Question

A light block hangs at rest from the end of a spring when it is pulled down \(10 \mathrm{cm}\) and released. Assume the block oscillates with an amplitude of \(10 \mathrm{cm}\) on either side of its rest position and with a period of 1.5 s. Find a function \(d(t)\) that gives the displacement of the block \(t\) seconds after it is released, where \(d(t)>0\) represents downward displacement.

Short answer

Answer: The displacement function for the block oscillating on a spring is given by \(d(t) = 10\sin\left(\frac{2\pi}{1.5}t + \frac{3\pi}{2}\right)\), where \(d(t)\) represents the displacement of the block in centimeters at any time \(t\) in seconds after it is released.

Step-by-step solution

Determine amplitude, period, and angular frequency

We know the amplitude is 10 cm and the period is 1.5 s. To find the angular frequency, we use the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the period. We have: \(\omega =\frac{2\pi}{1.5}\)

Choose phase angle

Since the block is initially pulled downward and released, the displacement function must have a phase angle of \(\phi = \frac{3\pi}{2}\) (270 degrees) to start with maximum positive displacement. Adding a phase angle of \(\frac{3\pi}{2}\) ensures the sine function starts with the maximum positive value (since the sine of \(\frac{3\pi}{2}\) is 1).

Write the displacement function

Using the given amplitude, angular frequency from step 1, and the phase angle from step 2, we can write the displacement function as: \(d(t) = 10\sin\left(\frac{2\pi}{1.5}t + \frac{3\pi}{2}\right)\) \(10 cm\) represents the maximum displacement downwards and \(-10 cm\) represents the maximum displacement upwards (swing direction). The displacement function \(d(t)\) gives the displacement of the block in centimeters at any time \(t\) in seconds after it is released.

Key concepts

Amplitude

The amplitude in oscillatory motion refers to the maximum extent of displacement from the object's rest position. It represents the farthest distance the object moves to one side of its equilibrium. In this context, the amplitude of the oscillating block is 10 cm, meaning it moves 10 cm up or down from its resting point.
The amplitude is vital because it shows how far the system can reach. It helps us understand the energetic scope of the oscillations.
This value is essential when writing equations for the oscillating motion, as it defines the peak values the function can reach.

Angular Frequency

Angular frequency (\( \omega \)) is a measure of how quickly an object rotates or oscillates through its cycle. It is given by the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of oscillation.
In our example, the block completes a full oscillation every 1.5 seconds, its period. Using the formula, we find \( \omega = \frac{2\pi}{1.5} \). This calculation tells us how rapidly the block is moving through its oscillation cycle, underpinning the motion's speed.
Angular frequency helps describe the temporal aspect of the oscillation, specifying how fast things happen in the cycle.

Displacement Function

The displacement function, denoted as \( d(t) \), provides the position of the oscillating object at any given time. It's a crucial element in describing oscillatory motion as it helps us predict where the object will be.
For our block, the displacement function is \( d(t) = 10\sin\left(\frac{2\pi}{1.5}t + \frac{3\pi}{2}\right) \). This function takes into account the amplitude, angular frequency, and phase angle, providing a comprehensive snapshot of the block's motion at any time \( t \) seconds after being released.
The displacement being positive indicates a downward motion, and negative indicates upward. The displacement function is the mathematical representation of the block's oscillatory path.

Phase Angle

The phase angle is a crucial component that shifts the oscillation function along the time axis. It determines the starting position of the oscillation cycle.
In our example, unless adjusted, the sine function wouldn't reflect the system's initial conditions. By setting the phase angle (\( \phi \)) to \( \frac{3\pi}{2} \), the displacement begins with maximum downward motion. This adjustment means that at time \( t = 0 \), the block is at its extreme lower position.
This phase adjustment allows the displacement function to line up mathematically with the system’s physical behavior, ensuring accuracy in tracking the motion from start to finish.