Chapter 6, Problem 13P

(a)

To determine

Find the given equation is dimensionally homogenous or not.

Answer to Problem 13P

The given equation is dimensionally homogenous and it is proved.

Explanation of Solution

Given data:

The equation is,

$F\left(x_2-x_1\right)=\frac{1}{2}m{V}_2^2-\frac{1}{2}m{V}_1^2$ (1)

Here,

$F$ is the force in N,

$m$ is the mass in kg,

$x_1,x_2$ is the distance in $\text{m}$, and

$V_1,V_2$ is the velocity in $\frac{\text{m}}{\text{s}}$,

Formula used:

The SI unit expression in terms of base units as follows,

$\text{N}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for $m$, m for $x_1$, m for $x_2$, $\frac{\text{m}}{\text{s}}$ for $V_1$ and $\frac{\text{m}}{\text{s}}$ for $V_2$ in equation (1).

$\begin{array}{l}\text{N}\left(\text{m}-\text{m}\right)=\frac{1}{2}\left(\text{kg}\right){\left(\frac{\text{m}}{\text{s}}\right)}^2-\frac{1}{2}\left(\text{kg}\right){\left(\frac{\text{m}}{\text{s}}\right)}^2\\ \text{N}\cdot \text{m}=\frac{1}{2}\left(\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}-\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}\right)\\ \text{N}\cdot \text{m}=\frac{1}{2}\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}\end{array}$ (2)

Here,

$\frac{1}{2}$ is constant (unitless).

Equation (2) becomes,

$\text{N}\cdot \text{m}=\left(\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}\right)\cdot \text{m}$ (3)

Substitute the unit $\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$ for $\text{N}$ in equation (3).

$\text{N}\cdot \text{m}=\text{N}\cdot \text{m}$ (4)

From equation (4), Left-hand side (LHS) is equal to Right-hand side (RHS). Thus, the given equation is dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is dimensionally homogenous and it is proved.

(b)

To determine

Find the given equation is dimensionally homogenous or not.

Answer to Problem 13P

The given equation is not dimensionally homogenous and it is proved.

Explanation of Solution

Given data:

The equation is,

$F=\frac{1}{2}m{V}_2^2-\frac{1}{2}m{V}_1^2$ (5)

Here,

$F$ is the force in N,

$m$ is the mass in kg, and

$V_1,V_2$ is the velocity in $\frac{\text{m}}{\text{s}}$.

Formula used:

The SI unit expression in terms of base units as follows,

$\text{N}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for $m$, $\frac{\text{m}}{\text{s}}$ for $V_1$ and $\frac{\text{m}}{\text{s}}$ for $V_2$ in equation (5).

$\begin{array}{c}\text{N}=\frac{1}{2}\left(\text{kg}\right){\left(\frac{\text{m}}{\text{s}}\right)}^2-\frac{1}{2}\left(\text{kg}\right){\left(\frac{\text{m}}{\text{s}}\right)}^2\\ \text{N}=\frac{1}{2}\left(\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}-\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}\right)\\ \text{N}=\frac{1}{2}\frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^{\text{2}}}\end{array}$ (6)

Here,

$\frac{1}{2}$ is constant (unitless).

Equation (6) becomes,

$\text{N}=\left(\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}\right)\cdot \text{m}$ (7)

Substitute the unit $\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$ for $\text{N}$ in equation (7).

$\text{N}=\text{N}\cdot \text{m}$ (8)

From equation (8), Left-hand side (LHS) is not equal to Right-hand side (RHS). Thus, the given equation is not dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is not dimensionally homogenous and it is proved.

(c)

To determine

Find the given equation is dimensionally homogenous or not.

Answer to Problem 13P

The given equation is not dimensionally homogenous and it is proved.

