Chapter 11.5, Problem 11.179P

The three-dimensional motion of a particle is defined by the cylindrical coordinates R = A/(t + 1), θ = Bt, and z = Ct/(t + 1). Determine the magnitudes of the velocity and acceleration when (a) t = 0, (b) t = ∞.

To determine

The magnitudes of the velocity $\left(v_0\right)$ and acceleration $\left(a_0\right)$ when time (t) is 0.

Answer to Problem 11.179P

The magnitudes of the velocity $\left(v_0\right)$ and acceleration $\left(a_0\right)$ when time (t) is 0 are $\underset{\_}{\sqrt{A^2+A^2{B}^2+C^2}}$ and $\underset{\_}{\sqrt{4A^2+A^2{B}^4+4C^2}}$ respectively.

Explanation of Solution

Given Information:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is $\frac{A}{\left(t+1\right)}$, $\left(\theta \right)$ is $Bt$ and (z) is $\frac{Ct}{\left(t+1\right)}$.

Calculation:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R):

$R=\frac{A}{\left(t+1\right)}$ (1)

The three dimensional motion of a particle is defined by the cylindrical coordinates $\left(\theta \right)$:

$\theta =Bt$ (2)

The three dimensional motion of a particle is defined by the cylindrical coordinates (z):

$z=\frac{Ct}{\left(t+1\right)}$ (3)

Differentiate the equation (1) with respective to time (t),

$\dot{R}=\frac{A}{{\left(t+1\right)}^2}$ (4)

Differentiate the equation (4) with respective to time (t),

$\ddot{R}=\frac{2A}{{\left(t+1\right)}^3}$ (5)

Differentiate the equation (2) with respective to time (t),

$\dot{\theta }=B$ (6)

Differentiate the equation (5) with respective to time (t),

$\ddot{\theta }=0$

Differentiate the equation (3) with respective to time (t),

$\begin{array}{c}\dot{z}=\frac{C\left(t+1\right)-Ct}{{\left(t+1\right)}^2}\\ =\frac{C}{{\left(t+1\right)}^2}\end{array}$ (7)

Differentiate the equation (6) with respective to time (t),

$\ddot{z}=-\frac{2C}{{\left(t+1\right)}^3}$ (8)

Calculate the value (R):

Substitute 0 for t in equation (1).

$\begin{array}{c}R=\frac{A}{\left(0+1\right)}\\ =A\end{array}$

Calculate the value $\left(\dot{R}\right)$:

Substitute 0 for t in equation (4).

$\begin{array}{c}\dot{R}=\frac{A}{{\left(0+1\right)}^2}\\ =A\end{array}$

Calculate the value $\left(\ddot{R}\right)$:

Substitute 0 for t in equation (4).

$\begin{array}{c}\ddot{R}=\frac{2A}{{\left(\left(0\right)+1\right)}^3}\\ =2A\end{array}$

Calculate the value $\left(\theta \right)$.

Substitute 0 for t in equation (2).

$\begin{array}{c}\theta =B\left(0\right)\\ =0\end{array}$

Calculate the value (z):

Substitute 0 for t in equation (3).

$\begin{array}{c}z=\frac{C\left(0\right)}{\left(\left(0\right)+1\right)}\\ =0\end{array}$

Calculate the value $\left(\dot{z}\right)$:

Substitute 0 for t in equation (7).

$\begin{array}{c}\dot{z}=\frac{C}{{\left(\left(0\right)+1\right)}^2}\\ =C\end{array}$

Calculate the value $\left(\ddot{z}\right)$:

Substitute 0 for t in equation (8).

$\begin{array}{c}\ddot{z}=-\frac{2C}{{\left(\left(0\right)+1\right)}^3}\\ =-2C\end{array}$

Write the expression for radial component of velocity $\left(v_r\right)$ as below:

$v_r=\dot{R}$

Substitute A for $\dot{R}$.

$v_r=A$

Write the expression for transverse component of velocity $\left(v_{\theta }\right)$:

$v_{\theta }=R\dot{\theta }$

Substitute A for R and $B$ for $\dot{\theta }$.

$v_{\theta }=AB$

Write the expression for axial component of velocity $\left(v_z\right)$:

$v_z=\dot{z}$

Substitute C for $\dot{z}$.

$v_z=C$

Calculate the magnitude of the velocity (v) using the relation:

$v_0^2=v_r^2+v_{\theta }^2+v_z^2$

Substitute A for $v_r$, AB for $v_{\theta }$ and C for $v_z$.

$\begin{array}{l}{v}_0^2=A^2+{\left(AB\right)}^2+C^2\\ v_0=\sqrt{A^2+A^2{B}^2+C^2}\end{array}$

Write the expression for radial component of acceleration $\left(a_r\right)$:

$a_r=\ddot{R}-R{\dot{\theta }}^2$

Substitute $2A$ for $\ddot{R}$, A for R and B for $\dot{\theta }$.

$\begin{array}{c}{a}_r=2A-A{\left(B\right)}^2\\ =2A-A{B}^2\end{array}$

Write the expression for transverse component of acceleration $\left(a_{\theta }\right)$:

$a_{\theta }=R\ddot{\theta }+2\dot{R}\dot{\theta }$

Substitute A for R, B for $\dot{\theta }$, $-A$ for $\dot{R}$, and 0 for $\ddot{\theta }$.

$\begin{array}{c}{a}_{\theta }=A\left(0\right)+2\left(-A\right)\left(B\right)\\ =-2AB\end{array}$

Write the expression for axial component of acceleration $\left(a_z\right)$:

$a_z=\ddot{z}$

Substitute $-2C$ for $\ddot{z}$.

