Simple Harmonic Motion Acceleration Calculation

Let's determine the acceleration of a mass undergoing Simple Harmonic Motion (SHM) based on the given parameters: maximum displacement, time period, and initial conditions. Simple Harmonic Motion is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

Understanding Key Parameters

• Mass (m): 2 kg. While important for force and energy calculations, it's not directly needed to find the acceleration equation itself, which depends on displacement and time.
• Maximum Displacement (Amplitude, A): 20 mm. This needs to be converted to meters for SI consistency: \(A = 20 \text{ mm} = 0.02 \text{ m}\).
• Time Period (T): 1.57 s. This is the time taken for one complete oscillation.
• Initial Condition: At time \(t = 0\), the displacement is 0. This condition helps us determine the phase constant of the motion.

Angular Frequency Determination

The angular frequency (\(\omega\)) is a crucial parameter in Simple Harmonic Motion, relating the time period to the oscillatory speed. It is calculated using the formula:

\[ \omega = \frac{2\pi}{T} \]

Given \(T = 1.57 \text{ s}\) and taking \(\pi \approx 3.14\):

\[ \omega = \frac{2 \times 3.14}{1.57} = \frac{6.28}{1.57} = 4 \text{ rad/s} \]

Displacement Equation Formulation

The general equation for displacement in Simple Harmonic Motion can be written as \(x(t) = A \sin(\omega t + \phi)\) or \(x(t) = A \cos(\omega t + \phi)\), where \(\phi\) is the phase constant.

We are given that at \(t = 0\), the displacement \(x = 0\). Let's use the sine function form:

\[ x(t) = A \sin(\omega t + \phi) \]

Substituting \(x = 0\) and \(t = 0\):

\[ 0 = A \sin(\omega \times 0 + \phi) \]

\[ 0 = A \sin(\phi) \]

Since the amplitude \(A\) cannot be zero, it must be that \(\sin(\phi) = 0\). This implies \(\phi = 0\) (or \(\phi = \pi\)). For the simplest representation starting from equilibrium, we choose \(\phi = 0\).

Therefore, the displacement equation becomes:

\[ x(t) = A \sin(\omega t) \]

Substituting the amplitude \(A = 0.02 \text{ m}\) and angular frequency \(\omega = 4 \text{ rad/s}\):

\[ x(t) = 0.02 \sin(4t) \text{ m} \]

Acceleration Equation Derivation

Acceleration in Simple Harmonic Motion is the second derivative of displacement with respect to time.

First, let's find the velocity \(v(t)\) by differentiating the displacement \(x(t)\) with respect to time:

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt} [A \sin(\omega t)] \]

\[ v(t) = A \omega \cos(\omega t) \]

Now, let's find the acceleration \(a(t)\) by differentiating the velocity \(v(t)\) with respect to time:

\[ a(t) = \frac{dv}{dt} = \frac{d}{dt} [A \omega \cos(\omega t)] \]

\[ a(t) = A \omega (-\omega \sin(\omega t)) \]

\[ a(t) = -A \omega^2 \sin(\omega t) \]

This is the general formula for acceleration in SHM when the displacement starts from equilibrium (sine function).

Substitute the values for \(A\) and \(\omega\):

\[ a(t) = -(0.02 \text{ m}) \times (4 \text{ rad/s})^2 \times \sin(4t) \]

\[ a(t) = -(0.02) \times (16) \times \sin(4t) \]

\[ a(t) = -0.32 \sin(4t) \text{ m/s}^2 \]

Summary of Results

The detailed calculations for the Simple Harmonic Motion parameters lead to the following results:

Thus, the acceleration of the oscillating mass at time \(t\) is given by \( -0.32 \sin(4t) \).