Problem 225 - Calculus Volume 1

Question

An object moving in simple harmonic motion is modeled by the function \(s(t)=-6 \cos \left(\frac{\pi t}{2}\right),\) where \(s\) is measured in inches and \(t\) is measured in seconds. Determine the first time when the distance moved is 4.5 in.

Step-by-Step Solution

Answer

The first time the distance is 4.5 inches is approximately 1.54 seconds.

Step 1: Set up the equation

Substitute the given distance into the function to find the time \( t \). The equation becomes \(-6\cos\left(\frac{\pi t}{2}\right) = 4.5\).

Step 2: Simplify the equation

Divide both sides of the equation by \(-6\) to isolate the cosine term, giving \(\cos\left(\frac{\pi t}{2}\right) = -0.75\).

Step 3: Solve for angle

Find the angle \( \frac{\pi t}{2} \) such that \( \cos\left(\frac{\pi t}{2}\right) = -0.75 \). Using a calculator, the principal value is \( \cos^{-1}(-0.75)\). This gives \( \frac{\pi t}{2} = \cos^{-1}(-0.75) \approx 2.4189 \) radians.

Step 4: Solve for time

Rearrange the equation to solve for \( t \): \[ t = \frac{2}{\pi} \times 2.4189 \approx 1.54 \text{ seconds} \].

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of triangles, especially right-angled triangles. They also model periodic phenomena like waves and oscillations, which is exactly why they're handy in describing simple harmonic motion. The basic trigonometric functions include sine (\( ext{sin}\)), cosine (\( ext{cos}\)), and tangent (\( ext{tan}\)). Each of these functions has a unique graph, showing periodic behavior.

You will often encounter these functions in equations that describe motion, each having peculiar properties and periods. They're mainly used to find unknown angles or sides in triangles, especially when other dimensions are given. Understanding trigonometric functions is essential if you're dealing with waves, circular motion, or repetitive patterns.

In simple harmonic motion, as seen in the function provided in the exercise, the cosine function is used to depict the repetitive back-and-forth motion of the object. It helps determine positions at any given point in time.

Solving Trigonometric Equations

Solving trigonometric equations is a key competency when dealing with problems in physics and engineering, especially when it involves periodic motion. The approach typically involves isolating the trigonometric function and solving for the angle or time.

To do this efficiently, follow these essential steps:

• First, express the equation in a form where the trigonometric function is isolated (e.g., \( ext{cos}(x) = c\)).
• Next, utilize known values or inverse trigonometric functions to find the principal angle(s) or values that satisfy the equation.
• Determine other possible angles based on the periodicity of the trigonometric function.
• Finally, solve any additional algebraic steps to find the desired variable, such as time or distance.

By understanding these steps, you can solve similar equations and predict the behavior of oscillating systems, like in our model of simple harmonic motion.

Cosine Function

The cosine function is one of the primary trigonometric functions that describe the horizontal coordinate in the unit circle. It's expressed as \( ext{cos}(x)\), where \(x\) is the angle. In the context of periodic motion, it often represents the displacement or position over time.

The function has a period of \(2\pi\) radians, meaning it repeats every \(360^\u00b0\). For simple harmonic motion, the cosine function helps depict the smooth, back-and-forth movement, starting from the maximum point of displacement.

The important characteristics of the cosine function include:

• Amplitude: Maximum value it reaches, which is the coefficient of \( ext{cos}\) in the equation (e.g., \(-6\) in the model).
• Periodicity: The length of the interval after which the function repeats.
• Phase Shift: Which indicates horizontal shifts along the time axis.

Understanding how the cosine function works means you can effectively predict positions at any time in a cycle, crucial for modeling anything that moves back and forth like pendulums or springs.