Problem 20 - Advanced Engineering Mathematics

Question

Find the curvature at \(t=\pi\) of the cycloid that is described by $$ \mathbf{r}(t)=a(t-\sin t) \mathbf{i}+a(1-\cos t) \mathbf{j}, a>0 $$

Step-by-Step Solution

Answer

The curvature at \( t=\pi \) is \( \frac{1}{4a} \).

Step 1: Understand the Problem

We need to find the curvature of a curve described by a parametric vector function \( \mathbf{r}(t)=a(t-\sin t) \mathbf{i}+a(1-\cos t) \mathbf{j} \) at \( t=\pi \). The formula for the curvature \( \kappa \) of a parametric curve \( \mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j} \) is \( \kappa = \frac{|x' y'' - y' x''|}{((x')^2 + (y')^2)^{3/2}} \).

Step 2: Compute the Derivatives

Calculate the first derivatives: \( x(t) = a(t - \sin t) \) and \( y(t) = a(1 - \cos t) \). Therefore, \( x'(t) = a(1 - \cos t) \) and \( y'(t) = a\sin t \). Next, calculate the second derivatives: \( x''(t) = a\sin t \) and \( y''(t) = a\cos t \).

Step 3: Evaluate Derivatives at \( t=\pi \)

Substitute \( t=\pi \) into the derivatives. \( x'(\pi) = a(1 - \cos \pi) = 2a \), \( y'(\pi) = a\sin \pi = 0 \), \( x''(\pi) = a\sin \pi = 0 \), and \( y''(\pi) = a\cos \pi = -a \).

Step 4: Apply Curvature Formula

Plug the values of the derivatives evaluated at \( t=\pi \) into the curvature formula: \( \kappa = \frac{|x' y'' - y' x''|}{((x')^2 + (y')^2)^{3/2}} = \frac{|2a(-a)-0\cdot0|}{((2a)^2 + 0^2)^{3/2}} = \frac{2a^2}{(4a^2)^{3/2}} = \frac{2a^2}{8a^3} = \frac{1}{4a} \).

Step 5: Conclude the Solution

Therefore, the curvature of the given cycloid at \( t=\pi \) is \( \frac{1}{4a} \).

Key Concepts

Parametric Curves

Parametric curves are a fascinating way to describe the geometry of curves using parameters, typically involving two or more equations to define each of the curve's coordinates. In our example, we have a cycloid given by the parametric equations:

• For the x-coordinate: \( x(t) = a(t - \sin t) \)
• For the y-coordinate: \( y(t) = a(1 - \cos t) \)

This means that for any value of \( t \), which is the parameter in this case, you can plug it into these equations and get a point \( (x, y) \) on the curve. Think of \( t \) as a kind of time parameter, where every value gives you a new point along the curve's path.
Choosing parametric representations often makes it easier to work with complex curves, especially when dealing with curves like cycloids that show interesting physical properties.

Cycloid

A cycloid is a curve generated by the path of a point on the rim of a circular wheel as the wheel rolls along a straight line. This results in a wavelike motion, making it an interesting object to study in terms of its geometry. The movement is captured using the parametric equations we've looked at:

• \( x(t) = a(t - \sin t) \)
• \( y(t) = a(1 - \cos t) \)

These equations reflect how the x and y coordinates change over the parameter \( t \). The cycloid is known for its intriguing properties in physics and engineering, such as being the curve of fastest descent, known as the brachistochrone problem.
The parameter \( a \) represents the radius of the generating circle, and it scales the size of the cycloid. Overall, cycloids provide great examples of how parametric equations can help describe complicated paths in a simpler way.

Derivatives

Derivatives play a crucial role in understanding and analyzing parametric curves like the cycloid. To calculate features like curvature, we need to know how the curve changes—or, more technically, how the position along the curve changes with respect to the parameter \( t \).
The derivatives of our parametric curve are:

• First derivative with respect to \( t \) for the x-coordinate: \( x'(t) = a(1 - \cos t) \)
• First derivative with respect to \( t \) for the y-coordinate: \( y'(t) = a\sin t \)
• Second derivative with respect to \( t \) for the x-coordinate: \( x''(t) = a\sin t \)
• Second derivative with respect to \( t \) for the y-coordinate: \( y''(t) = a\cos t \)

These derivatives tell us how fast the coordinates are changing as \( t \) changes, which in this context is crucial for computing the curvature at specific points.

Curvature Formula

The curvature of a curve at a particular point gives us a measure of how sharply it is bending. For parametric curves, we use a specialized curvature formula. This is given by:
\[\kappa = \frac{|x' y'' - y' x''|}{((x')^2 + (y')^2)^{3/2}}\]
In our specific example of the cycloid, we evaluated these derivatives specifically at \( t = \pi \), leading us to find the curvature as:

• Substitute the derivatives into the formula at \( t = \pi \):
• \( x' = 2a, y' = 0, x'' = 0, y'' = -a \)
• \( \kappa = \frac{|2a(-a) - 0\cdot 0|}{((2a)^2 + 0^2)^{3/2}} = \frac{2a^2}{(4a^2)^{3/2}} = \frac{1}{4a} \)

Finding this curvature not only confirms how the cycloid bends at \( t = \pi \) but also showcases how parametric representations let us apply calculus tools to understand complex curves thoroughly.