Chapter 13

Find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) \begin{equation}r=a(1-\cos \theta) \quad \text { and } \quad \frac{d \theta}{d t}=3\end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves $$ \mathbf{r}(t)=t \mathbf{i}+(\ln \cos t) \mathbf{j}, \quad-\pi / 2< t <\pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t \mathbf{j})+\sqrt{5} t \mathbf{k}, \quad 0 \leq t \leq \pi $$

Evaluate the integrals. $$\int_{0}^{1}\left[t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right] d t$$

In Exercises \(1-4, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of \(t .\) \begin{equation} \mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1 \end{equation}

Write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathrm{N}\) without finding \(\mathrm{T}\) and \(\mathrm{N} .\) \(\mathbf{r}(t)=(1+3 t) \mathbf{i}+(t-2) \mathbf{j}-3 t \mathbf{k}\)

Find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) \begin{equation}r=a \sin 2 \theta \quad \text { and } \quad \frac{d \theta}{d t}=2 t\end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves $$ \mathbf{r}(t)=(\ln \sec t) \mathbf{i}+t \mathbf{j}, \quad-\pi / 2< t <\pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k}, \quad 0 \leq t \leq \pi $$

Evaluate the integrals. $$\int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t \mathbf{j}}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t$$

\({r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of \(t .\) \begin{equation} \mathbf{r}(t)=\frac{t}{t+1} \mathbf{i}+\frac{1}{t} \mathbf{j}, \quad t=-\frac{1}{2} \end{equation}

Write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) at the given value of \(t\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) \(\mathbf{r}(t)=(t+1) \mathbf{i}+2 t \mathbf{j}+t^{2} \mathbf{k}, \quad t=1\)

Find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) \begin{equation}r=e^{a \theta} \quad \text { and } \quad \frac{d \theta}{d t}=2\end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves $$ \mathbf{r}(t)=(2 t+3) \mathbf{i}+\left(5-t^{2}\right) \mathbf{j} $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=t \mathbf{i}+(2 / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq 8 $$

Evaluate the integrals. $$\int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t$$

\({r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of \(t .\) \begin{equation} \mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3 \end{equation}

Find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) \begin{equation}r=a(1+\sin t) \quad \text { and } \quad \theta=1-e^{-t}\end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad t > 0 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2+t) \mathbf{i}-(t+1) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 3 $$

Evaluate the integrals. $$\int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t$$

\({r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of \(t .\) \begin{equation} \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0 \end{equation}

Find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) \begin{equation}r=2 \cos 4 t \quad \text { and } \quad \theta=2 t\end{equation}

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{j}+\left(\sin ^{3} t\right) \mathbf{k}, \quad 0 \leq t \leq \pi / 2 $$

Evaluate the integrals. $$ \int_{1}^{4}\left[\frac{1}{t} \mathbf{i}+\frac{1}{5-t} \mathbf{j}+\frac{1}{2 t} \mathbf{k}\right] d t $$

Exercises \(5-8\) give the position vectors of particles moving along various curves in the \(x y\) -plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. \begin{equation} \mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4 \text { and } \pi / 2 \end{equation}

Write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) at the given value of \(t\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) \(\mathbf{r}(t)=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+\sqrt{2} e^{t} \mathbf{k}, \quad t=0\)

A formula for the curvature of a parametrized plane curve $$ \begin{array}{l}{\text { a. Show that the curvature of a smooth curve } \mathbf{r}(t)=f(t) \mathbf{i}+} \\ {g(t) \mathbf{j} \text { defined by twice- differentiable functions } x=f(t) \text { and }} \\ {y=g(t) \text { is given by the formula }}\end{array} $$ $$ \kappa=\frac{|\dot{x} \ddot{y}-\dot{y} \ddot{x}|}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2}} $$ The dots in the formula denote differentiation with respect to \(t\) one derivative for each dot. Apply the formula to find the curvatures of the following curves. $$ \begin{array}{l}{\text { b. } \mathbf{r}(t)=t \mathbf{i}+(\ln \sin t) \mathbf{j}, \quad 0< t <\pi} \\ {\text { c. } \mathbf{r}(t)=\left[\tan ^{-1}(\sinh t)\right] \mathbf{i}+(\ln \cosh t) \mathbf{j}}\end{array} $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=6 t^{3} \mathbf{i}-2 t^{3} \mathbf{j}-3 t^{3} \mathbf{k}, \quad 1 \leq t \leq 2 $$

Evaluate the integrals. $$ \int_{0}^{1}\left[\frac{2}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{\sqrt{3}}{1+t^{2}} \mathbf{k}\right] d t $$

Give the position vectors of particles moving along various curves in the \(x y\) -plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the circle \(x^{2}+y^{2}=16\) \begin{equation} \mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2 \end{equation}

Circular orbits Show that a planet in a circular orbit moves with a constant speed. (Hint: This is a consequence of one of Kepler's laws.)

