Question

An object is in simple harmonic motion with frequency \(\omega_{0}\), with \(y(0)=y_{0}\) and \(y^{\prime}(0)=v_{0}\). Find its displacement for \(t>0 .\) Also, find the amplitude of the oscillation and give formulas for the sine and cosine of the initial phase angle.

Short answer

The displacement of the object in SHM at time t can be represented by the equation: \(y(t) = y_0\cos(\omega_0 t) + \frac{v_0}{\omega_0}\sin(\omega_0 t)\) The amplitude of the oscillation is given by: \(C = \sqrt{y_0^2 + \frac{v_0^2}{\omega_0^2}}\) The sine and cosine of the initial phase angle can be expressed as: \(\sin\phi = -\frac{v_0}{\omega_0 C}\) and \(\cos\phi = \frac{y_0}{C}\)

Step-by-step solution

Write the Differential Equation for Simple Harmonic Motion

In order to find the displacement of the object, we need to solve the differential equation governing simple harmonic motion. This equation can be written as: \begin{equation} y''(t) + \omega_0^2 y(t) = 0 \end{equation} This is a second-order linear homogeneous differential equation.

Solve the Differential Equation Given the Initial Conditions

The general solution for this differential equation can be written as: \begin{equation} y(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) \end{equation} where \(A\) and \(B\) are constants to be determined from the initial conditions. The initial conditions are: \begin{equation} y(0) = y_0 \quad \text{and} \quad y'(0) = v_0 \end{equation} First, let's find the derivative of \(y(t)\): \begin{equation} y'(t) = -\omega_0 A\sin(\omega_0 t) + \omega_0 B\cos(\omega_0 t) \end{equation} Now, we can apply the initial conditions: \begin{equation} y(0) = A\cos(0) + B\sin(0) \Rightarrow y_0 = A, \\ y'(0) = -\omega_0 A\sin(0) + \omega_0 B\cos(0) \Rightarrow v_0 = \omega_0 B \end{equation} Thus, the solution for the displacement is: \begin{equation} y(t) = y_0\cos(\omega_0 t) + \frac{v_0}{\omega_0}\sin(\omega_0 t) \end{equation}

Calculate the Amplitude and Find the Formulas for the Sine and Cosine of the Initial Phase Angle

In order to find the amplitude and initial phase angle of the oscillation, we express the displacement \(y(t)\) in the form: \begin{equation} y(t) = C\cos(\omega_0 t - \phi) \end{equation} where \(C\) is the amplitude and \(\phi\) is the initial phase angle. To find the amplitude \(C\), we use the trigonometric identity: \begin{equation} \cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta \end{equation} Comparing the expressions for \(y(t)\), we get: \begin{equation} y_0\cos(\omega_0 t) + \frac{v_0}{\omega_0}\sin(\omega_0 t) = C\cos(\omega_0 t - \phi) \end{equation} To find \(C\), we look at the squares of the coefficients: \begin{equation} C^2 = y_0^2 + \frac{v_0^2}{\omega_0^2} \end{equation} Thus, the amplitude of the oscillation is given by: \begin{equation} C = \sqrt{y_0^2 + \frac{v_0^2}{\omega_0^2}} \end{equation} To find the sine and cosine of the initial phase angle, we can write the displacement for \(t=0\): \begin{equation} y(0) = y_0\cos(0) + \frac{v_0}{\omega_0}\sin(0) = C\cos(-\phi) \Rightarrow \cos\phi = \frac{y_0}{C}, \end{equation} and for \(t'=0\): \begin{equation} y'(0) = -\omega_0 y_0\sin(0) + v_0\cos(0) = C\omega_0\sin(-\phi) \Rightarrow \sin\phi = -\frac{v_0}{\omega_0 C} \end{equation} Thus, the formulas for the sine and cosine of the initial phase angle are: \begin{equation} \sin\phi = -\frac{v_0}{\omega_0 C}, \quad \cos\phi = \frac{y_0}{C} \end{equation}

Key concepts

Differential Equations

In physics and mathematics, differential equations are fundamental for modeling how systems evolve over time. The function y(t) describes the position of an object at time t, and its derivatives represent the object's velocity and acceleration. As seen in the exercise, simple harmonic motion is represented by the second-order differential equation \[\begin{equation} y''(t) + \omega_0^2 y(t) = 0 \text\end{equation}\] which showcases a direct relationship between the second derivative of displacement and the displacement itself. Here, \( \omega_0 \) is the angular frequency, defining how rapid the oscillation is. A key aspect is the linear and homogeneous nature of this equation, implying that solutions can be superimposed to form new solutions. This property is crucial when solving for constants using initial conditions.

Initial Conditions

To fully solve a differential equation for a specific scenario, we must consider initial conditions, which are the known values of the function (and its derivatives) at the start of observation. For instance, in simple harmonic motion, the initial position \( y(0) \) and initial velocity \( y'(0) \) are critical to finding a unique solution. These initial values anchor the general solution to a particular physical context, allowing us to express the constants in the solution in terms of known quantities, such as \( y_0 \) for initial position and \( v_0 \) for initial velocity. By applying these conditions, one can personalize the solution to reflect the behavior of the oscillating object from the onset of motion.

Amplitude of Oscillation

The amplitude of oscillation represents the maximum displacement of an object from its equilibrium position in simple harmonic motion. It's a measure of how 'large' the oscillations are. After solving the differential equation with initial conditions, the amplitude, denoted as \( C \), is the positive square root of the sum of the squares of the initial position and the initial velocity (scaled by the angular frequency). Mathematically, this is expressed as \[\begin{equation} C = \sqrt{y_0^2 + \left(\frac{v_0}{\omega_0}\right)^2} \text\end{equation}\] The amplitude, a crucial parameter, determines the energy stored in the oscillating system, with greater amplitudes implying more energy.

Phase Angle

The phase angle, often denoted as \( \phi \), is a measure that indicates the initial state of the oscillation. Essentially, it describes how much the function's wave pattern is shifted horizontally from a standard cosine wave. To find the phase angle of a simple harmonic oscillator, we use the trigonometric identities to relate the initial conditions to the form \[\begin{equation} y(t) = C\cos(\omega_0 t - \phi) \text\end{equation}\] To extract the phase angle, we manipulate the solution to get expressions for \( \sin(\phi) \) and \( \cos(\phi) \) in terms of the initial displacement and initial velocity. This reflective process reveals how the initial conditions shape not just the amplitude but also the starting phase of the oscillating object, a concept that's vital in understanding wave phenomena and vibrations.