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Engineering Mathematics Through Applications

Kuldeep Singh

Chapter 5

Logarithmic, Exponential and Hyperbolic Functions - all with Video Answers

Educators

Section 1

Indices revisited

02:14

Problem 1

Do [thermodynamics] A gas with an initial volume $V_{1}=0.3 \mathrm{~m}^{3}$ is compressed from $P_{1}=100 \mathrm{kPa}$ to $P_{2}=350 \mathrm{kPa} .$
From the formula
$$
P_{1}\left(V_{1}\right)^{5 / 3}=P_{2}\left(V_{2}\right)^{5 / 3}
$$
evaluate the final volume $V_{2}$. (Pascal, $\mathrm{Pa}=\mathrm{N} / \mathrm{m}^{2}$, is the named SI unit of pressure.)

Mukesh Devi
Mukesh Devi
Numerade Educator
02:14

Problem 2

[W] [thermodynamics] A gas in a cylinder with an initial volume $V_{1}=0.25 \mathrm{~m}^{3}$ and pressure $P_{1}=100 \mathrm{kPa}$ is compressed to a final pressure $P_{2}=450 \mathrm{kPa}$. Given that
$$
P_{1}\left(V_{1}\right)^{1.33}=P_{2}\left(V_{2}\right)^{1.33}
$$
determine the final volume $V_{2}$.

Mukesh Devi
Mukesh Devi
Numerade Educator
02:14

Problem 3

[D_[thermodynamics] A volume, $0.1 \mathrm{~m}^{3}$, of gas at a pressure of $200 \mathrm{kN} / \mathrm{m}^{2}$ is compressed to a pressure of $632 \mathrm{kN} / \mathrm{m}^{2}$. If the gas obeys the law
$$
P_{1} V_{1}{ }^{1.5}=P_{2} V_{2}{ }^{1.5}
$$
then find the new volume.

Mukesh Devi
Mukesh Devi
Numerade Educator
01:02

Problem 4

|D|[thermodynamics] For an ideal gas, the $P V T$ equations are given by
$$
P_{1}\left(V_{1}\right)^{k}=P_{2}\left(V_{2}\right)^{k} \text { and } \frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}
$$
where $P, V, T$ are pressure, volume, temperature respectively and $k$ is a constant. Show that
$\mathbf{i} \frac{T_{2}}{T_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{k-1}$
ii $\frac{T_{1}}{T_{2}}=\left(\frac{V_{2}}{V_{1}}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
01:04

Problem 5

Show that $x^{\frac{1}{\Psi}}\left(y^{\frac{\psi-1}{\psi}}-x^{\frac{\psi-1}{\psi}}\right)$ can be simplified to $x\left[\left(\frac{y}{x}\right)^{\frac{\psi-1}{\psi}}-1\right]$.

Aman Gupta
Aman Gupta
Numerade Educator
03:07

Problem 6

[fluid mechanics] The velocity, $v$, of a fluid in a channel of slope $s$ and radius $r$ is defined as $v=\frac{r^{1 / 6}}{\eta} \sqrt{r s}$ where $\eta$ is a constant. Show that
$$
v=\frac{r^{2 / 3} s^{1 / 2}}{\eta}
$$

Narayan Hari
Narayan Hari
Numerade Educator