Question

The system in Fig. 12-38 is in equilibrium. A concrete block of mass$\mathbf{225}\mathbf{}\mathit{k}\mathit{g}$hangs from the end of the uniform strut of mass$\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{}\mathit{k}\mathit{g}$. A cable runs from the ground, over the top of the strut, and down to the block, holding the block in place. For angles$\mathit{\varphi }\mathbf{}\mathbf{=}\mathbf{}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$and$\mathit{\theta }\mathbf{=}\mathbf{}\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{°}$, find (a) the tension Tin the cable and the (b) horizontal and (c) vertical components of the force on the strut from the hinge.

Short answer

a)The tension in the cableis.$6.63\times {10}^{3}N$

b)Horizontalcomponent of the forceis$5.74\times {10}^{3}N$

c) Vertical component of the force is$5.96\times {10}^{3}N$

Step-by-step solution

Listing the given quantities

Theconcreteblock of mass$225kg$

Strut of mass$45.0kg$

$$\begin{gathered} \varphi =30.0° \\ \theta =45.0° \end{gathered}$$

Understanding the concept of the horizontal and vertical componentof the force

In the free body diagram of the given situation, we notice the acting torques on the body both horizontally and vertically.Here, the acting torque results from the applied force along the length of the cable hanging from the strut and along the weight of the block to balance it at the strut.The vertical force acting on the body about the hinge is due to the tension resulting from the hanging to the strut, also the downward pull due to the weight of the block, and the downward pull due to the weight of the strut that acts at the center of the strut. Then using Newton's laws of motion, we balance the acting torques in an equation separately for both the horizontal and the vertical directions. Then, accordingly, calculate the required values as per the problem

Calculation of the tension in the cable

We note that the angle between the cable and the strut is

$\begin{array}{rcl}\alpha & =& \theta -\varphi \\ & =& 45°-30°\\ & =& 15°\\ & & \end{array}$

The angle between the strut and any vertical force(like the weights in the problem) is

$\begin{array}{rcl}\beta & =& 90°-45°\\ & =& 45°\\ & & \end{array}$

Denoting$M=225kgandm=45.0kg$and $\ell$as the length of the boom, we compute torques about the hinge and find

$\begin{array}{rcl}T& =& \frac{Mg\ell sin\beta +mg\frac{\ell }{2}sin\beta }{\ell sin\alpha }\\ & =& \frac{Mgsin\beta +mgsin\frac{\beta }{2}}{sin\alpha }\\ & & \end{array}$

Theunknown lengthcancels out and we obtain$T=6.63\times {10}^{3}N$

The tension in the cable is.$6.63\times {10}^{3}N$

 Calculation of thehorizontal component of the force

(b)

Since the cable is at$30°$from horizontal, then horizontal equilibrium of forces requires that the horizontal hinge force be

$\begin{array}{rcl}Fx& =& Tcos30°\\ & =& 5.74\times {10}^{3}N\\ & & \end{array}$

Horizontal component of the force is$5.74\times {10}^{3}N$

Calculation of thevertical component of the force

(c)

Vertical equilibrium of forces gives the vertical hinge force component

$\begin{array}{rcl}Fy& =& Mg+mg+Tsin30°\\ & =& 5.96\times {10}^{3}N\\ & & \end{array}$

Vertical component of the force is$5.96\times {10}^{3}N$