Problem 40 - Thomas Calculus

Question

\(\begin{array}{l}{\text { Motion along a cycloid A particle moves in the } x y \text { -plane in }} \\ {\text { such a way that its position at time } t \text { is }}\end{array}\) $$\quad \mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}$$ \(\begin{array}{l}{\text { a. Graph } \mathbf{r}(t) . \text { The resulting curve is a cycloid. }} \\ {\text { b. Find the maximum and minimum values of }|\mathbf{v}| \text { and }|\mathbf{a}| \text { . }} \\ {\text { (Hint: Find the extreme values of }|\mathbf{v}|^{2} \text { and }|\mathbf{a}|^{2} \text { first and }} \\ {\text { take square roots later.) }}\end{array}\)

Step-by-Step Solution

Answer

Graph the cycloid from its equations. Maximum \(|\mathbf{v}| = 2\), minimum \(|\mathbf{v}| = 0\); both \(|\mathbf{a}_\max|\) and \(|\mathbf{a}_\min|\) are 1.

Step 1: Understand the Parametric Equations

The position of the particle in the xy-plane is given by \( \mathbf{r}(t) = (t - \sin t) \mathbf{i} + (1 - \cos t) \mathbf{j} \). This represents a cycloid, which is the path traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.

Step 2: Graph the Cycloid

To graph the cycloid, plot the parametric equations \( x(t) = t - \sin t \) and \( y(t) = 1 - \cos t \) over a range of \( t \) values. This will show the typical shape of a cycloid, which consists of a series of arches.

Step 3: Find the Velocity Vector

The velocity \( \mathbf{v}(t) \) is the derivative of the position vector \( \mathbf{r}(t) \). Compute \( \mathbf{v}(t) = \frac{d}{dt}\big((t - \sin t) \mathbf{i} + (1 - \cos t) \mathbf{j}\big) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j} \).

Step 4: Compute Speed \( |\mathbf{v}| \)

The magnitude of the velocity \( |\mathbf{v}| \) is given by \(|\mathbf{v}| = \sqrt{(1 - \cos t)^2 + (\sin t)^2} \). Simplify to \(|\mathbf{v}| = \sqrt{2 - 2\cos t} = 2\sin\left(\frac{t}{2}\right) \).

Step 5: Find the Maximum and Minimum of \( |\mathbf{v}|^2 \)

Since \( |\mathbf{v}| = 2\sin\left(\frac{t}{2}\right) \), we have \( |\mathbf{v}|^2 = 4\sin^2\left(\frac{t}{2}\right) \). The extreme values are when \( \sin\left(\frac{t}{2}\right) \) is maximum and minimum, giving maximum \( |\mathbf{v}|^2 = 4 \) and minimum \( |\mathbf{v}|^2 = 0 \).

Step 6: Find the Acceleration Vector

The acceleration \( \mathbf{a}(t) \) is the derivative of the velocity vector \( \mathbf{v}(t) \). Compute \( \mathbf{a}(t) = \frac{d}{dt}\big((1 - \cos t) \mathbf{i} + \sin t \mathbf{j}\big) = \sin t \mathbf{i} + \cos t \mathbf{j} \).

Step 7: Compute Magnitude of \( |\mathbf{a}| \)

The magnitude of the acceleration \( |\mathbf{a}| \) is given by \(|\mathbf{a}| = \sqrt{(\sin t)^2 + (\cos t)^2} = \sqrt{1} = 1 \).

Step 8: Analyze Maximum and Minimum of \( |\mathbf{a}|^2 \)

Since \( |\mathbf{a}| = 1 \), the magnitude squared \( |\mathbf{a}|^2 = 1 \) is constant, meaning both the maximum and minimum values are 1.

Key Concepts

Parametric Equations

In the study of cycloid motion, parametric equations play a crucial role. They allow us to trace the path of a particle moving in the xy-plane. Specifically, the equations \( x(t) = t - \sin t \) and \( y(t) = 1 - \cos t \) describe the cycloid path. This path is generated by a point on a circle as it rolls along a straight line.

These equations separate the motion into horizontal and vertical components as functions of \( t \), the time variable.

• The equation \( x(t) = t - \sin t \) accounts for the horizontal displacement and includes the effect of the rolling circle's arc, represented by \( \sin t \).
• The equation \( y(t) = 1 - \cos t \) describes the vertical displacement, showing how far the point is from the line the wheel is rolling on, where \( \cos t \) captures oscillations around a midline.

Understanding and graphing these equations gives us a visual image of the cycloid, helping to comprehend the nature of this particular type of motion.

Velocity Vector

The velocity vector illustrates how fast the position of the particle changes over time. It is essentially the derivative of the position vector, \( \mathbf{r}(t) \). In our cycloid scenario, the velocity vector is \( \mathbf{v}(t) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j} \).

Here's what each part represents:

• \( 1 - \cos t \): Describes the change in horizontal position, showing that the object's speed depends on the angle \( t \).
• \( \sin t \): Indicates vertical movement, which oscillates similarly to how a wave might move.

Calculating the velocity vector is a key step in understanding the dynamics of the motion because it directly links to how the particle traverses its path over time.

Acceleration Vector

Acceleration is the rate of change of the velocity with respect to time. For our cycloid motion, we derive the acceleration vector \( \mathbf{a}(t) = \sin t \mathbf{i} + \cos t \mathbf{j} \). This vector provides insight into how the speed and direction of the particle's motion alter over time.

Each component tells us about the adjustments in motion:

• \( \sin t \): Affects horizontal acceleration, hinting at the rolling effect as it peaks and dips.
• \( \cos t \): Manages vertical acceleration, showing how the vertical motion stabilizes and changes.

Recognizing the acceleration vector allows us to predict future positions and velocities, and understand forces acting on the particle.

Maximum and Minimum Values

Identifying the maximum and minimum values of various quantities helps us to understand the boundaries and limits of the motion. In this context, it refers specifically to the speed of the particle and its acceleration.

For the velocity magnitude \( |\mathbf{v}| \), the extremes are calculated as:

• Maximum when the velocity vector magnitude, \( |\mathbf{v}|^2 = 4 \).
• Minimum when \( |\mathbf{v}|^2 = 0 \).

Meanwhile, the acceleration magnitude \( |\mathbf{a}| \) is constant:

• Both maximum and minimum value at \( 1 \), demonstrating consistent acceleration throughout the motion.

Knowing these values helps in predicting and comparing different segments of the motion.

Magnitude of Vectors

Magnitude of a vector represents the size or length irrespective of its direction. For vectors like velocity and acceleration, understanding their magnitudes explains the intensity of motion in cycloid paths.

To compute the magnitude for the velocity vector, we use:

• Velocity \( |\mathbf{v}| = \sqrt{(1 - \cos t)^2 + (\sin t)^2} = 2\sin\left(\frac{t}{2}\right) \), indicating how the speed varies as the wheel rolls.

For the acceleration vector:

• Acceleration constant \( |\mathbf{a}| = 1 \), reflecting uniform circular motion properties, aligning with trigonometric identities \( \sin^2 t + \cos^2 t = 1 \).

Grasping vector magnitudes provides a clearer picture of kinetic energy and motion dynamics throughout the cycle of a rolling wheel.