People's Education Press High School Physics: Selective Mandatory Course 1: Vibrations in Daily Life and the Spring Oscillator Model

Welcome to the fascinating world of mechanical vibrations. In our daily lives, vibrations are everywhere: fromthe trembling of a guitar stringbringing us beautiful music, tothe back-and-forth swinging of a pendulum, eventhe swaying of tree branches in a gentle breeze. Physics unifies these complex phenomena into a fundamental form of motion.

Definition: Mechanical Vibration
We define the back-and-forth motion of an object or part of an object around a specific position asmechanical vibration (mechanical vibration), commonly referred to as vibration.

Figure 2.2-1: Horizontal Spring Oscillator Model

Idealized Model: The Spring Oscillator

To study more deeply, we construct an idealized physical model—the spring oscillator (spring oscillator). It consists of a small mass and a spring. In this model, we apply scientific simplifications:

Ignoring the mass of the spring.
Neglecting friction at contact surfaces and air resistance.

Equilibrium Position (equilibrium position) is the core of study: when the spring is undeformed, the net force on the mass is zero, making this point the center of oscillation. Whether it’s a floating buoy in water or a steel ball hanging vertically, their dynamic essence lies in the fact that the object experiences arestoring forcethat always points toward this equilibrium point.

Model Pitfall (Pitfall)

It must be emphasized: the spring oscillator is aidealized model. It forms the foundation for studying general vibrations. When dealing with real-world problems (such as the wobbling of a bamboo pole), we must recognize that neglecting mass and friction is only an approximation under specific conditions.

QUESTION 1

Figure 2.2-9 shows the vibration graphs of two simple harmonic motions, A and B. Given that A has an amplitude of 0.5 cm and a period of 0.4 s, and B has an amplitude of 0.2 cm and a period of 0.8 s, write the displacement-time relationship equations for both simple harmonic motions based on the graph.

x_A = 0.5 sin(5πt), x_B = 0.2 sin(2.5πt)

x_A = 0.5 sin(0.4t), x_B = 0.2 sin(0.8t)

x_A = 0.4 sin(0.5πt), x_B = 0.8 sin(0.2πt)

QUESTION 2

As shown in Figure 2.6-6, when a vertical disk rotates, a small cylinder fixed on the disk drives a T-shaped bracket to vibrate vertically. If the disk rotates uniformly at frequency f, what is the frequency of the mass’s vibration once it reaches steady state?

Equal to the natural frequency f₀ of the mass

Equal to the rotation frequency f of the disk

Equal to (f + f₀)/2

QUESTION 3

What is the core criterion for mechanical vibration?

The object must undergo circular motion

The object moves back and forth near the equilibrium position

The net external force on the object remains constant

QUESTION 4

Which of the following examples does not fall within the scope of the idealized spring oscillator model?

A spring-mass system sliding on a horizontal rod with negligible friction

The self-vibration of a heavy helical spring whose mass cannot be ignored

A steel ball suspended vertically by a spring with negligible air resistance

QUESTION 5

Which physical quantities reach their maximum values when a particle undergoing simple harmonic motion passes through the equilibrium position?

Displacement, acceleration

Restoring force, potential energy

Velocity, kinetic energy