Question

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=7 ; \quad T=5 \pi \text { seconds } $$

Short answer

The function is \( d(t) = 7 \, \cos \left( \frac{2}{5} t \right) \).

Step-by-step solution

- Understand Simple Harmonic Motion

Simple harmonic motion can be described by a sinusoidal function such as sine or cosine. The general equation for displacement in simple harmonic motion is given by \[ d(t) = A \, \cos(\omega t + \varphi) \] where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \varphi \) is the phase constant.

- Identify the Amplitude

The amplitude is the maximum displacement from the rest position. Given that the object is pulled down a distance \( a = 7 \) units, the amplitude \( A = 7 \).

- Calculate the Angular Frequency

The angular frequency \( \omega \) is related to the period \( T \) by the formula \( \omega = \frac{2 \pi}{T} \). Given \( T = 5 \pi \) seconds, we have \[ \omega = \frac{2 \pi}{5 \pi} = \frac{2}{5} \text{ radians per second} \]

- Determine the Phase Constant

Since the object is pulled down and then released from the lowest point, the appropriate function to use is the cosine function with a phase shift. At \( t = 0 \), displacement is at maximum negative, so using the cosine function is straightforward without any additional phase constant \( \varphi = 0 \). Hence, the function can be written as: \[ d(t) = 7 \, \cos \left( \frac{2}{5} t \right) \]

- Final Function

The final function relating the displacement \( d \) of the object from its rest position after \( t \) seconds is given by: \[ d(t) = 7 \, \cos \left( \frac{2}{5} t \right) \]

Key concepts

displacement function

In Simple Harmonic Motion (SHM), the displacement function provides a way to describe the position of an object relative to its rest position over time. For SHM, this function is usually a sinusoidal function, either sine or cosine. The general form of the displacement function is:
\[ d(t) = A \, \cos(\omega t + \varphi) \] Here, \(d(t)\) represents the displacement at time \(t\), \(A\) is the amplitude, \(\omega\) the angular frequency, and \(\varphi\) the phase constant.
The function effectively maps how far the object is from its equilibrium position (rest position) at any given time, following a path determined by the characteristics of SHM.
To summarize:

• Displacement function is essential to understand the motion path.
• It is derived based on initial conditions and SHM properties.

amplitude in SHM

The amplitude \(A\) in SHM is the maximum displacement of the object from its rest (equilibrium) position. In the provided exercise, the object is pulled down a distance of 7 units from rest, making the amplitude \(A = 7\).
Amplitude is a crucial parameter as it depicts the extent of motion on either side of the rest position. Greater the amplitude, greater is the energy of the object in motion.
Remember:

• Amplitude is always a positive quantity.
• It represents the furthest distance covered from the equilibrium.

angular frequency

Angular frequency \(\omega\) determines how fast the object oscillates back and forth in SHM. It is related to the period \(T\) of oscillation by the formula: \[ \omega = \frac{2 \pi}{T} \] Given the period \(T = 5 \pi\) seconds, the angular frequency \(\omega\) is calculated as: \[ \omega = \frac{2 \pi}{5 \pi} = \frac{2}{5} \] Angular frequency is typically measured in radians per second and provides insight into the speed of oscillation.
Keep in mind:

• The period \(T\) is the time for one complete oscillation.
• Higher angular frequency means faster oscillation.

cosine function in SHM

The cosine function is commonly used in SHM to describe motion, especially when the initial displacement is at its maximum. The formula \(d(t) = A \cos(\omega t + \varphi)\) often features the cosine function due to its convenient initial conditions.
In this exercise, since the object begins from a maximum displacement, the cosine function is used without a phase shift \(\varphi = 0\). Thus, the displacement function becomes: \[ d(t) = 7 \cos \left( \frac{2}{5} t \right) \] Points to consider:

• Cosine starts at the maximum value when \(t = 0\), making it suitable for such initial conditions.
• No need for phase shift \(\varphi\) in this case.