Question

(a) When the displacement is one-half the amplitude \(x_{m}\), what fraction of the total energy is kinetic and what fraction is potential in simple harmonic motion? (b) At what displacement is the energy half kinetic and half potential?

Short answer

For(a), when \(x = 0.5 x_m\), the kinetic energy is \(K/E = 0.75\) and the potential energy is \(U/E = 0.25\). For(b), the energy is half kinetic and half potential when the displacement is \(x = x_m / sqrt{2}\)

Step-by-step solution

Understand Energy Distribution in Simple Harmonic Motion

In simple harmonic motion, energy continually transfers back and forth between kinetic and potential energy, however the total energy is conserved. At maximum displacement, i.e. at amplitude \(x_m\), all energy is potential, and at equilibrium (where displacement is zero), all energy is kinetic. The kinetic energy K and the potential energy U at any point can be described using these formulas:\(K = 0.5 m v^2\), \(U = 0.5 k x^2\), where m denotes the mass, v the speed, k the spring constant and x the displacement

Find the Kinetic and Potential Energy Fraction with Displacement \(0.5 x_m\)

When the displacement is half the amplitude (\(x = 0.5 x_m\)), using formulas for kinetic and potential energy, the total energy E is \(E = K + U = 0.5 k x^2 + 0.5 m v^2\). But the total energy is also equal to the potential energy at maximum displacement (\(x = x_m\)), so \(E = 0.5 k x_m^2\). Therefore, the fractions of kinetic and potential energy are \(K/E = 0.5 m v^2 / 0.5 k x_m^2\) and \(U/E = 0.5 k x^2 / 0.5 k x_m^2\), respectively. Note here that \(v^2 = ω^2 (x_m^2 - x^2)\), where ω is the angular frequency.

Determine the Displacement for Half Kinetic and Half Potential Energy

When the energy is half kinetic and half potential, it implies \(K = U\). Using the formulas \(K = 0.5 m v^2\) and \(U = 0.5 k x^2\), you find \(0.5 m v^2 = 0.5 k x^2\). Remember \(v = ω sqrt{x_m^2 - x^2}\), then solve for x. This gives the result \(x = x_m / sqrt{2}\).

Key concepts

Kinetic Energy in Simple Harmonic Motion

In simple harmonic motion, kinetic energy represents the energy of motion. It highlights how fast an object moves back and forth as it oscillates. The formula used to determine the kinetic energy (\(K\)) is crucial:

• \(K = 0.5 \cdot m \cdot v^2\)

where:

• \(m\) is the mass of the object,
• \(v\) is the velocity at any given point.

At the equilibrium position of simple harmonic motion, all the energy is kinetic because the object moves at its fastest.
No force acts in the direction of displacement, resulting in a zero net force.
This energy decreases as the object approaches the maximum displacement (amplitude) since the speed reduces.
Understanding how kinetic energy changes with position is key to mastering the dynamics of oscillating systems.

Potential Energy in Simple Harmonic Motion

Potential energy in simple harmonic motion arises from the object's position and the force trying to restore it to equilibrium.
This energy is stored when the object is displaced from equilibrium due to the spring's stiffness in spring-mass systems or gravity in pendulums.
The potential energy (\(U\)) in such systems is calculated using:

• \(U = 0.5 \cdot k \cdot x^2\)

where:

• \(k\) is the spring constant (how stiff the spring is),
• \(x\) is the displacement from the equilibrium position.

At maximum displacement, all energy is potential since velocity is zero.
As the object moves back to equilibrium, some potential energy turns into kinetic energy.
Therefore, potential energy helps us understand the potential for movement in any system exhibiting simple harmonic motion.

Amplitude in Simple Harmonic Motion

Amplitude is a crucial concept in simple harmonic motion, defining the maximum extent of an object's oscillation from its equilibrium point. One can think of it as the 'reach' of the motion.

• Amplitude, denoted as \(x_m\), is the furthest point the oscillating object can be displaced.

The total energy in simple harmonic motion depends on the amplitude, since at this point, all the energy is potential.

• This initial potential energy is \(U = 0.5 \cdot k \cdot x_m^2\)

As the amplitude increases, the total energy increases, resulting in taller oscillations.
Conversely, lower amplitudes lead to lower energy and shorter oscillations.
Understanding amplitude is essential because it helps indicate the energy conservation and distribution throughout the cycle of motion.