Problem 31 - College Physics

Question

\(\bullet\bullet\) Show that the total energy of a mass-spring system in simple harmonic motion is given by \(\frac{1}{2} m \omega^{2} A^{2}\).

Step-by-Step Solution

Answer

The total energy is \( \frac{1}{2} m \omega^2 A^2 \).

Step 1: Understanding the Energy Components

In simple harmonic motion, the total energy is the sum of kinetic energy and potential energy. The kinetic energy at displacement \( x \) is \( E_k = \frac{1}{2} m v^2 \), and the potential energy is \( E_p = \frac{1}{2} k x^2 \). Here, \( m \) is the mass, \( k \) is the spring constant, and \( v \) is the velocity.

Step 2: Relate Velocity and Displacement

The velocity \( v \) in simple harmonic motion is given by \( v = -rac{dx}{dt} = -A\omega\sin(\omega t) \), where \( A \) is the amplitude of motion, and \( \omega \) is the angular frequency \( \omega = \sqrt{\frac{k}{m}} \). Substituting, the velocity becomes \( v = A\omega \cos(\omega t) \).

Step 3: Substitute Velocity into Kinetic Energy

Substituting \( v = A\omega \cos(\omega t) \) into kinetic energy, we get \( E_k = \frac{1}{2} m (A\omega \cos(\omega t))^2 = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \).

Step 4: Substitute Displacement into Potential Energy

The displacement \( x \) is \( x = A \cos(\omega t) \). Substituting into potential energy, we get \( E_p = \frac{1}{2} k (A \cos(\omega t))^2 = \frac{1}{2} k A^2 \cos^2(\omega t) \).

Step 5: Use Relationship between \( k \) and \( \omega \)

Since \( \omega = \sqrt{\frac{k}{m}} \), it follows that \( k = m \omega^2 \). Substitute this into the expression for potential energy: \( E_p = \frac{1}{2} m \omega^2 A^2 \cos^2(\omega t) \).

Step 6: Summing Kinetic and Potential Energy

Now, sum the kinetic and potential energies: \( E_k + E_p = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) + \frac{1}{2} m \omega^2 A^2 \cos^2(\omega t) = \frac{1}{2} m \omega^2 A^2 (\cos^2(\omega t) + \sin^2(\omega t)) \).

Step 7: Simplify Using Trigonometric Identity

Using the identity \( \cos^2(\omega t) + \sin^2(\omega t) = 1 \), the total energy simplifies to \( \frac{1}{2} m \omega^2 A^2 \). This is a constant, showing energy conservation in the system.

Key Concepts

Mass-Spring System

A mass-spring system is a classic example of simple harmonic motion, which is oscillatory by nature. Imagine a mass attached to a spring, where the spring is either compressed or stretched from its equilibrium position. This system can move back and forth due to the restoring force exerted by the spring. The restoring force follows Hooke's law, which states that the force is directly proportional to the displacement, mathematically expressed as

• Force, \( F = -kx \)

Here, \( k \) is the spring constant, and \( x \) is the displacement from equilibrium.
The negative sign indicates the force direction is opposite to the displacement. In essence, the energy transformations in this system oscillate between kinetic and potential energy, allowing the mass to undergo simple harmonic motion.

Kinetic Energy

Kinetic energy in a mass-spring system represents the energy of motion when the mass is moving. As the mass moves back and forth, its velocity—and hence its kinetic energy—gets continuously transformed. The equation for kinetic energy is:

• \( E_k = \frac{1}{2} m v^2 \)

Here, \( m \) is the mass of the object, and \( v \) is the velocity at a particular displacement. This velocity, under simple harmonic motion, can be derived from the system's angular frequency and amplitude, expressed as:

• \( v = A\omega \cos(\omega t) \)

This equation shows how the velocity is dependent on the position in the oscillation cycle, recomputing the kinetic energy over time as the mass moves through its path.

Potential Energy

Potential energy in a mass-spring system is stored energy due to the spring's position. When the spring is extended or compressed from its equilibrium position, potential energy becomes vital. The potential energy is calculated using:

• \( E_p = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant and \( x \) is the displacement. In terms of simple harmonic motion, the displacement \( x \) is often expressed as

• \( x = A \cos(\omega t) \)

As the system oscillates, potential energy transforms into kinetic energy and vice-versa. Thus, despite changes in the individual energies, the total energy remains constant in an ideal scenario.

Angular Frequency

Angular frequency denotes how fast an object oscillates in simple harmonic motion. It is a crucial component in describing mass-spring systems, linking spring characteristics with the motion's frequency. Defined as

• \( \omega = \sqrt{\frac{k}{m}} \)

ANGULAR FREQUENCY integrates both the spring constant \( k \) and the mass \( m \) to predict oscillation behavior. The equation reveals that larger spring constants or smaller masses lead to increased oscillation rates.
\( \omega \) not only defines how quickly the mass moves back to equilibrium but also ties into both kinetic and potential energy calculations. Thus, understanding angular frequency is crucial for grasping the entire energy dynamic in a mass-spring system.