Question

The spring constant for a spring-mass system undergoing simple harmonic motion is doubled. If the total energy remains unchanged, what will happen to the maximum amplitude of the oscillation? Assume that the system is underdamped. a) It will remain unchanged. b) It will be multiplied by 2 . c) It will be multiplied by \(\frac{1}{2}\) d) It will be multiplied by \(1 / \sqrt{2}\).

Short answer

Answer: The maximum amplitude of the oscillation will be multiplied by \(\frac{1}{\sqrt{2}}\).

Step-by-step solution

Write down the equation for the total energy of a spring-mass system

The total energy (E) in a spring-mass system is given by the equation E = \(\frac{1}{2} k A^2\), where k is the spring constant and A is the maximum amplitude of the oscillation.

Set up the equation for the system with the doubled spring constant

The problem states that the spring constant is doubled. Let's call the doubled spring constant \(k'\), so we have \(k' = 2k\). If E remains unchanged, then we can write a new equation for the total energy with the doubled spring constant: E = \(\frac{1}{2} k' A'^2\), where A' is the maximum amplitude of the oscillation after the spring constant has been doubled.

Equate the two energy equations

Since the total energy is unchanged, we can set these two equations equal to each other: \(\frac{1}{2} k A^2 = \frac{1}{2} k' A'^2\)

Substitute the doubled spring constant in the equation and solve for the new amplitude

We know that \(k' = 2k\), so we can substitute this into the equation: \(\frac{1}{2} k A^2 = \frac{1}{2} (2k) A'^2\) Now we can solve for the relationship between A and A': \(A^2 = 2 A'^2\) Taking the square root of both sides: \(A = A' \sqrt{2}\)

Determine the change in amplitude

From the previous step, we now have: \(A' = \frac{A}{\sqrt{2}}\) This indicates that when the spring constant is doubled and the total energy remains unchanged, the maximum amplitude of the oscillation will be multiplied by \(\frac{1}{\sqrt{2}}\). The correct answer is (d), the maximum amplitude of the oscillation will be multiplied by \(\frac{1}{\sqrt{2}}\).

Key concepts

Spring Constant

The spring constant, represented by the variable 'k', is a measure of the stiffness of a spring. In the context of a spring-mass system undergoing simple harmonic motion (SHM), the spring constant directly affects the oscillation frequency—the greater the spring constant, the stiffer the spring, and the faster the system oscillates. Its unit is Newton per meter (N/m).

When we double the spring constant, it doesn't simply double the force required to compress or extend the spring but also fundamentally alters how the mass attached to the spring moves. This is described by Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, or mathematically, F = -kx. Thus, a higher 'k' value means it takes more force to displace the spring the same distance, leading to a change in the system's dynamic properties.

Oscillation Amplitude

The amplitude of an oscillation is the maximum displacement from its equilibrium position that the mass can reach during its motion. Amplitude is a key descriptor in SHM, as it relates to the extent of the motion—the larger the amplitude, the more energy stored in the system. However, during simple harmonic motion, the amplitude does not change over time unless external forces or damping effects come into play.

It is crucial to understand that while we might alter the spring constant, the amplitude's relationship to energy is governed by the total energy formula for a spring-mass system, E = (1/2)kA^2. If we maintain the same amount of total energy but increase 'k,' the amplitude must adjust accordingly to keep the energy constant, as reflected in the solution to the provided exercise.

Underdamped System

In the context of SHM, an underdamped system refers to a scenario where the damping force is present but not strong enough to prevent oscillations. In an underdamped spring-mass system, the mass will oscillate with a gradually decreasing amplitude over time, eventually coming to a stop due to energy dissipation – like a swinging pendulum slowing down due to air resistance.

Interestingly, being underdamped is characterized by a damping ratio that is less than one but greater than zero. In these systems, the amplitude of each oscillation decreases exponentially. However, the exercise assumes a constant amplitude, indicating that the system is experiencing oscillations before noteworthy damping effects have diminished the motion.

Total Energy of a Spring-Mass System

The total energy in a spring-mass system undergoing simple harmonic motion is the sum of its potential and kinetic energy. This system's energy remains constant if there are no external forces or energy losses. The formula for the total energy, E = (1/2)kA^2, where 'E' stands for energy, 'k' for spring constant, and 'A' for amplitude, shows the relationship between these quantities.

The significance of this relationship is that energy conservation dictates how changing one quantity must lead to compensatory changes in others to keep the total energy unchanged. For example, doubling 'k' would naturally impact 'A' to ensure that E remains the same, as illustrated in the given exercise solution where the amplitude changes by a factor of 1/√2—ensuring energy is conserved in spite of the change in spring constant.