Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Harmonic Motion A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by \(y=\frac{1}{4}(\cos 8 t-3 \sin 8 t)\) where \(y\) is the displacement (in meters) and \(t\) is the time (in seconds). Find the times at which the weight is at the point of equilibrium \((y=0)\) for \(0 \leq t \leq 1\)

Step-by-Step Solution

Answer

The times at which the weight is at the point of equilibrium for \(0 \leq t \leq 1\) are \(t = \frac{1}{8} \tan^{-1}(3)\) and \(t = \frac{1}{8} \tan^{-1}(3) + \frac{\pi}{8}\).

Step 1: Identify the Equation

The statement provides an equation of motion: \(y= \frac{1}{4}(\cos 8t - 3\sin 8t)\), and we are to find the time \(t\) when the displacement \(y=0\). It means we set \(y\) to zero and solve for \(t\).

Step 2: Set y to zero

Setting \(y = 0\) we get the equation 0 = \frac{1}{4}(\cos 8t - 3\sin 8t)

Step 3: Simplify the Equation

Multiplying the equation by 4, to eliminate the fraction, we get 0 = \cos 8t - 3\sin 8t

Step 4: Solve for t

The equation 0 = \cos 8t - 3\sin 8t is equivalent to \cos 8t = 3\sin 8t. Dividing both sides by \cos 8t, we obtain \tan 8t = 3. Now, \(t = \frac{1}{8} \tan^{-1}(3) + \frac{n\pi}{8}\), for \(n\) being an integer. Therefore, within the interval \(0 \leq t \leq 1\), there are two solutions for \(t\): \(t = \frac{1}{8} \tan^{-1}(3)\) and \(t = \frac{1}{8} \tan^{-1}(3) + \frac{\pi}{8}\).

Key Concepts

Equation of Motion in Harmonic Systems

In physics, the equation of motion for a system describes how the position of that system changes in time. In the context of harmonic motion, such as a weight oscillating on a spring, the motion is periodic and can be described by trigonometric functions. The general form for the vertical position, y, in harmonic motion can be represented as a combination of sine and cosine functions:

\[\begin{equation} y(t) = A \times \text{cos}( \beta t + \theta ) \tag{1}ewline \end{equation}\]Here, A is the amplitude, β is the angular frequency (twice the number of oscillations per unit time), and θ is the phase angle at t = 0. The equilibrium point is the position at which the net force on the oscillating object is zero, typically at y = 0. Solving for the time t when the weight reaches this equilibrium point involves setting equation (1) to zero and solving for t. This task not only reflects an understanding of the physical system but also requires a grasp of trigonometry and algebra to find the solution.

Trigonometric Identities and Their Role in Physics

A trigonometric identity is an equation that is true for all values within its domain. These identities are crucial in simplifying expressions and solving equations that involve trigonometric functions. Common identities include the Pythagorean identities, sum and difference formulas, double angle formulas, and others.

In the context of the harmonic motion equation:\[\begin{equation} y = A(\text{cos}( \beta t) - B\text{sin}( \beta t)) \tag{2}ewline \end{equation}\]where A and B are constants, a student can use trigonometric identities to combine the cosine and sine terms into a single trigonometric function. This simplification can make it easier to solve for t when y = 0, which corresponds to the equilibrium point. Knowledge and proper application of these identities enable the student to solve complex equations and understand the principles governing harmonic motion.

Solving Trigonometric Equations

The process of solving trigonometric equations often involves isolating the trigonometric function and then using inverse trigonometric functions to find the angle or time t. When we had the equation 0 = \text{cos}( 8t ) - 3\text{sin}( 8t ), we arranged it to isolate a single trigonometric function:\[\begin{equation} \text{tan}( 8t ) = 3 \tag{3}ewline \end{equation}\]Then, using the inverse tangent function, tan^-^1, we can find the specific times at which the weight passes through the equilibrium. There's a catch, though: trigonometric functions are periodic, meaning they repeat their values at regular intervals. Because of this, there are potentially many solutions to equation (3). To find the specific solutions within a given time interval, such as 0 ≤ t ≤ 1, we apply constraints to the range of acceptable solutions, usually resulting from the physical context of the problem such as the limits of one complete oscillation. Thus, understanding how to solve these equations and apply proper constraints is fundamental for analyzing harmonic motion.