Problem 45 - Precalculus with Calculus Previews

Question

A blue circle of radius \(a\) rolls to the right without slipping on a horizontal line which we take to be the \(x\) -axis. The parametric equations $$ x=a t-b \sin t, y=a-b \cos t $$ describe a family of curves, called trochoids, that are traced out by a point \(P\) that is located a distance \(b\) from the center of the blue circle. Note that the equations of a cycloid, (3) in Example 7 , define a special trochoid when \(P\) lies on the circumference of the circle, that is, when \(b=a\). If \(b

Step-by-Step Solution

Answer

Plot the parametric equations for each scenario over \([0, 8\pi]\) to visualize the different trochoids.

Step 1: Identify the Type of Trochoid

Examine each set of given parameters \(a\) and \(b\) to determine whether they describe a curtate cycloid, a cycloid (when \(b = a\)), or a prolate cycloid (when \(b > a\)).

Step 2: Set Up Parametric Equations for (a)

For (a) with \(a = 2\) and \(b = 1\), use the parametric equations: \(x = 2t - \sin t\) and \(y = 2 - \cos t\). This is a curtate cycloid since \(b < a\).

Step 3: Graph the Parametric Equations for (a)

Plot the parametric equations from Step 2 using a graphing utility over the interval \([0, 8\pi]\). Observe the curtate cycloid pattern, characterized by loops that do not cross below the x-axis.

Step 4: Set Up Parametric Equations for (b)

For (b) with \(a = 1\) and \(b = 0.9\), use the parametric equations: \(x = t - 0.9\sin t\) and \(y = 1 - 0.9\cos t\). This describes another curtate cycloid.

Step 5: Graph the Parametric Equations for (b)

Plot the parametric equations from Step 4 using a graphing utility over the interval \([0, 8\pi]\). Analyze the smaller loops that appear due to a smaller \(b\) value compared to \(a\).

Step 6: Set Up Parametric Equations for (c)

For (c) with \(a = 5\) and \(b = \frac{3}{2}\), use the parametric equations: \(x = 5t - \frac{3}{2}\sin t\) and \(y = 5 - \frac{3}{2}\cos t\). Again, this is a curtate cycloid.

Step 7: Graph the Parametric Equations for (c)

Plot the parametric equations from Step 6 over the interval \([0, 8\pi]\). Notice the longer loops due to the larger \(b\) compared to previous examples.

Step 8: Set Up Parametric Equations for (d)

For (d) with \(a = 3\) and \(b = 6\), use the parametric equations: \(x = 3t - 6\sin t\) and \(y = 3 - 6\cos t\). This condition \(b > a\) results in a prolate cycloid.

Step 9: Graph the Parametric Equations for (d)

Plot the parametric equations from Step 8 over the interval \([0, 8\pi]\). Identify the looping, which dips below the x-axis due to \(b > a\).

Key Concepts

cycloids

Cycloids are fascinating curves that result from the path traced by a point located on the circumference of a circle as it rolls along a straight line. This rolling motion without slipping can be visualized as a "path of travel" similar to a bicycle wheel rolling along the ground. Cycloids have a very distinctive shape, characterized by their repetitive "arching" segments.
Cycloids have found importance in various fields, especially in physics and engineering. They have practical applications in areas like the design of pendulum paths or calculating the optimal paths in roller coasters for maximum speed.
Understanding cycloids helps in grasping more complex curve formations, including trochoids, which broaden from the basic cycloid form to include other interesting variations. This fundamental nature of cycloids, being a special form of trochoids where the tracing point \( P \) lies exactly on the circle's edge, makes them a building block for understanding further curve renditions.

parametric equations

Parametric equations represent a set of equations that express the coordinates of the points of a curve by using a separate parameter, often denoted as \(t\). For curves like cycloids and their trochoid relatives, parametric equations are instrumental as they efficiently describe motion and path.

• For cycloids, the parametric representation typically involves sine and cosine functions reflecting the circular nature of the wheel's edge motion.
• The equation's parameters \( (a, b) \) further define the circle's size and the offset distance from the circle's center, respectively.

Utilizing parametric equations allows for dynamic and visual understanding when plotted over a specific interval, like \([0, 8\pi]\), providing a clear picture of how the curve develops. As a tool, parametric equations also enable us to simulate and calculate various curve paths using software and graphing utilities, which are very handy in educational and practical real-world scenarios.

curtate cycloid

Curtate cycloids are specific forms of trochoids that occur when the tracing point is located inside the rolling circle, rather than on the edge, which is when \(b < a \). This positioning leads to a curve where loops keep closely tied to the cycloid path without dropping below the initial line where the cycle rolls.

• This arithmetic difference in parameter values results in a more "concave" appearance that keeps the loops above the x-axis, creating a series of arches rather than full loops.
• Such a pattern is seen in objects like the reflectors in a bicycle's spokes, which create curtate cycloids as they keep internal to the edge of the tire.
• Curtate cycloids are aesthetically appealing when used in designs, exemplified by some musical instruments' curvature that uses this precise creation.

prolate cycloid

Prolate cycloids expand upon the trochoid family as curves traced with the point positioned outside the circle's circumference, indicated when \(b > a \). This configuration results in elegant wave-like loops that invariably cross the base axis.

• In contrast to curtate cycloids, these loops feature larger arcs that extend below the line that the circle rolls along.
• This property makes prolate cycloids suitable for modeling systems where crossing below the base line is necessary, offering insight into more intricate or expansive motion paths.
• Studying prolate cycloids can introduce applications where mechanical or physical movement extends beyond simple planar constraints.

Prolate cycloids thus provide a rich field for exploration in both theoretical studies and practical implementations where open, extended loops are useful.