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+(2 \sqrt{2} / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq \pi $$

Give the position vectors of particles moving along various curves in the \(x y\) -plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the cycloid \(x=t-\sin t, \quad y=1-\cos t\) \begin{equation} \mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2 \end{equation}

Find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t\) . Then find equations for the osculating, normal, and rectifying planes at that value of \(t\) . \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}-\mathbf{k}, \quad t=\pi / 4\)

Find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t\) . Then find equations for the osculating, normal, and rectifying planes at that value of \(t\) . \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad t=0\)

Suppose that \(\mathbf{r}\) is the position vector of a particle moving along a plane curve and \(d A / d t\) is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the necessary derivatives exist, give a geometric argument based on increments and limits for the validity of the equation \begin{equation} \frac{d A}{d t}=\frac{1}{2}|\mathbf{r} \times \dot{\mathbf{r}}|. \end{equation}

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \sin t+\cos t) \mathbf{i}+(t \cos t-\sin t) \mathbf{j}, \quad \sqrt{2} \leq t \leq 2 $$

Evaluate the integrals. $$ \int_{1}^{\ln 3}\left[t e^{t} \mathbf{i}+e^{\ell} \mathbf{j}+\ln t \mathbf{k}\right] d t $$

Give the position vectors of particles moving along various curves in the \(x y\) -plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the parabola \(y=x^{2}+1\) \begin{equation} \mathbf{r}(t)=t \mathbf{i}+\left(t^{2}+1\right) \mathbf{j} ; \quad t=-1,0, \text { and } 1 \end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves $$ \mathbf{r}(t)=(3 \sin t) \mathbf{i}+(3 \cos t) \mathbf{j}+4 t \mathbf{k} $$

Find the point on the curve $$ \mathbf{r}(t)=(5 \sin t) \mathbf{i}+(5 \cos t) \mathbf{j}+12 t \mathbf{k} $$ at a distance 26\(\pi\) units along the curve from the point \((0,5,0)\) in the direction of increasing arc length.

Evaluate the integrals. $$ \int_{0}^{\pi / 2}\left[\cos t \mathbf{i}-\sin 2 t \mathbf{j}+\sin ^{2} t \mathbf{k}\right] d t $$

In Exercises \(9-14, \mathbf{r}(t)\) is the position of a particle in space at time \(t .\) Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of \(t\) . Write the particle's velocity at that time as the product of its speed and direction. \begin{equation} \mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad t=1 \end{equation}

Section 13.4, you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa .\) Now, in the following Exercises 9-16, find \(\mathbf{B}\) and \(\tau\) for these space curves. \(\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}+3 \mathbf{k}\)

Do the data in the accompanying table support Kepler's third law? Give reasons for your answer. \begin{equation}\begin{array}{\underline{\phantom{xx}}}\hline & {\text { Semimajor axis }} & \\\ {\text { Planet }} & {a\left(10^{10} \mathbf{m}\right)} & {\text { Period } T \text { (years) }} \\ \hline \text { Mercury } & {5.79} & {0.241} \\ {\text { Venus }} & {10.81} & {0.615} \\ {\text { Mars }} & {22.78} & {1.881} \\\ {\text { Saturn }} & {142.70} & {29.457} \\ \hline\end{array}\end{equation}

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}+3 \mathbf{k} $$

Find the point on the curve $$ \mathbf{r}(t)=(12 \sin t) \mathbf{i}-(12 \cos t) \mathbf{j}+5 t \mathbf{k} $$ at a distance 13\(\pi\) units along the curve from the point \((0,-12,0)\) in the direction opposite to the direction of increasing arc length.

Evaluate the integrals. $$ \int_{0}^{\pi / 4}\left[\sec t \mathbf{i}+\tan ^{2} t \mathbf{j}-t \sin t \mathbf{k}\right] d t $$

\({r}(t)\) is the position of a particle in space at time \(t .\) Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of \(t\) . Write the particle's velocity at that time as the product of its speed and direction. \begin{equation} \mathbf{r}(t)=(1+t) \mathbf{i}+\frac{t^{2}}{\sqrt{2}} \mathbf{j}+\frac{t^{3}}{3} \mathbf{k}, \quad t=1 \end{equation}