Question

Figure 5-66a shows a mobile hanging from a ceiling; it consists of two metal pieces(${\mathit{m}}_{\mathit{1}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{kg}\mathbf{}\mathbf{and}\mathbf{}{\mathit{m}}_{\mathit{2}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{kg}$) that are strung together by cords of negligible mass. What is the tension in (a) the bottom cord and (b) the top cord? Figure 5-66b shows a mobile consisting of three metal pieces. Two of the masses are${\mathit{m}}_{\mathit{3}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{8}\mathbf{kg}\mathbf{}\mathbf{and}\mathbf{}{\mathit{m}}_{\mathit{5}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{kg}$. The tension in the top cord is 199 N. What is the tension in (c) the lowest cord and (d) the middle cord?

Short answer

a)$T_2=44\mathrm{N}$

b)$T_1=78\mathrm{N}$

c)$T_5=54\mathrm{N}$

d)$T_4=152\mathrm{N}$

Step-by-step solution

Given Data

$$\begin{gathered} m_1=3.5\mathrm{kg} \\ {m}_2=4.5\mathrm{kg} \\ {m}_3=4.8\mathrm{kg} \\ {m}_5=5.5\mathrm{kg} \end{gathered}$$

The tension in the top cord is 199 N.

Understanding the concept

In this problem we have mobiles consisting of masses connected by cords. We apply Newton’s second law to calculate the tensions in the cords.

Draw the free body diagrams and write the force equations

The free-body diagrams for$m_1$and$m_2$for part (a) are shown to the right.

The bottom cord is only supporting $m_2=4.5\mathrm{kg}$against gravity, so its tension is $T_2=m_{2}g$.

On the other hand, the top cord is supporting a total mass of

$$\begin{gathered} m_1+m_2=\left(3.5\mathrm{kg}+4.5\mathrm{kg}\right) \\ =8.0\mathrm{kg} \end{gathered}$$

against gravity. Applying Newton’s second law gives

$T_1-T_2-m_{1}g=0$

So the tension is

$T_1=m_{1}g+T_2=(m_1+m_2)g$

(a) Calculate the tension in the bottom of the cord

From the equations above, we find the tension in the bottom cord to be

$\begin{array}{l}{T}_2=m_{2}g\\ =\left(4.5\mathrm{kg}\right)(9.8\mathrm{m}/{\mathrm{s}}^2)\\ =44\mathrm{N}\end{array}$

Hence the solution is$T_2=44\mathrm{N}$

(b) Calculate the tension in the top of the cord

Similarly, the tension in the top cord is

$$\begin{gathered} T_1=\left(m_1+m_2\right)g \\ =\left(8.0\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right) \\ =78\mathrm{N} \end{gathered}$$

Hence the solution is$T_1=78\mathrm{N}$

(c) Calculate the tension in the lowest cord

The free-body diagrams for$m_3,m_4\mathrm{and}{m}_5$for part (b) are shown below

From the diagram, we see that the lowest cord supports a mass of$m_5=5.5\mathrm{kg}$against gravity and consequently has a tension of

$$\begin{gathered} T_5=m_{5}g \\ =\left(5.5\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right) \\ =54\mathrm{N} \end{gathered}$$

Hence the solution is $T_5=54\mathrm{N}$

(d) Calculate the tension in the middle cord

The top cord, as we are told, has a tension$T_3=199\mathrm{N}$which supports a total of $\left(199\mathrm{N}\right)\left(9.80\mathrm{m}/{\mathrm{s}}^2\right)=20.3\mathrm{kg}$, 10.3 kg of which is already accounted for in the figure. Thus, the unknown mass in the middle must be

$$\begin{gathered} {\mathrm{m}}_4=20.3\mathrm{kg}-10.3\mathrm{kg} \\ =10.0\mathrm{kg} \end{gathered}$$,

and the tension in the cord above it must be enough to support

$$\begin{gathered} m_4+m_5=(10.0\mathrm{kg}+5.50\mathrm{kg}) \\ =15.5\mathrm{kg} \end{gathered}$$

So

$T_4=(15.5\mathrm{kg})\left(9.8{\mathrm{mm}}^2\right)$

Hence the solution is$T_4=152\mathrm{N}$