Problem 71 - Algebra and Trigonometry with Analytic Geometry

Question

A point \(P\) in simple harmonic motion has a period of \(3 \mathrm{sec}-\) onds and an amplitude of 5 centimeters. Express the motion of \(P\) by means of an equation of the form \(d=a \cos \omega t\).

Step-by-Step Solution

Answer

The equation is \(d = 5 \cos \left(\frac{2\pi}{3} t\right)\).

Step 1: Identify Given Information

We have a point in simple harmonic motion with a period of 3 seconds and an amplitude of 5 cm. We also need to express this motion with the form \(d = a \cos \omega t\).

Step 2: Determine the Amplitude

The amplitude \(a\) is given directly in the problem as 5 cm.

Step 3: Calculate the Angular Frequency \(\omega\)

The period \(T\) is given as 3 seconds. The formula for angular frequency is \(\omega = \frac{2\pi}{T}\). Substituting the known value, we get:\[ \omega = \frac{2\pi}{3} \text{ radians per second} \]

Step 4: Write the Equation of Motion

Substitute the amplitude \(a = 5\) and angular frequency \(\omega = \frac{2\pi}{3}\) into the equation form \(d = a \cos \omega t\). Thus, the equation becomes:\[ d = 5 \cos \left(\frac{2\pi}{3} t\right) \]

Key Concepts

Angular Frequency

In simple harmonic motion, angular frequency plays a vital role in describing how quickly the motion occurs. Angular frequency, typically represented by the Greek letter \(\omega\), is a measure of how many cycles a system completes in a unit of time.

It is specifically calculated by the formula:

• \(\omega = \frac{2\pi}{T}\)
• \(T\) is the period of motion, which is the time it takes to complete one full cycle.

In our exercise, the period \(T\) was provided as 3 seconds. By substituting this into the formula, we calculated the angular frequency as \(\omega = \frac{2\pi}{3}\) radians per second.

Understanding angular frequency helps us predict how quickly the position of a point in simple harmonic motion will change over time. This concept is fundamental in fields like physics and engineering, where wave properties and oscillations are analyzed.

Amplitude

Amplitude in simple harmonic motion refers to the maximum displacement of the object from its equilibrium position.

In simpler terms, it measures how far the object moves from its center point as it oscillates back and forth. Amplitude is usually denoted by the letter \(a\), and it is a determining factor in how 'strong' or 'intense' the motion appears.

In our given exercise, the amplitude was straightforwardly provided as 5 cm. This value indicates that the farthest point from the equilibrium position that the point \(P\) can reach is 5 centimeters. An amplitude can help characterize the energy of the motion, as larger amplitudes often imply more energetic motion. Recognizing amplitude is crucial in understanding vibrations and waves, as it directly relates to characteristics such as volume in sound waves or brightness in light waves.

Period of Motion

The period of motion is the duration required for an object in simple harmonic motion to complete one full cycle of its path.

Represented by the letter \(T\), the period tells us the time it takes for the repeating motion to return to its initial position.

In our case, the period is 3 seconds, which means every three seconds, the point \(P\) completes a full oscillation. Understanding the period is essential for recognizing patterns in repetitive motions, such as the swinging of a pendulum or oscillations in electrical circuits.

Having knowledge of the period along with the angular frequency and amplitude, provides a comprehensive understanding of the dynamics of simple harmonic motion. It helps us calculate other parameters, such as frequency, and analyze periodic behaviors in natural and engineered systems.