Question

Use a computer algebra system to graph the space curve. Then find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathrm{T}},\) and \(a_{\mathrm{N}}\) at the given time \(t\). Sketch \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) on the space curve. $$ \mathbf{r}(t)=4 t \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k} \quad t=\frac{\pi}{2} $$

Short answer

The unit tangent vector \(\mathbf{T}(t = \frac{\pi}{2})\) is \(\frac{4}{5} \mathbf{i}\), the unit normal vector \(\mathbf{N}(t = \frac{\pi}{2})\) is \(-\mathbf{j}\), the tangential acceleration \(a_{\mathrm{T}}\) is 0 and the normal acceleration \(a_{\mathrm{N}}\) is 3.

Step-by-step solution

Derivation of the Velocity Vector

Find derivative of space curve, \( \mathbf{r'}(t) = 4 \mathbf{i} - 3\sin(t) \mathbf{j} + 3 \cos(t) \mathbf{k} \) which is the velocity vector.

Calculation of Speed

Speed is the magnitude of the velocity vector. It is given as \( ||\mathbf{r'}(t)|| = \sqrt{4^2 + (-3\sin(t))^2 + (3 \cos(t))^2} = 5 \). Hence the speed at \(t=\frac{\pi}{2}\) will remain the same and is equal to 5.

Computation of Unit Tangent Vector \(\mathbf{T}(t)\)

The unit tangent vector is defined as the velocity vector divided by the speed. Therefore, \(\mathbf{T}(t) = \frac{\mathbf{r'}(t)}{||\mathbf{r'}(t)||} = \frac{4}{5} \mathbf{i} - \frac{3}{5} \sin(t) \mathbf{j} + \frac{3}{5} \cos(t) \mathbf{k} \). When \(t=\frac{\pi}{2}\), \(\mathbf{T} ( \frac{\pi}{2} ) = \frac{4}{5} \mathbf{i}\).

Derivation of Unit Normal Vector \(\mathbf{N}(t)\)

The unit normal vector \(\mathbf{N}(t)\) is the derivative of the unit tangent vector normalized. So, first compute \(\mathbf{T'}(t)\) then normalize by dividing by the magnitude of \(\mathbf{T'}(t)\). \(\mathbf{T'}(t) = -\frac{3}{5} \cos(t) \mathbf{j} - \frac{3}{5} \sin(t) \mathbf{k} \) and \(||\mathbf{T'}(t)|| = \frac{3}{5}\). Hence, \(\mathbf{N}(t) = \frac{\mathbf{T'}(t)}{|| \mathbf{T'}(t) ||} = -\cos(t) \mathbf{j} -\sin(t) \mathbf{k} \). At \(t=\frac{\pi}{2}\), \(\mathbf{N} ( \frac{\pi}{2} ) = - \mathbf{j}\).

Calculation of Tangential and Normal Acceleration

Tangential acceleration \(a_{\mathrm{T}}\) is the derivative of the speed and since speed does not change with \(t\), \(a_{\mathrm{T}} = 0\). The normal acceleration is defined as \(a_{\mathrm{N}} = ||\mathbf{T'}(t)|| \times ||\mathbf{r'}(t)|| = \frac{3}{5} \times 5 = 3\).

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on differential and integral calculus with vector functions. It involves a variety of operations, such as dot and cross products, along with differentiation and integration of vector fields.

When we deal with space curves like the one in this exercise, \(\mathbf{r}(t)=4 t \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k}\), vector calculus helps us find multiple characteristics of the curve's motion along the path. The function \(\mathbf{r}(t)\) is a vector function representing the position of a particle in three dimensions mapped over time \(t\).

By deriving and analyzing \(\mathbf{r}(t)\), we can compute several significant features such as the velocity (first derivative), acceleration (second derivative), and specific vectors like the tangent \(\mathbf{T}(t)\) and normal \(\mathbf{N}(t)\) to the curve at any given point.

Tangent and Normal Vectors

Tangent and normal vectors play a crucial role in understanding the motion and geometry of a space curve.

A tangent vector \(\mathbf{T}(t)\) is a vector that touches the curve at a particular point and points in the direction of the curve's motion at that instant. It is obtained by normalizing the velocity vector, meaning it will have a magnitude of 1. This means no matter how fast the particle is moving, the tangent vector only describes the direction of the motion.

On the other hand, the normal vector \(\mathbf{N}(t)\) is perpendicular to the tangent vector at the same point on the curve. It is found by normalizing the derivative of the tangent vector, hence indicating the direction in which the curve is turning.

Visualizing Tangent and Normal Vectors

When we sketch these vectors on a space curve, we can visually ascertain not only the direction the object is moving but also the curvature of its path at any point. This visualization enhances the interpretation of an object's path in physics and engineering, providing insights into real-world scenarios like vehicle turning on a road or the path a planet takes around the sun.

Acceleration in Calculus

Acceleration is a fundamental concept in calculus that describes how an object's velocity changes over time. In the world of vector calculus, we distinguish between tangential and normal acceleration.

Tangential Acceleration

\(a_{\mathrm{T}}\) represents how the speed of an object changes along the curve and is the derivative of speed with respect to time. If the speed is constant, as in this exercise, the tangential acceleration is zero since there's no change in the magnitude of the velocity.

Normal Acceleration

\(a_{\mathrm{N}}\) however, relates to how the direction of the velocity changes and is linked to the curve's curvature. It's calculated as the product of the speeds of the change in the direction of the unit tangent vector (\(||\mathbf{T'}(t)||\)) and the speed of the particle (\(||\mathbf{r'}(t)||\)). It is often conceptualized as the 'centripetal acceleration' in circular motion, pulling the moving object toward the center of its curved path.

Understanding each type of acceleration is crucial for many practical applications, whether it's for analyzing forces on a roller coaster or determining the stress on an airplane during a turn.