91Ó°ÊÓ

In a mass-spring system, we deal with a setup where a mass is attached to a spring that can be compressed or stretched. This system is fundamental to understanding simple harmonic motion (SHM) because it naturally exhibits oscillations. The spring exerts a force proportional to the displacement of the mass from its equilibrium position, a behavior described by Hooke's Law:
\[ F = -kx \] where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement from equilibrium. This force always acts to restore the mass to its equilibrium position.
In our example, the mass is \( 0.1 kg \) and the spring constant is \( 15 \text{ N/m} \). The system is set up so the object can slide frictionlessly, ensuring pure SHM. This arrangement allows us to study the periodic motion and how the mass moves back and forth in response to the spring's restoring force.

The equilibrium position in a mass-spring system is the point where the spring exerts no force because it is neither compressed nor stretched. When the object is at equilibrium, the net force acting on it is zero, meaning there's no acceleration if friction or other forces are absent.
Understanding the equilibrium position is crucial because it acts as a reference point for oscillations. The object passes through this position twice during a complete cycle: once while moving in each direction. In this scenario, the equilibrium position helps us calculate when the object returns to center after being displaced. The first time the object reaches equilibrium after release is at half its period \( T/2 \), and it revisits there consistently every half period after that, predicting points of zero displacement within the oscillation cycle.

Angular frequency, denoted by \( \omega \), is a measure of how quickly an object rotates or oscillates in a circle. For SHM, this translates to how fast the oscillations occur. It is related to the mass and spring constant like this:
\[ \omega = \sqrt{\frac{k}{m}} \]where \( k \) is the spring constant and \( m \) is the mass. The angular frequency has units of radians per second. It allows us to calculate the period \( T \) and subsequently predict the timing of events like when the object will return to the equilibrium position or reach particular displacements.
In our setup, the period \( T \) is given by: \[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} \approx 0.516 \text{ seconds} \] This calculation helps set the framework for understanding motion because angular frequency directly influences how fast the periodic oscillations occur.

Displacement in simple harmonic motion describes how far the object is from the equilibrium position at any given time. It is often modeled by a sine or cosine function because the motion is periodic.
The general formula is: \[ x(t) = A \cos(\omega t + \phi) \] where:

• \( x(t) \) is the displacement at time \( t \)
• \( A \) is the amplitude, the maximum displacement from equilibrium
• \( \omega \) is the angular frequency
• \( \phi \) is the phase constant, which adjusts based on the initial conditions

In this exercise, when calculating for when the object is \(10 \text{ cm} \) to the left or \( 5 \text{ cm} \) to the right, we adjust our cosine function to solve for the specific times these displacements occur. Understanding this displacement formula allows you to predict positions over time, describing the object's location during its oscillatory motion.