Question

Block on a spring 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 with a period of 1.5 s. Find a trigonometric 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 trigonometric function representing the displacement of the block is \(d(t) = 10\cos\left(\frac{2\pi}{1.5}t\right)\).

Step-by-step solution

Identify the given information

We are given the following information: 1. The amplitude of oscillation is 10 cm. 2. The period of oscillation is 1.5 s. 3. The block is initially pulled down 10 cm.

Choose either sine or cosine function

Let's choose a cosine function, since at the initial point (t=0), the displacement is at its maximum (10 cm). If we were to choose a sine function, we would need to add a phase shift to achieve the desired initial displacement.

Determine the angular frequency

The angular frequency (\(\omega\)) is related to the period of oscillation (T) by: \(\omega = \frac{2\pi}{T}\) Plugging in the given period (T = 1.5 s), we get: \(\omega = \frac{2\pi}{1.5}\)

Write down the general form of the cosine function

The general form of the cosine function representing the displacement of the block (d(t)) is given by: \(d(t) = A\cos(\omega t)\) where A is the amplitude and \(\omega\) is the angular frequency.

Plug in the given values

Now we can plug in the amplitude (A = 10 cm) and the angular frequency (\(\omega = \frac{2\pi}{1.5}\)) into the general form of the cosine function: \(d(t) = 10\cos\left(\frac{2\pi}{1.5}t\right)\)

Final answer

The final trigonometric function representing the displacement of the block is: \(d(t) = 10\cos\left(\frac{2\pi}{1.5}t\right)\)

Key concepts

Amplitude of Oscillation

The amplitude of oscillation describes the maximum displacement of an object from its position of equilibrium in a periodic motion, such as a block on a spring or a pendulum. It's the distance from the center of motion to either extreme. In the example provided, the amplitude is given as 10 cm, indicating the distance the block travels above or below its rest position during each oscillation. Understanding amplitude is crucial as it helps determine the energy stored in a system; generally, the greater the amplitude, the more energy the system possesses.

For instance, in the given exercise, the block's amplitude of oscillation indicates the maximum extent it moves away from the rest position when released. This value is essential when we construct the trigonometric function to model the block's motion, as it will dictate the peak values of the function.

Angular Frequency

Angular frequency, denoted by the Greek letter \(\omega\), is a measure of how many oscillations occur in a unit of time. It is directly related to the frequency of the oscillation, which tells us how many cycles occur per second. Angular frequency, however, is measured in radians per second, providing a link between linear and rotational motion.

In our problem, we find angular frequency by taking the ratio of \(2\pi\) to the period of oscillation, as \(\omega = \frac{2\pi}{T}\). With the period given as 1.5 seconds, we calculate \(\omega\) to be \(\frac{2\pi}{1.5}\) radians per second. This measurement is vital for formulating the cosine function model of the block's motion, allowing us to know how quickly the block oscillates.

Cosine Function

The cosine function is one of the principal trigonometric functions, often used to model periodic motion, like the oscillation of a block on a spring. It starts with a maximum value at zero, which intuitively matches scenarios where an object begins at an extreme position. In trigonometric terms, the cosine function has the form \( A\cos(\omega t + \phi) \) where \(A\) is the amplitude, \(\omega\) the angular frequency, \(t\) the time, and \(\phi\) the phase shift.

For the exercise, we use the cosine function without a phase shift because the block starts at its maximum displacement. Consequently, our cosine function directly reflects the oscillation without any horizontal translation, making it simpler to apply and understand. The cosine function's characteristics align perfectly with the movement of a spring-block system at its most stretched or compressed point at time zero.

Period of Oscillation

The period of oscillation (represented by the symbol \(T\)) is the time it takes for one complete cycle of periodic motion, such as one full swing of a pendulum or one complete oscillation of a spring. In our example, the period is 1.5 seconds, meaning that every 1.5 seconds, the block completes one oscillation. The period is directly related to frequency, with frequency being the reciprocal of the period. Moreover, in the context of our trigonometric function, the period helps us determine the angular frequency which is crucial for predicting the timing of the oscillations.

Understanding the period of oscillation allows us to fathom the speed of the oscillating system and, for a student of physics, to connect the period to other properties of the system, such as mass or spring constant, especially in systems following Hooke's Law. It tells us how fast or slow a system oscillates and can be critical when comparing the dynamic behavior of different oscillating systems.