15.5
Frequency of Spring-Mass System
One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and…
Consider a block of mass m connected to a horizontal spring, placed over a frictionless surface.
The net force on the block is the sum of the force due to its weight, the normal force, and the force due to the spring.
Since the weight and the normal force are of equal magnitude and opposite in direction, they cancel each other, and the net force becomes equal to the force due to the spring.
Here, the magnitude of force is proportional to the first power of displacement. Because of this, the spring-mass system is called a linear simple harmonic oscillator.
Using Newton's second law, the force can be expressed in terms of acceleration.
Substituting the expressions for acceleration and displacement, the equation for angular frequency is obtained.
The angular frequency is also defined as 2π over the period of oscillation.
Also, the inverse of the period is the frequency of oscillation.
A stiff spring produces rapid oscillations and a short period. In comparison, a heavy object tends to produce sluggish oscillations and a large period.
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Q1: Why is a spring-mass system called a linear simple harmonic oscillator?
A spring-mass system is called a linear simple harmonic oscillator because the spring force is proportional to the first power of displacement, following Hooke's law. On a frictionless surface, weight and normal force cancel, leaving only the spring force as the net force. This linear relationship between force and displacement defines the system's oscillatory behavior.
Q2: What factors determine the angular frequency of a spring-mass system?
Angular frequency depends only on the mass and the spring's force constant, not on amplitude or initial conditions. Using Newton's second law with expressions for acceleration and displacement yields the angular frequency equation. A stiffer spring increases angular frequency, while greater mass decreases it, producing faster or slower oscillations respectively.
Q3: How does spring stiffness affect the period of oscillation?
A stiffer spring produces a shorter period and rapid oscillations, while a more flexible spring produces a longer period. The period depends only on mass and spring force constant. Mathematically, period is inversely related to angular frequency, so increased stiffness reduces the time for one complete oscillation.
Q4: What is the relationship between period and frequency in spring-mass oscillations?
Frequency is the inverse of the period of oscillation. If a spring-mass system completes one full oscillation in time T, its frequency is 1/T. Angular frequency relates to period through the equation ω = 2π/T, connecting these fundamental oscillation parameters.
Q5: How does mass affect the oscillation behavior of a spring-mass system?
A heavier mass produces sluggish oscillations with a longer period, while a lighter mass produces rapid oscillations with a shorter period. Since period is proportional to the square root of mass, doubling the mass increases the period by a factor of √2, slowing the system's oscillatory motion.
Q6: Why do weight and normal force not affect the spring-mass system's oscillation?
On a frictionless horizontal surface, weight and normal force are equal in magnitude and opposite in direction, so they cancel completely. The net force becomes equal only to the spring force, which acts parallel to the surface. This cancellation simplifies the system to depend solely on spring force and mass.
Q7: How can you derive the angular frequency equation for a spring-mass system?
Start with Newton's second law, F = ma, and substitute Hooke's law (F = -kx) for spring force and the acceleration expression from kinematics. Solving the resulting differential equation yields the angular frequency ω = √(k/m), where k is the force constant and m is mass.