Question

A 15 -centimeter pendulum moves according to the equation \(\theta=0.2 \cos 8 t,\) where \(\theta\) is the angular displacement from the vertical in radians and \(t\) is the time in seconds. Determine the maximum angular displacement and the rate of change of \(\theta\) when \(t=3\) seconds.

Short answer

The maximum angular displacement is \(0.2\) radians and the rate of change at \(t=3\) is \(-0.2 * 8 * \sin(24)\) radians per second.

Step-by-step solution

Determine the Maximum Angular Displacement

The maximum angular displacement corresponds to the amplitude of the cosine function. The amplitude can be identified as the coefficient of the cosine function in the equation. Therefore, the maximum angular displacement is \(0.2\) radians.

Compute the Derivative of the Function

The rate of change of \(\theta\) at \(t=3\) seconds requires the computation of the derivative of the function with respect to time \(t\). Therefore, the derivative \(\frac{d\theta}{dt} = -0.2 * 8 * \sin(8t)\).

Evaluate the Derivative at \(t=3\) seconds

To get the rate of change at \(t=3\) seconds, substitute \(t=3\) into the expression from step 2: \(-0.2 * 8 * \sin(8*3) = -0.2 * 8 * \sin(24)\). Compute this value to get the rate of change at \(t=3\) seconds.

Key concepts

Harmonic Motion

Harmonic motion refers to a type of movement where the restoring force is directly proportional to the displacement and acts in the opposite direction. This kind of motion is characteristically repetitive and can be described by sinusoidal functions, such as sine and cosine.

In the context of the exercise, the pendulum exhibits harmonic motion. It swings back and forth, with its position at any time described by the cosine function \( \theta=0.2 \cos 8t \). This equation illustrates that the angular displacement varies over time in a smooth, wave-like pattern — one of the hallmarks of harmonic motion. The amplitude, or maximum displacement, is constant at 0.2 radians, indicating a predictable and steady range of motion for the pendulum.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They have pivotal roles in various areas of science and engineering, particularly in describing periodic phenomena.

In our exercise, the cosine function is instrumental in defining the pendulum's displacement. The general form of a cosine function is \( A \cos(Bt+C) + D \), where A is the amplitude, B affects the period of the wave, C is the phase shift, and D is the vertical shift. For the pendulum's motion described by \( \theta=0.2 \cos 8t \), there's no phase or vertical shift, and the frequency is determined by the coefficient 8 in the argument of the cosine function.

Derivatives

Derivatives in mathematics signify the rate at which a function is changing at any given point and are foundational in calculus. They are highly useful in understanding motion, among other things, since they provide instantaneous rates of change, akin to finding the velocity of an object when given its position over time.

In the provided exercise, we find the derivative of the angular displacement, \( \theta \), with respect to time, \( t \), to determine how fast the pendulum's angle is changing at a specific moment. This derivative is computed as \( \frac{d\theta}{dt} = -0.2 \cdot 8 \cdot \sin(8t) \), and tells us the angular velocity of the pendulum at any time \( t \). Evaluating this derivative at \( t=3 \), gives us the specific rate of change of the angle at that exact moment in time.

Cosine Function

The cosine function is one of the primary trigonometric functions, depicting the relationship between the angle of a right-angled triangle and the ratio of the adjacent side to the hypotenuse. In the context of periodic motion, such as a pendulum's swing, the cosine function can model periodic behavior over time with a starting position of maximum displacement.

Within our exercise, \( \theta=0.2 \cos 8t \) represents the angular displacement of the pendulum using the cosine function. The maximum value of the cosine function is 1, and so the maximum angular displacement of the pendulum is the amplitude, 0.2 radians. This demonstrates the ease with which trigonometric functions can model physical phenomena with simple and elegant expressions.