Problem 17 - Precalculus Essentials

Question

An object is attached to a coiled spring. In Exercises \(17-18,\) the object is pulled down (negative direction from the rest position and then released. In Exercises \(19-20\), the object is propelled downward from its rest position at time \(t=0 .\) Write an equation for the distance of the object from its rest position after \(t\) seconds. Distance from Rest Position at \(t=0\) 6 centimeters Amplitude 6 sentimeters Period 4 seconds

Step-by-Step Solution

Answer

The equation for the distance of the object from its rest position after \(t\) seconds is \(-6 \cos(\frac{\pi}{2}t)\) cm.

Step 1: Identify the given parameters

The given parameters are the distance from the rest position at \(t=0\) which is also the amplitude and is 6 cm, and the period of the motion which is 4 seconds.

Step 2: Formulate the formula for the motion

The standard formula for the motion of an object propelled downwards from the equilibrium position is \(-A \cos (\frac{2\pi}{T}t)\), where \(A\) is the amplitude of the motion, \(T\) is the period and \(t\) is the time in seconds.

Step 3: Substitute the given values in the formula

Substitute \(A=6\) cm and \(T=4\) seconds into the formula. The resulting function for the motion of the object is \(-6 \cos(\frac{2\pi}{4}t)\) cm.

Step 4: Simplify the equation

Simplify the equation further to get the final equation for the motion of the object: \(-6 \cos(\frac{\pi}{2}t)\) cm, where \(t\) is the time in seconds since the object was released.

Key Concepts

Amplitude

Amplitude represents the maximum distance an object moves from its rest position in harmonic motion. In our exercise, the amplitude is given as 6 centimeters. This tells us that the object moves 6 cm away from its resting point at its furthest extent.
This distance defines the scale of motion and helps us understand how far the object swings in either direction.

• Why Amplitude Matters: Amplitude directly affects how wide the oscillation is. Greater amplitudes mean larger swings.
• Real-life Examples: A pendulum clock's pendulum may have a small amplitude, while a child on a swing can have a much larger amplitude.

Period

The period of a harmonic motion describes how long it takes for an object to complete one full cycle of oscillation. In our example, the period is 4 seconds. This means that every 4 seconds, the object's motion repeats itself.
The period determines the speed of the oscillation.

• Relationship to Frequency: Frequency, which is how often cycles occur in one second, is the inverse of the period. For instance, if the period is 4 seconds, the frequency is \( \frac{1}{4} \) or 0.25 cycles per second.
• Significance in Physics: Periodic motion is a fundamental concept, applicable to wave forms, electrical circuits, and spring systems, making it critical to various fields of science.

Cosine Function

The cosine function is frequently used in modeling harmonic motion, such as the one described in our problem. In this case, the formula \( -6 \cos(\frac{\pi}{2}t) \) describes how the object's position changes over time.
The cosine function starts at its maximum value when \( t = 0 \), making it ideal for systems at their peak displacement at the start.

• Function Behavior: The cosine wave moves from maximum to minimum, then back to maximum, repeating this pattern over its period.
• Equation Components: In the equation given, the negative sign indicates direction, the amplitude (6 cm) shows maximum displacement, and \( \frac{\pi}{2} \) helps define how quickly these cycles occur, based on the period of 4 seconds.

Understanding these components of the cosine function is essential in predicting how motion evolves in time.