Question

Figure (a) shows a horizontal uniform beam of mass${\mathbf{m}}_{\mathbf{b}}$and length$\mathbf{L}$that is supported on the left by a hinge attached to a wall and on theright by a cable at angle $\mathbf{\theta }$ with the horizontal. A package of mass ${\mathbf{m}}_{\mathbf{p}}$ is positioned on the beam at a distance $\mathbf{x}$ from the left end. The total mass is${\mathbf{m}}_{\mathbf{b}}\mathbf{+}{\mathbf{m}}_{\mathbf{p}}\mathbf{=}\mathbf{61.22}\mathbf{\text{kg}}$. Figure (b) gives the tension $\mathbf{\text{T}}$ in the cable as a function of the package’s position given as a fraction $\mathbf{x}\mathbf{/}\mathbf{L}$ of the beam length. The scale of the $\mathbf{T}$ axis is set by ${\mathbf{T}}_{\mathbf{a}}\mathbf{=}\mathbf{500}\mathbf{\text{N}}$ and ${\mathit{T}}_{\mathbf{b}}\mathbf{=}\mathbf{700}\mathbf{\text{N}}$.

(a) Evaluate angle$\mathbf{\text{θ}}$ ,

(b) Evaluate mass${\mathbf{\text{m}}}_{\mathbf{b}}$ , and

(c) Evaluate mass ${\mathbf{\text{m}}}_{\mathbf{p}}$.

Short answer

1. Angle is${30.02}^0$
2. Mass of bar is $51\mathrm{kg}$
3. Mass of package is $10.2kg$

Step-by-step solution

Listing the given quantities       

${\mathrm{M}}_{\mathrm{b}}+{\mathrm{M}}_{\mathrm{p}}=61.22\mathrm{kg}$

Understanding the concept of force, tension and torque

By using the equilibrium condition along a vertical direction, we can write the equation in terms of tension T. Then compare that equation with the standard equation of line. From that, we can write the slope and y intercept. Once we know the slope and y intercept, then we can find the mass of rod, package and angle.

Equations:

$\sum {\mathrm{F}}_{\mathrm{x}}=0$

$\sum {\mathrm{F}}_{\mathrm{y}}=0$

$\sum \mathrm{\tau }=0$

Free Body Diagram

(a) Calculation of Angle (b) Mass of bar (c) Mass of package

Using condition of equilibrium, we can write
$\sum {\mathrm{F}}_{\mathrm{y}}=0$ (i)

Consider point A as pivot point. We have weight of the beam and weight of the package acting downwards at distance$\mathrm{L}/2$and $\mathrm{x}$ respective. The torque caused by these two forces would be balanced by the vertical component of the tension. Using equation for the torque, we have

$\mathrm{TL}\text{sin}\mathrm{\theta }=\frac{{\mathrm{M}}_{\mathrm{b}}\mathrm{gL}}{2}+{\mathrm{M}}_{\mathrm{p}}\mathrm{gx}$ (ii)

Now divide both sides of the equation (ii) by $\mathrm{Lsin\theta }$

$\begin{array}{c}\mathrm{T}=\frac{{\mathrm{M}}_{\mathrm{b}}\mathrm{gL}}{2\text{sin}\mathrm{\theta L}}+\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{gx}}{\mathrm{L}\text{sin}\mathrm{\theta }}\\ =\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{gx}}{\mathrm{L}\text{sin}\mathrm{\theta }}+\frac{{\mathrm{M}}_{\mathrm{b}}\mathrm{g}}{2\text{sin}\mathrm{\theta }}\end{array}$

We can write the slope as,

role="math" localid="1663335112681" $\mathrm{Slope}=\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{\text{sin}\mathrm{\theta }}$ (iii)

Therefore, the equation for $\text{T}$ can be written as,

$\mathrm{T}=\mathrm{slope}\left(\frac{\mathrm{X}}{\mathrm{L}}\right)+\frac{{\mathrm{M}}_{\mathrm{b}}\mathrm{g}}{2\text{sin}\mathrm{\theta }}$