$a_z=-2C$

Calculate the magnitude of the acceleration (a) using the relation:

$a_0^2=a_r^2+a_{\theta }^2+a_z^2$

Substitute $2A-A{B}^2$ for $a_r$, $-2AB$ for $a_{\theta }$ and $-2C$ for $a_z$.

$\begin{array}{c}{a}_0^2={\left(2A-A{B}^2\right)}^2+{\left(-2AB\right)}^2+{\left(-2C\right)}^2\\ a_0^2=4A^2+A^2{B}^4-4A^2{B}^2+4A^2{B}^2+4C^2\\ a_0^2=4A^2+A^2{B}^4+4C^2\\ a_0=\sqrt{4A^2+A^2{B}^4+4C^2}\end{array}$

Therefore, the magnitudes of the velocity (v) and acceleration (a) when time (t) is 0 are $\underset{\_}{\sqrt{A^2+A^2{B}^2+C^2}}$ and $\underset{\_}{\sqrt{4A^2+A^2{B}^4+4C^2}}$ respectively.

To determine

The magnitudes of the velocity $\left(v_{\infty }\right)$ and acceleration $\left(a_{\infty }\right)$ when time (t) is $\infty$.

Answer to Problem 11.179P

The magnitudes of the velocity $\left(v_{\infty }\right)$ and acceleration $\left(a_{\infty }\right)$ when time (t) is $\infty$ are $\underset{\_}{0}$ and $\underset{\_}{0}$ respectively.

Explanation of Solution

Given Information:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is $\frac{A}{\left(t+1\right)}$, $\left(\theta \right)$ is $Bt$ and (z) is $\frac{Ct}{\left(t+1\right)}$.

Calculation:

Calculate the value (R):

Substitute $\infty$ for t in equation (1).

$\begin{array}{c}R=\frac{A}{\left(\infty +1\right)}\\ =0\end{array}$

Calculate the value $\left(\dot{R}\right)$:

Substitute $\infty$ for t in equation (4).

$\begin{array}{c}\dot{R}=\frac{A}{{\left(\infty +1\right)}^2}\\ =0\end{array}$

Calculate the value $\left(\ddot{R}\right)$:

Substitute $\infty$ for t in equation (4).

$\begin{array}{c}\ddot{R}=\frac{2A}{{\left(\left(\infty \right)+1\right)}^3}\\ =0\end{array}$

Calculate the value $\left(\theta \right)$.

Substitute $\infty$ for t in equation (2).

$\begin{array}{c}\theta =B\left(\infty \right)\\ =\infty \end{array}$

Calculate the value (z):

Substitute $\infty$ for t in equation (3).

$\begin{array}{c}z=\frac{C\left(\infty \right)}{\left(\left(\infty \right)+1\right)}\\ =C\end{array}$

Calculate the value $\left(\dot{z}\right)$:

Substitute $\infty$ for t in equation (7).

$\begin{array}{c}\dot{z}=\frac{C}{{\left(\left(\infty \right)+1\right)}^2}\\ =0\end{array}$

Calculate the value $\left(\ddot{z}\right)$ :

Substitute $\infty$ for t in equation (8).

$\begin{array}{c}\ddot{z}=-\frac{2C}{{\left(\left(\infty \right)+1\right)}^3}\\ =0\end{array}$

Write the expression for radial component of velocity $\left(v_r\right)$:

$v_r=\dot{R}$

Substitute 0 for $\dot{R}$.

$v_r=0$

Write the expression for transverse component of velocity $\left(v_{\theta }\right)$:

$v_{\theta }=R\dot{\theta }$

Substitute 0 for R and $B$ for $\dot{\theta }$.

$\begin{array}{c}{v}_{\theta }=\left(0\right)B\\ =0\end{array}$

Write the expression for axial component of velocity $\left(v_z\right)$:

$v_z=\dot{z}$

Substitute 0 for $\dot{z}$.

$v_z=C$

Calculate the magnitude of the velocity (v) using the relation:

$v^2=v_r^2+v_{\theta }^2+v_z^2$

Substitute 0 for $v_r$, 0 for $v_{\theta }$ and 0 for $v_z$.

$\begin{array}{l}{v}^2={\left(0\right)}^2+{\left(0\right)}^2+{\left(0\right)}^2\\ v=0\end{array}$

Write the expression for radial component of acceleration $\left(a_r\right)$:

$a_r=\ddot{R}-R{\dot{\theta }}^2$

Substitute 0 for $\ddot{R}$, 0 for R and B for $\dot{\theta }$.

$\begin{array}{c}{a}_r=0-0{\left(B\right)}^2\\ =0\end{array}$

Write the expression for transverse component of acceleration $\left(a_{\theta }\right)$:

$a_{\theta }=R\ddot{\theta }+2\dot{R}\dot{\theta }$

Substitute 0 for R, B for $\dot{\theta }$, 0 for $\dot{R}$ and 0 for $\ddot{\theta }$.

$\begin{array}{c}{a}_{\theta }=0+2\left(0\right)\left(B\right)\\ =0\end{array}$

Write the expression for axial component of acceleration $\left(a_z\right)$:

$a_z=\ddot{z}$

Substitute 0 for $\ddot{z}$.

$a_z=0$

Calculate the magnitude of the acceleration (a) using the relation:

$a^2=a_r^2+a_{\theta }^2+a_z^2$

Substitute 0 for $a_r$, 0 for $a_{\theta }$ and 0 for $a_z$.

$\begin{array}{l}{a}^2={\left(0\right)}^2+{\left(0\right)}^2+{\left(0\right)}^2\\ a=0\end{array}$

Therefore, the magnitudes of the velocity $\left(v_{\infty }\right)$ and acceleration $\left(a_{\infty }\right)$ when time (t) is $\infty$ are $\underset{\_}{0}$ and $\underset{\_}{0}$ respectively.