Question

When the amplitude of the oscillation of a mass on a stretched string is increased, why doesn't the period of oscillation also increase?

Short answer

Answer: The period of oscillation for a mass on a stretched string does not increase when the amplitude of oscillation is increased because the period depends only on the mass and the spring constant of the system, not the amplitude. As amplitude increases, the object experiences a larger restoring force; however, this results in a larger acceleration, balancing out the effect of the increased amplitude and keeping the period of oscillation constant.

Step-by-step solution

Understand the concept of oscillation

An oscillation is the repetitive motion of a system or object between two points, in this case, the oscillation of a mass on a stretched string. The main properties of an oscillation are amplitude, which is the maximum displacement from the equilibrium position, and period, which is the time taken to complete one full cycle of oscillation.

Identify the equation for the period of the oscillation

The period of a mass on a stretched string can be found using the equation: T = 2π √(m/k) where T is the period of oscillation, m is the mass of the object, and k is the spring constant of the stretched string. The spring constant depends on the stiffness and length of the string. Notice that the equation does not include the amplitude of the oscillation.

Show that the amplitude does not affect the period of the oscillation

From the equation mentioned above, we can see that the period (T) depends only on the mass (m) and the spring constant (k) of the system. There is no term representing the amplitude in this equation, indicating that the amplitude does not affect the period of oscillation. Therefore, when the amplitude of oscillation is increased, the period remains unchanged because the period of oscillation is determined solely by the mass and the spring constant.

Explain the physical reasoning behind this phenomenon

Based on Hooke's Law, the mass on the stretched string experiences a restoring force proportional to its displacement from the equilibrium position. As the amplitude increases, the maximum displacement from the equilibrium position will increase, resulting in a larger maximum restoring force. However, the rate at which the force increases will be the same, causing the mass to accelerate at the same rate as when the oscillations had a smaller amplitude. Thus, even though the amplitude is increased, causing the mass to travel a longer distance in each oscillation, it also experiences a larger restoring force that results in a larger acceleration. These two factors balance each other out, keeping the period of oscillation constant.

Key concepts

Oscillation Amplitude

Oscillation amplitude refers to the maximum extent of displacement that an object moves from its equilibrium position during oscillation. It's a measure of how far the object travels in either direction along its path of motion. Imagine a child swinging back and forth on a swing—the highest points they reach on either side represent the amplitude of their swinging motion.

Importantly, it is a common misconception that changing the amplitude of an oscillation affects its period, but as demonstrated in our step-by-step solution, the amplitude does not factor into the period of a mass-spring system. Increasing the amplitude means the mass covers more distance in each swing, but the time it takes to complete a cycle (the period) remains the same. This is crucial for students to understand, as it underlies the behavior of harmonic oscillators in many physical systems.

Hooke's Law

Hooke's Law is a fundamental principle that relates the force needed to extend or compress a spring to the distance of that extension or compression. It's mathematically represented as F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.

The negative sign indicates that the force is a restoring force, which acts in opposition to the direction of displacement. This law is the cornerstone of understanding harmonic motion, as it describes how the force that brings oscillating objects back to their starting point is directly proportional to the distance they've been displaced.

Harmonic Motion

Harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to the displacement. An object in harmonic motion will oscillate around an equilibrium position in a predictable pattern, such as a mass on a spring or a pendulum swinging back and forth.

In harmonic motion, the system will continue to oscillate with a constant period as long as there is no net external force acting to dampen or drive the oscillation. The period of harmonic motion is determined by properties like mass and spring constant, not the amplitude of the oscillation, which allows for the predictability of these systems.

Spring Constant

The spring constant, denoted as k, is a measure of a spring's stiffness. It's a key factor in determining the period of oscillation for a mass-spring system. A higher spring constant means a stiffer spring, and thus a greater force is required to extend or compress it by some displacement. According to Hooke's Law, the spring constant is the ratio of the force exerting the restoring force to the displacement caused.

To put it another way, the spring constant relates how 'tough' or 'flexible' a spring is. When considering the period of oscillation (T), the spring constant appears in the denominator in the equation T = 2π √(m/k), which signifies that as the spring constant increases, the period decreases, and vice versa. Importantly, this is independent of amplitude—the spring constant is intrinsic to the material properties and dimensions of the spring itself.