Question

Find \(a_{\mathrm{T}}\) and \(a_{\mathrm{N}}\) given \(\vec{r}(t) .\) Sketch \(\vec{r}(t)\) on the indicated interval, and comment on the relative sizes of \(a_{\mathrm{T}}\) and \(a_{\mathrm{N}}\) at the indicated \(t\) values. $$ \begin{array}{l} \vec{r}(t)=\langle a \cos t, a \sin t, b t\rangle \text { on }[0,2 \pi], \text { where } a, b>0 ; \text { con- } \\ \text { sider } t=0 \text { and } t=\pi / 2 \end{array} $$

Short answer

At both \( t=0 \) and \( t=\pi/2 \), \( a_{\mathrm{T}} = 0 \) and \( a_{\mathrm{N}} = a \). The acceleration is entirely normal due to the helical path.

Step-by-step solution

Determine the Velocity Vector

The velocity vector \( \vec{v}(t) \) is obtained by differentiating \( \vec{r}(t) \) with respect to \( t \). Calculate \( \vec{v}(t) = \frac{d}{dt} \langle a \cos t, a \sin t, b t \rangle \).The derivatives are:\( \frac{d}{dt} (a \cos t) = -a \sin t \)\( \frac{d}{dt} (a \sin t) = a \cos t \)\( \frac{d}{dt} (b t) = b \)Thus, \( \vec{v}(t) = \langle -a \sin t, a \cos t, b \rangle \).

Determine the Acceleration Vector

The acceleration vector \( \vec{a}(t) \) is the derivative of the velocity vector \( \vec{v}(t) \). Compute \( \vec{a}(t) = \frac{d}{dt} \langle -a \sin t, a \cos t, b \rangle \).The derivatives are:\( \frac{d}{dt} (-a \sin t) = -a \cos t \)\( \frac{d}{dt} (a \cos t) = -a \sin t \)\( \frac{d}{dt} (b) = 0 \)So, \( \vec{a}(t) = \langle -a \cos t, -a \sin t, 0 \rangle \).

Calculate Tangential and Normal Components of Acceleration

The tangential acceleration \( a_{\mathrm{T}} \) is given by \( a_{\mathrm{T}} = \frac{\vec{a}(t) \cdot \vec{v}(t)}{\|\vec{v}(t)\|} \) and normal acceleration \( a_{\mathrm{N}} \) is given by \( \| \vec{a}(t) \|_{\perp} \), which can be determined values when these vectors are perpendicular.Calculate:\( \vec{a}(t) \cdot \vec{v}(t) = (-a \cos t)(-a \sin t) + (-a \sin t)(a \cos t) \times 0 = 0\), since the second component doesn't affect this because it's orthogonal.\( \| \vec{v}(t) \| = \sqrt{(-a \sin t)^2 + (a \cos t)^2 + b^2} = \sqrt{a^2 \sin^2 t + a^2 \cos^2 t + b^2} = \sqrt{a^2 + b^2} \).Thus, \( a_{\mathrm{T}} = 0 \).\( a_{\mathrm{N}} = \| \vec{a}(t) \| = \sqrt{(-a \cos t)^2 + (-a \sin t)^2} = \sqrt{a^2 \cos^2 t + a^2 \sin^2 t} = \sqrt{a^2} = a \).

Sketch and Analyzing Relative Sizes at t=0 and t=π/2

The vector \( \vec{r}(t) \) traces a helical path, with a circular motion in the \(xy\)-plane and linear motion in the \(z\)-direction. The interval \([0, 2\pi]\) completes one revolution of the circle.At \( t = 0 \), the particle is at \( (a, 0, 0) \), and at \( t = \pi/2 \), it is at \( (0, a, b\pi/2) \).Both \( a_{\mathrm{T}} = 0 \) and \( a_{\mathrm{N}} = a \) holds at both \( t=0 \) and \( t=\pi/2 \). This indicates that there is no change in speed along the path (hence \( a_{\mathrm{T}} = 0 \)), with normal acceleration maintaining the circular motion.

Key concepts

Tangent Vector

A tangent vector gives insight into the direction and rate of change of a curve at a particular point. In context, for the given vector function \( \vec{r}(t) = \langle a \cos t, a \sin t, b t \rangle \), the tangent vector is obtained by differentiating \( \vec{r}(t) \). This derivative, known as the velocity vector \( \vec{v}(t) = \langle -a \sin t, a \cos t, b \rangle \), describes how the position of the particle changes with time.

Key points about tangent vectors:

• Tangent vectors are perpendicular to the radius of circular motion in \(xy\)-plane.
• They provide an essential glimpse into how fast and in what direction an object moves along a path.

This directional indicator is paramount when analyzing helical or circular motions, as it shows how the motion unfolds over time.

Normal Acceleration

Normal acceleration refers to the component of acceleration perpendicular to the velocity vector. In curved paths like in the given helical path, it's responsible for changing the direction of the velocity vector without altering its magnitude. For our function, the normal acceleration \( a_{\mathrm{N}} \) is determined to be \( a \), showing that it's constant and influences the circular motion of the object.

Normal acceleration reveals:

• How the path curvature affects the rate of change in direction.
• Its independent role from tangential acceleration since our example yields \( a_{\mathrm{T}} = 0 \).

Notably, it maintains the helical path's steady motion around its axis by accounting for the circular path in the \(xy\)-plane.

Helical Path

A helical path combines circular and linear motion, much like a spring. In our example, \( \vec{r}(t) = \langle a \cos t, a \sin t, b t \rangle \) describes such a path, where the circle rotates in the \(xy\)-plane and simultaneously progresses linearly along the \(z\)-axis. Some critical points about helical paths are:

• The linear motion component (\( b t \)) in the \(z\)-direction describes the extent of the path.
• The parameters \( a \) and \( b \) define the geometry and the pitch (distance between turns), respectively.

Understanding a helical path involves recognizing its dual nature of continuous circular rotation and linear forward progress along a distinct axis.

Velocity Vector

A velocity vector describes the speed and direction of an object moving along a path. In the example we have \( \vec{v}(t) = \langle -a \sin t, a \cos t, b \rangle \), which tells us how quickly and in which direction the object moves at any time \( t \).

Attributes of the velocity vector include:

• Its magnitude, which can be calculated using \( \sqrt{a^2 + b^2}\), shows the combined effect of speed from both rotational (\( a \)) and linear (\( b \)) components.
• It consists of both horizontal (arising from the circular path) and vertical (arising from the linear path in \(z\)-direction) components.

The velocity vector, therefore, is essential for understanding motion dynamics, with its components indicating how different forces shape the path taken by an object.