Displacement
In the context of simple harmonic motion (SHM), displacement refers to the distance of the oscillating body from its central or equilibrium position at any given moment in time. The displacement changes cyclically as the object moves back and forth, and it's typically described by a sine or cosine function involving time, as seen in our example.
In our exercise, the equation of displacement is provided as \( x = 6.12 \cos(8.38t + 1.92) \). By substituting the time \( t = 1.90 \, \text{s} \) into this equation, we can find the exact position of the body at that specific moment. This calculation is an essential step in understanding SHM as it sets the foundation for finding other important aspects such as velocity and acceleration.
Velocity
Velocity in SHM represents the rate at which the displacement is changing with respect to time and is always directed towards the equilibrium position. Since displacement in SHM is a sine or cosine function, the velocity is the first derivative of displacement with respect to time.
For our exercise, we can calculate the velocity by differentiating our displacement equation to get \( v = -wA \sin(wt + \phi) \). By plugging in the values for angular frequency \( w \), amplitude \( A \), phase angle \( \phi \), and time \( t \), we would derive the body's velocity at a specific time. This calculation helps us understand how fast the body is moving and in which direction during its oscillation cycle.
Acceleration
Acceleration in SHM is the rate of change of velocity with time and is proportional to the displacement but in the opposite direction. This means that the acceleration is greatest when the body is at its furthest from equilibrium and zero when it passes through equilibrium. In SHM, the acceleration can be found by taking the second derivative of the displacement equation or the derivative of the velocity equation.
According to our given problem, the acceleration at any time can be calculated by \( a = -w^2 A \cos(wt + \phi) \). Substituting the amplitude \( A \), angular frequency \( w \), phase angle \( \phi \), and the specific time into the equation, we find the instantaneous acceleration of the oscillating body.
Angular Frequency
Angular frequency, often denoted by \( w \) or \( \omega \), is a measure of how quickly an object oscillates in terms of the angular movement per unit time. It's related to the frequency, which tells us how many cycles per second the motion completes. For granular clarity, if we think of SHM as circular motion projected onto a single dimension, angular frequency would represent the 'speed' of rotation in radians per second.
The given equation tells us that the angular frequency for our oscillating body is \( 8.38 \,\text{rad/s} \). This value is highly relevant as it helps to calculate not just displacement, but also plays a central role in finding out velocity, acceleration, frequency, and period of the motion.
Period of Motion
The period of motion, denoted as \( T \), is the duration of time it takes for one complete cycle of SHM to occur. In other words, it's the time for the body to return to the same position in its motion path. The period is inversely proportional to the frequency, which is the number of cycles per second.
To compute the period in our exercise, we first find the frequency using the angular frequency \( f = w / (2π) \), and then calculate the period by taking the reciprocal of the frequency \( T = 1 / f \). This value tells us how long it takes for the oscillation to repeat itself, a fundamental concept in understanding the timing of SHM cycles.
Amplitude
Amplitude in SHM refers to the maximum displacement from the equilibrium position. It determines the range or extent of the oscillation and is a measure of how far the oscillating body moves from its central position. The amplitude remains constant over time in ideal SHM, as energy is conserved throughout the motion.
The amplitude can be easily extracted from the standard SHM equation \( x = A \cos(wt + \phi) \) as the coefficient of the cosine function. In our case, the amplitude of the body's motion is given as \( 6.12 \,\text{m} \), and it depicts the maximum displacement the body achieves during its oscillation.
Phase Angle
The phase angle in SHM, typically denoted by \( \phi \), represents the initial angle at the start of the motion if we were to visualize SHM as a projection of uniform circular motion. It determines the starting point of the oscillation in its cycle.
The phase angle is an integral part of the displacement equation, and it can alter where the body begins its motion in the oscillatory path. Based on our problem, the phase angle is \( 1.92 \,\text{rad} \), which signifies where the body is positioned in its SHM cycle at time \( t = 0 \). Understanding the phase angle is important as it helps us predict the state of motion at any given time by using the complete displacement equation.
