Problem 42 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The maximum \(|\mathbf{v}|\) is 2, the minimum is 0; \(|\mathbf{a}|\) is always 1.

Step 1: Understand the motion

The position vector \( \mathbf{r}(t) \) describes the motion of a particle along a cycloid in the \( xy \)-plane, expressed as \( \mathbf{r}(t) = (t - \sin t) \mathbf{i} + (1 - \cos t) \mathbf{j} \). We want to understand its trajectory over time.

Step 2: Graph the motion

Plot the parametric equations \( x(t) = t - \sin t \) and \( y(t) = 1 - \cos t \) over a range of \( t \) values. This visualizes the cycloid path. The graph demonstrates how the horizontal motion advances as a cycloid path is traced horizontally along the x-axis while vertically along the y-axis.

Step 3: Calculate velocity \(\mathbf{v}(t)\)

The velocity is the derivative of the position with respect to time \( t \). Thus, \( \mathbf{v}(t) = \frac{d}{dt}(t - \sin t) \mathbf{i} + \frac{d}{dt}(1 - \cos t) \mathbf{j} \). Compute this to obtain \( \mathbf{v}(t) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j} \).

Step 4: Calculate \(|\mathbf{v}|^2\)

The magnitude of the velocity vector is \(|\mathbf{v}| = \sqrt{(1 - \cos t)^2 + (\sin t)^2}\). Calculate the square \(|\mathbf{v}|^2 = (1 - \cos t)^2 + \sin^2 t \). Using the identity \( \sin^2 t + \cos^2 t = 1 \), simplify this to \(|\mathbf{v}|^2 = 2(1 - \cos t) = 4 \sin^2(\frac{t}{2})\).

Step 5: Find extremes of \(|\mathbf{v}|^2\) by analyzing \(\sin^2\left(\frac{t}{2}\right)\)

Since \( \sin^2(\frac{t}{2}) \) varies from 0 to 1, \( |\mathbf{v}|^2 = 4 \sin^2(\frac{t}{2}) \) ranges from 0 to 4. Therefore, the maximum \(|\mathbf{v}|\) is 2 and the minimum is 0.

Step 6: Calculate acceleration \(\mathbf{a}(t)\)

Differentiate the velocity \( \mathbf{v}(t) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j} \) to find the acceleration. \( \mathbf{a}(t) = \frac{d}{dt}(1 - \cos t) \mathbf{i} + \frac{d}{dt}(\sin t) \mathbf{j} \) gives \( \mathbf{a}(t) = \sin t \mathbf{i} + \cos t \mathbf{j} \).

Step 7: Calculate \(|\mathbf{a}|^2\)

The magnitude of the acceleration vector is \( |\mathbf{a}| = \sqrt{\sin^2 t + \cos^2 t} = 1 \). Thus, \(|\mathbf{a}|^2 = 1^2 = 1\) and since this is constant, both the minimum and maximum values of \(|\mathbf{a}|\) are 1.

Key Concepts

Parametric Equations

In cycloid motion, particles travel along a specific type of curve generated by equations that define both horizontal and vertical movement. These equations are called parametric equations. They allow us to specify both coordinates, \(x\) and \(y\), in terms of a third variable, often time \(t\).
For the cycloid described, the parametric equations are:\[ x(t) = t - \sin t\] and \[y(t) = 1 - \cos t.\]
This setup means at any given time \(t\), we can pinpoint the exact position \((x, y)\) of the particle based on these equations. This is essential for analyzing movements like cycles or paths where direct equations of \(y\) in terms of \(x\) aren't easily obtained.
To visualize the motion, you can graph these equations for various values of \(t\). This results in a cycloid, showcasing the path traced out as the particle moves. The curve resembles a series of arches, illustrating both forward and upward/downward movements characteristic of cycloids.

Velocity Calculation

Velocity in motion refers to both the speed and direction at which a particle moves. To find the velocity of a particle on a cycloid, we need to differentiate the position functions.
The velocity vector \(\mathbf{v}(t)\) is the derivative of the position vector \(\mathbf{r}(t)\):\[\mathbf{v}(t) = \frac{d}{dt}(t - \sin t) \mathbf{i} + \frac{d}{dt}(1 - \cos t) \mathbf{j}.\]
When calculated, this results in:\[\mathbf{v}(t) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j}.\]
The magnitudes \(|\mathbf{v}|\) capture how fast the particle travels without regard to direction. To determine it, calculate:\[|\mathbf{v}| = \sqrt{(1 - \cos t)^2 + (\sin t)^2}.\]
Further simplification using trigonometric identities reveals:\[|\mathbf{v}|^2 = 4 \sin^2\left(\frac{t}{2}\right).\]
This reveals how the particle's speed changes over time.

Acceleration Calculation

Acceleration describes how a particle's velocity changes over time. For our cycloid motion, it involves differentiating the velocity function
\(\mathbf{v}(t) = (1 - \cos t) \mathbf{i} + \sin t \mathbf{j}\).
This results in the acceleration vector:\[\mathbf{a}(t) = \sin t \mathbf{i} + \cos t \mathbf{j}.\]
Understanding the acceleration is crucial as it affects both how the particle speeds up or slows down, and how it changes its direction.
The magnitude of the acceleration is simply calculated as\[|\mathbf{a}| = \sqrt{(\sin t)^2 + (\cos t)^2} = 1.\]
Since this magnitude is constant, the acceleration is uniform in speed, while the direction follows the changing path of the cycloid.

Extreme Values Analysis

In mathematics and physics, extreme values help us understand significant limits of behavior, like maxima and minima. For cycloid motion, exploring extreme values of velocity \(|\mathbf{v}|\) and acceleration \(|\mathbf{a}|\) informs about peak speeds and consistent motion.
Starting with velocity, observed from:\[|\mathbf{v}|^2 = 4 \sin^2\left(\frac{t}{2}\right),\]
we see it varies between 0 and 4. Therefore, taking the square root, the maximum velocity \(|\mathbf{v}|\) is 2, while the minimum is 0. The velocity changes as the sine function does, peaking when \(\sin(\frac{t}{2})\) reaches its maximum.
For acceleration, due to:\[|\mathbf{a}|^2 = 1^2 = 1,\]
both maximum and minimum values are constantly 1. This reflects the consistent circular motion, where centripetal acceleration spins the particle evenly throughout the path.
Analyzing these extremes offers insights into how a particle behaves over its cycloidal path, guiding applications in mechanical and motion studies.