Explanation of Solution

Given data:

The equation is,

$F\left(V_2-V_1\right)=\frac{1}{2}m{x}_2^2-\frac{1}{2}m{x}_1^2$ (9)

Here,

$F$ is the force in N,

$m$ is the mass in kg,

$x_1,x_2$ is the distance in $\text{m}$, and

$V_1,V_2$ is the velocity in $\frac{\text{m}}{\text{s}}$,

Formula used:

The SI unit expression in terms of base units as follows,

$\text{N}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for $m$, m for $x_1$, m for $x_2$, $\frac{\text{m}}{\text{s}}$ for $V_1$ and $\frac{\text{m}}{\text{s}}$ for $V_2$ in equation (9).

$\begin{array}{l}\text{N}\left(\frac{\text{m}}{\text{s}}-\frac{\text{m}}{\text{s}}\right)=\frac{1}{2}\left(\text{kg}\right){\left(\text{m}\right)}^2-\frac{1}{2}\left(\text{kg}\right){\left(\text{m}\right)}^2\\ \text{N}\cdot \frac{\text{m}}{\text{s}}=\frac{1}{2}\left(\text{kg}\cdot {\text{m}}^{\text{2}}-\text{kg}\cdot {\text{m}}^{\text{2}}\right)\\ \text{N}\cdot \frac{\text{m}}{\text{s}}=\frac{1}{2}\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$ (10)

Here,

$\frac{1}{2}$ is constant (unitless).

Equation (10) becomes,

$\text{N}\cdot \frac{\text{m}}{\text{s}}=\text{kg}\cdot {\text{m}}^{\text{2}}$ (11)

Substitute the unit $\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$ for $\text{N}$ in equation (11).

$\begin{array}{l}\left(\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}\right)\cdot \frac{\text{m}}{\text{s}}=\text{kg}\cdot {\text{m}}^{\text{2}}\\ \frac{\text{kg}\cdot {\text{m}}^{\text{2}}}{{\text{s}}^3}=\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$ (12)

From equation (12), Left-hand side (LHS) is not equal to Right-hand side (RHS). Thus, the given equation is not dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is not dimensionally homogenous and it is proved.

(d)

To determine

Find the given equation is dimensionally homogenous or not.

Answer to Problem 13P

The given equation is dimensionally homogenous and it is proved.

Explanation of Solution

Given data:

The equation is,

$F\left(t_2-t_1\right)=m{V}_2-m{V}_1$ (13)

Here,

$F$ is the force in N,

$m$ is the mass in kg,

$t_1,t_2$ is the time in $\text{s}$, and

$V_1,V_2$ is the velocity in $\frac{\text{m}}{\text{s}}$,

Formula used:

The SI unit expression in terms of base units as follows,

$\text{N}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for $m$, s for $t_1$, s for $t_2$, $\frac{\text{m}}{\text{s}}$ for $V_1$ and $\frac{\text{m}}{\text{s}}$ for $V_2$ in equation (13).

$\begin{array}{l}\text{N}\left(\text{s}-\text{s}\right)=\left(\text{kg}\right)\left(\frac{\text{m}}{\text{s}}\right)-\left(\text{kg}\right)\left(\frac{\text{m}}{\text{s}}\right)\\ \text{N}\cdot \text{s}=\frac{\text{kg}\cdot \text{m}}{\text{s}}-\frac{\text{kg}\cdot \text{m}}{\text{s}}\\ \text{N}\cdot \text{s}=\frac{\text{kg}\cdot \text{m}}{\text{s}}\end{array}$ (14)

Substitute the unit $\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}$ for $\text{N}$ in equation (14).

$\begin{array}{l}\text{N}\cdot \text{s}=\frac{\text{kg}\cdot \text{m}}{\text{s}}\\ \left(\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}\right)\cdot \text{s}=\frac{\text{kg}\cdot \text{m}}{\text{s}}\end{array}$

$\frac{\text{kg}\cdot \text{m}}{\text{s}}=\frac{\text{kg}\cdot \text{m}}{\text{s}}$ (15)

From equation (15), Left-hand side (LHS) is equal to Right-hand side (RHS). Thus, the given equation is dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is dimensionally homogenous and it is proved.