91Ó°ÊÓ

Trigonometric functions, such as the sine and cosine functions, play a crucial role in describing harmonic motion. Harmonic motion often entails oscillations that can be effectively modeled using these functions due to their repetitive wave-like properties. In this exercise, the displacement is given as a combination of cosine and sine terms, specifically \( y = \frac{1}{3} \cos 12t - \frac{1}{4} \sin 12t \). Trigonometric identities help in simplifying expressions and calculating precise values for specific points in time.

• The cosine function represents the horizontal component of the oscillation.
• The sine function represents the vertical component of the oscillation.
• Each function is scaled (by \( \frac{1}{3} \) and \( \frac{1}{4} \) respectively) to adjust the amplitude, meaning how high or low the wave peaks are.
• The frequency is affected by the argument \(12t\), which determines how often the oscillation occurs.

In our context, when substituting \( t = \pi/8 \), we compute specific instances of cosine and sine to determine the precise position of the object in its oscillatory path.

Differentiation is a key concept in calculus. It helps in finding rates of change, such as velocity, from position functions. In harmonic motion, to find the object's velocity, we differentiate the displacement function concerning time \(t\).
In this exercise, the displacement function is:
\[ y = \frac{1}{3} \cos 12t - \frac{1}{4} \sin 12t \]
The velocity is given by its derivative with respect to \(t\):
\[ v(t) = -\frac{1}{3} \cdot 12 \sin 12t - \frac{1}{4} \cdot 12 \cos 12t \]

• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The chain rule is applied since the functions have \(12t\) as their argument. The derivative of \( t \) with respect to itself is \( 1 \).

This operation transforms the position function into a velocity function, describing how fast the position changes at any point in time.

Velocity describes how quickly the position of an object changes over time. In the context of harmonic motion, it's derived from the displacement or position function. To determine this velocity specifically at \( t = \pi/8 \), we substitute this value into our previously derived velocity function:
The velocity function is:
\[ v(t) = -\frac{1}{3} \cdot 12 \sin 12t - \frac{1}{4} \cdot 12 \cos 12t \]
Substitute \( t = \pi/8 \):
\[ v(\pi/8) = -\frac{1}{3} \cdot 12 \sin(12 \cdot \pi/8) - \frac{1}{4} \cdot 12 \cos(12 \cdot \pi/8) \]

• Compute the values of \( \sin(\pi/2) \) and \( \cos(\pi/2) \), since \( 12\cdot\pi/8 = \pi/2 \).
• Utilize the knowledge that \( \sin(\pi/2) = 1 \) and \( \cos(\pi/2) = 0 \) to simplify the expression.
• Combine and simplify further to reach the result.

Using these calculations, we find out how fast the object in harmonic motion moves when \( t = \pi/8 \). Understanding the velocity helps us predict future movements and analyze the motion's energy dynamics.