Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. What is the amplitude of the simple harmonic motion described by \(d=3 \sin \frac{\pi}{2} r ?\)

Step-by-Step Solution

Answer

The amplitude of the simple harmonic motion is 3.

Step 1: Identify the amplitude

For simple harmonic motion of the form \(d = A \sin(\omega t)\), the amplitude is \(|A|\).

Step 2: State the answer

In \(d = 3\sin\frac{\pi}{2}t\), the amplitude is \(|3| = 3\).

Key Concepts

Amplitude

In the context of simple harmonic motion, amplitude plays a crucial role. It represents the maximum displacement from the equilibrium position. In simpler terms, it is how far the object moves from its rest position, either to the left or to the right. Plus, the amplitude is always positive since it measures a distance from the center point of motion.

In the function provided: \[d = 3 \sin \frac{\pi}{2} r\],the amplitude is the number directly in front of the sine function, which is 3. Here's what that means:

• The object can move up to 3 units away from the center point or equilibrium position.
• The amplitude is the peak point it can reach in either direction.
• It defines the 'size' of the oscillations.

Understanding amplitude helps you visualize the motion path of the object, whether it's a swinging pendulum or a vibrating string.

Trigonometric Functions

Trigonometric functions, like sine (\sin ), are fundamental in describing simple harmonic motion. They allow us to model waves and oscillations with precision. Let's break down how they work in our simple harmonic motion equation:

1. **Sine Function**: - The sine function oscillates between -1 and 1. - This oscillatory behavior makes it perfect for modeling waves and cyclic phenomena.

2. **Angular Frequency**: - In \(d = 3 \sin \frac{\pi}{2} r\),the expression \(\frac{\pi}{2} r\) determines how fast the oscillations occur. - This part of the equation is tied to the rate of motion, known as angular frequency.
Together, these components create a pattern of repeated motion, mimicking natural cycles like the swinging of a pendulum or the vibration of a string. By understanding these parts, you can predict how the object will move over time.

Equilibrium Position

The equilibrium position is a key concept in simple harmonic motion. It refers to the position where the object naturally rests, undisturbed by outside forces. When an object is set into motion, it oscillates around this equilibrium point.

Consider it as the center point of motion, and here's why it's important:

• The object passes through this position when it's moving fastest.
• It is the balance point, where forces are neutralized.
• Amplitude is measured from this position, showing maximum displacement.

This concept helps us visualize where energy transitions happen in motion. As the object passes through equilibrium, potential energy is minimal, and kinetic energy peaks, propelling the object onward.