Chapter 9, Problem 9P

To determine

The vertical force on each of the supports.

Answer to Problem 9P

Solution: $F_1=2891\text{N}$ downwards and $F_2=1323\text{N}$ downwards.

Explanation of Solution

Explanation:Given: Mass of the horizontal beam $M_1=110\text{kg}$

Mass of the piano $M_2=320\text{kg}$

Formula Used:

System is in equilibrium. Hence,

  ${\displaystyle \sum F=0}$

Calculations:

  

Let length of the beam = l

Weight of the beam $W_1=\left(110\times 9.8\right)\text{N}\text{=}\text{1078}\text{N}$

Weight of piano $W_2=\left(320\times 9.8\right)\text{N}\text{=}\text{3136}\text{N}$

Since system is in equilibrium, ${\displaystyle \sum F=0}$

From the diagram:

  $\begin{array}{l}{F}_1+F_2=W_1+W_2\\ =1078\text{N}+3136\text{N}\text{=}\text{4214}\text{N}\end{array}$

Sum of the moments about left support (clockwise) = Sum of the moments about right support (anti-clockwise)

  $\begin{array}{l}{F}_2\times l=W_2\times l/4+W_1\times l/2\\ \Rightarrow F_2=W_1/2+W_2/4\\ \Rightarrow F_2=\frac{1078\text{N}}{2}+\frac{3136\text{N}}{4}=1323\text{N}\\ F_1=4214\text{N}-1323\text{N}=2891\text{N}\end{array}$

Conclusion: The vertical force on each of the supports is $F_1=2891\text{N}$ downwards and $F_2=1323\text{N}$ downwards.