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) Positions: \((1.88, 0.62)\), \((-1.88, -0.62)\), \((0, 2)\), start: \((0, 2)\). (b) Swap sine and cosine terms. (c) At \(t = 2.5\) seconds.

Step 1: Understanding Parametric Circular Motion

The parametric equations \( x(t) = r \sin(\omega t) \) and \( y(t) = r \cos(\omega t) \) model circular motion, where \( r \) is the radius and \( \omega \) is the angular frequency. If the wheel has a 4-foot diameter, then its radius \( r = 2 \) feet. For one complete clockwise rotation in 10 seconds, \( \omega = \frac{2 \pi}{10} = \frac{\pi}{5} \).

Step 2: Matching the Model

To model the wheel's motion, use \( x(t) = 2 \sin\left(\frac{\pi t}{5}\right) \) and \( y(t) = 2 \cos\left(\frac{\pi t}{5}\right) \). These equations correctly incorporate the radius and angular frequency for clockwise motion, matching the given conditions.

Step 3: Calculating Positions at Specific Times

Plug in the values of \( t \) into the parametric equations: - For \( t = 2: x(2) = 2 \sin\left(\frac{2\pi}{5}\right), y(2) = 2 \cos\left(\frac{2\pi}{5}\right) \).- For \( t=6: x(6) = 2 \sin\left(\frac{6\pi}{5}\right), y(6) = 2 \cos\left(\frac{6\pi}{5}\right) \).- For \( t=10: x(10) = 2 \sin\left(2\pi\right) = 0, y(10) = 2 \cos\left(2\pi\right) = 2 \).- For \( t=0 \) (starting point): \( x(0) = 2 \sin(0) = 0, y(0) = 2 \cos(0) = 2 \).

Step 4: Adjusting for Counterclockwise Rotation

For counterclockwise rotation, swap \( \sin \) and \( \cos \) in the equations to \( x(t) = 2 \cos\left(\frac{\pi t}{5}\right) \) and \( y(t) = 2 \sin\left(\frac{\pi t}{5}\right) \). This change reflects reversing the direction of rotation.

Step 5: Finding When the Point is at (2,0)

Solve \(x(t) = 2\) and \(y(t) = 0\) using the equations for the clockwise rotation:- \(2 \sin\left(\frac{\pi t}{5}\right) = 2\) gives \(\sin\left(\frac{\pi t}{5}\right) = 1\).- This occurs at \(\frac{\pi t}{5} = \frac{\pi}{2}\), so \(t = \frac{5}{2} = 2.5 \text{ seconds}\).- Check \(y(t): 2 \cos\left(\frac{\pi t}{5}\right) = 0\) confirms this condition is met at the same time.

Key Concepts

Circular Motion

Circular motion occurs when an object moves along a circular path. This can be seen in many everyday examples, like a ferris wheel, a merry-go-round, or a planet orbiting the sun. In the context of the given problem, circular motion is described using parametric equations. These equations model the object’s position on a plane over time.
In a parametric representation, two equations are used to define the horizontal (\( x(t) \)) and vertical (\( y(t) \)) positions of a point. For circular motion, these equations are expressed as \(x(t) = r\sin(\omega t)\) and \(y(t) = r\cos(\omega t)\), where \(r\) is the radius of the path and \(\omega\) is the angular frequency. This setup allows us to simulate the motion of the object over a complete circle by adjusting the parameter \(t\), which often represents time.
Understanding these equations helps you predict the exact position of a point in circular motion at any given time.

Angular Frequency

Angular frequency \(\omega\) is a measure of how fast something is rotating or moving around in a circle. It tells you the number of complete rotations per unit of time, similar to how you might think about speed in linear motion.
In the problem provided, you determine \(\omega\) by dividing the total angle, \(2\pi\) radians (a full circle), by the time it takes to complete one full rotation. For a wheel that makes one complete clockwise rotation in 10 seconds, \(\omega\) becomes \(\frac{2\pi}{10} = \frac{\pi}{5}\). Angular frequency is crucial in the parametric equations for circular motion because it dictates how quickly the point moves around the circle.
The role of angular frequency in the motion equations is to scale the time variable \(t\) within the sine and cosine functions, ensuring that the motion accurately reflects the speed of rotation.

Clockwise and Counterclockwise Rotation

In circular motion, understanding the direction of rotation is important. Clockwise rotation moves in the same direction as the hands of a clock, while counterclockwise is the opposite direction. The direction is determined by how you incorporate the trigonometric functions in your parametric equations.
For clockwise rotation, you follow the form \(x(t) = r \sin(\omega t)\) and \(y(t) = r \cos(\omega t)\), as seen in the problem. If the motion were counterclockwise, you would switch the functions of the \(x(t)\) and \(y(t)\) equations to \(x(t) = r \cos(\omega t)\) and \(y(t) = r \sin(\omega t)\).
This swap effectively reverses the direction, displaying how trigonometric order affects the path. Choosing the correct format ensures the modeled path accurately reflects the desired rotation direction.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, form the backbone of modeling circular motion through parametric equations. These functions smoothly transition between values as they move over their input angle, ideal for representing cyclic phenomena like circles.
Sine and cosine generate outputs ranging from -1 to 1, meaning they perfectly simulate an object's oscillation across the circle’s diameter. In parametric equations, \(r\sin(\omega t)\) controls horizontal motion, while \(r\cos(\omega t)\) dictates vertical motion.
Understanding these patterns helps you predict how changes in variables like radius, angular frequency, and direction will affect the motion's path. This knowledge is especially useful when solving for specific positions or trying to reverse the rotation, as evidenced by swapping the trigonometric functions between \(x(t)\) and \(y(t)\) for direction change.