Now, the slope can be calculated from the graph as,

$\begin{array}{c}\mathrm{Slope}=\frac{700-500}{1}\\ =200\end{array}$

Substitute the value of slope in equation (iii), we get

$200=\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{\text{sin}\mathrm{\theta }}$

$\text{sin}\mathrm{\theta }=\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}$ (iv)

The y intercept on the graph is written as,

$\mathrm{y}\mathrm{intercept}=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\text{sin}\mathrm{\theta }}$

(v)

But from the graph,$\mathrm{y}\mathrm{intercept}=500$ . Substitute the value in equation (v)

$500=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\text{sin}\mathrm{\theta }}$

Simplifying this equation, we get

$\text{sin}\mathrm{\theta }=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\times 500}$ (vi)

Equating equation (iv) and (vi), we get

$\begin{array}{c}\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\times 500}\\ {\mathrm{M}}_{\mathrm{b}}=\text{5}{\mathrm{M}}_{\mathrm{p}}\end{array}$

But we know

$\begin{array}{c}{\mathrm{M}}_{\mathrm{b}}+{\mathrm{M}}_{\mathrm{p}}=61.22\mathrm{kg}\\ 5{\mathrm{M}}_{\mathrm{p}}+{\mathrm{M}}_{\mathrm{p}}=61.22\text{kg}\\ {\mathrm{M}}_{\mathrm{p}}=10.2\text{kg}\end{array}$

Therefore, the mass of the package is $10.2\mathrm{kg}$.

We can calculate the mass of bar as,

$\begin{array}{c}{\mathrm{M}}_{\mathrm{b}}+{\mathrm{M}}_{\mathrm{p}}=61.22\mathrm{kg}\\ {\mathrm{M}}_{\mathrm{b}}=61.22\text{kg}-{\mathrm{M}}_{\mathrm{p}}\\ =61.22\text{kg}-10.2\text{kg}\\ \text{=51kg}\end{array}$

Therefore, the mass of bar is $51\mathrm{kg}$.

Now, use equation (vi) to calculate the angle.

$\begin{array}{c}\text{in}\mathrm{\theta }=\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}\\ \mathrm{\theta }={\text{sin}}^{-1}\left(\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}\right)\\ ={\text{sin}}^{-1}\left(\frac{10.2\text{kg}\times 9.81\text{m}/{\text{s}}^2}{200}\right)\\ =30.02°\end{array}$

Therefore, the angle is $30.02°$.

(b) Calculation of Mass of bar

Equating equation (iv) and (vi), we get

$\begin{array}{c}\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\times 500}\\ {\mathrm{M}}_{\mathrm{b}}=\text{5}{\mathrm{M}}_{\mathrm{p}}\\ {\mathrm{M}}_{\mathrm{p}}=\frac{{\mathrm{M}}_{\mathrm{b}}}{5}\end{array}$

But we know

$\begin{array}{c}{\mathrm{M}}_{\mathrm{b}}+{\mathrm{M}}_{\mathrm{p}}=61.22\mathrm{kg}\\ {\mathrm{M}}_{\mathrm{b}}+\frac{{\mathrm{M}}_{\mathrm{b}}}{5}=61.22\text{kg}\\ {\mathrm{M}}_{\mathrm{b}}=51\text{kg}\end{array}$

Therefore, the mass of the bar is $51\mathrm{kg}$.

(c) Calculation of Mass of package

Equating equation (iv) and (vi), we get

$\begin{array}{c}\frac{{\mathrm{M}}_{\mathrm{p}}\mathrm{g}}{200}=\frac{\left({\mathrm{M}}_{\mathrm{b}}\mathrm{g}\right)}{2\times 500}\\ {\mathrm{M}}_{\mathrm{b}}=\text{5}{\mathrm{M}}_{\mathrm{p}}\end{array}$

But we know

$\begin{array}{c}{\mathrm{M}}_{\mathrm{b}}+{\mathrm{M}}_{\mathrm{p}}=61.22\mathrm{kg}\\ 5{\mathrm{M}}_{\mathrm{p}}+{\mathrm{M}}_{\mathrm{p}}=61.22\text{kg}\\ {\mathrm{M}}_{\mathrm{p}}=10.2\text{kg}\end{array}$

Therefore, the mass of the package is $10.2\mathrm{kg}$.