Importantly, this equality occurs twice per cycle, allowing both energies to shift seamlessly between each form.

Understanding this equilibrium is essential for analyzing and predicting oscillating systems' time characteristics, like the occurrence frequency of energy equality throughout an oscillation cycle.