Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Circular motion can be modeled by using the parametric representations of the form \(x(t)=\sin t\) and \(y(t)=\cos t\) (A parametric representation means that a variable, \(t\) in this case, determines both \(x(t)\) and \(y(t) .)\) This will give the full circle for \(0 \leq t \leq 2 \pi .\) If we consider a 4 -foot- diameter wheel making one complete rotation clockwise once every 10 seconds, show that the motion of a point on the rim of the wheel can be represented by \(x(t)=2 \sin (\pi t / 5)\) and \(y(t)=2 \cos (\pi t / 5)\) (a) Find the positions of the point on the rim of the wheel when \(t=2\) seconds, 6 seconds, and 10 seconds. Where was this point when the wheel started to rotate at \(t=0\) ? (b) How will the formulas giving the motion of the point change if the wheel is rotating counterclockwise. (c) At what value of \(t\) is the point at \((2,0)\) for the first time?

Step-by-Step Solution

Answer

(a) (0, 2) at t=0, calculate for t=2, 6; at t=10 again (0, 2). (b) Swap x and y to x=2cos(πt/5), y=2sin(πt/5). (c) t=2.5 seconds.

Step 1: Understanding the parametric equations

The parametric equations for the motion of the point on the rim are given as \(x(t)=2 \sin(\pi t / 5)\) and \(y(t)=2 \cos(\pi t / 5)\). These equations describe a circular motion with a circle of radius 2 (since diameter is 4, radius = diameter/2 = 2). The factor \(\pi t / 5\) in the trigonometric functions accounts for the wheel completing a full rotation (\(2\pi\)) in 10 seconds.

Step 2: Finding positions of the point at specified times

For \(t = 0\):- \( x(0) = 2 \sin(\pi \times 0 / 5) = 0 \)- \( y(0) = 2 \cos(\pi \times 0 / 5) = 2 \)For \(t = 2\):- \( x(2) = 2 \sin(\pi \times 2 / 5) \)- \( y(2) = 2 \cos(\pi \times 2 / 5) \)Use a calculator to find these values.For \(t = 6\):- \( x(6) = 2 \sin(\pi \times 6 / 5) \)- \( y(6) = 2 \cos(\pi \times 6 / 5) \)Again, use a calculator.For \(t = 10\):- \( x(10) = 2 \sin(\pi \times 10 / 5) = 0 \)- \( y(10) = 2 \cos(\pi \times 10 / 5) = 2 \)

Step 3: Adjusting for counterclockwise rotation

To convert the parametric equations to represent counterclockwise rotation, swap the sine and cosine functions to have:- \(x(t) = 2 \cos(\pi t / 5)\)- \(y(t) = 2 \sin(\pi t / 5)\)

Step 4: Finding the first occurrence of the point at (2,0)

We need \(x(t) = 2\) and \(y(t) = 0\).- For \(x(t) = 2\), \(2 \sin(\pi t / 5) = 2\), gives \(\sin(\pi t / 5) = 1\), implying \(\pi t / 5 = \pi / 2\), so \(t = 2.5\) seconds.- Check \(y(t)\): \(y(2.5) = 2 \cos(\pi \times 2.5 / 5) = 2 \cos(\pi / 2) = 0\). Thus, the conditions \(x(t)=2\) and \(y(t)=0\) are satisfied at \(t=2.5\) seconds.

Key Concepts

circular motion

Circular motion is a movement of an object along the circumference of a circle. In the context of parametric equations, circular motion is modeled using equations that describe the position of the point along the circle over time. This motion is often uniform, meaning the object moves at a constant speed. However, circular motion can also vary in speed, depending on the forces involved.
Every point on the circumference of a wheel in circular motion traces out a circular path. For example, if a wheel rotates with a consistent speed, the time it takes to complete one full circle will remain constant in each rotation. Consider a scenario where a 4-foot diameter wheel completes one full rotation every 10 seconds. Each point on the rim of the wheel then follows the same path repeatedly over equal intervals of time.
Understanding circular motion is crucial, as it forms the basis of many physical phenomena and is used in various applications from engineering to everyday life.

parametric representation

Parametric representation allows us to define a set of related quantities as functions of an independent parameter. In our circular motion problem, we use parametric equations to represent the position of a point on a rotating wheel.
The parametric equations used here are:

• \(x(t) = 2 \sin(\frac{\pi t}{5})\)
• \(y(t) = 2 \cos(\frac{\pi t}{5})\)

These equations describe the position of the point on the circle in terms of the parameter \(t\), which represents time. As time progresses, the values of \(x\) and \(y\) change according to the sine and cosine functions, effectively giving a complete representation of the circular path.
Parametric equations are particularly useful in cases where a single function cannot effectively describe the behavior of the system, providing a clear description of the trajectory as a whole.

trigonometric functions

Trigonometric functions are the mathematical functions based on the ratios of the sides of a right triangle. They are widely used in modeling periodic phenomena, such as waves and circular motions. The most common trigonometric functions are sine, cosine, and tangent.
In our scenario, the sine and cosine functions are used in the parametric equations to model the circular motion of a point on a wheel:

• \( x(t) = 2 \sin(\frac{\pi t}{5}) \)
• \( y(t) = 2 \cos(\frac{\pi t}{5}) \)

The sine function gives the horizontal component \(x(t)\), and the cosine function provides the vertical component \(y(t)\). This combination ensures the point moves in a perfect circular path.
These functions are periodically repeating over intervals, thus accurately representing the cyclical nature of circular motion. Understanding these concepts can help decode many types of oscillatory movements in both theoretical and practical applications.

counterclockwise rotation

Counterclockwise rotation, as it pertains to circles, means the object rotates in the opposite direction of a clock's hands. To model a counterclockwise rotation using parametric equations, we modify the trigonometric functions used to describe the motion.
In the example of the wheel, the initial parametric equations for clockwise rotation are:

• \(x(t) = 2 \sin(\frac{\pi t}{5})\)
• \(y(t) = 2 \cos(\frac{\pi t}{5})\)

To represent counterclockwise rotation, simply swap the sine and cosine to achieve:

• \( x(t) = 2 \cos(\frac{\pi t}{5}) \)
• \( y(t) = 2 \sin(\frac{\pi t}{5}) \)

This adjustment effectively changes the direction of rotation, by altering the phase shift; the sine and cosine functions dictate the direction of the point traveling along its circular path. Understanding these modifications can help recognize how clockwise and counterclockwise motions are represented differently, which is crucial in many areas of physics and engineering.