Q4.6-34PE - College Physics (Urone)

Question

Figure 4.39 shows Superhero and Trusty Sidekick hanging motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty Sidekick’s is 55.0 kg, and the mass of the rope is negligible. 

(a) Draw a free-body diagram of the situation showing all forces acting on Superhero, Trusty Sidekick, and the rope. 

(b) Find the tension in the rope above Superhero. 

(c) Find the tension in the rope between Superhero and Trusty Sidekick. Indicate on your free-body diagram the system of interest used to solve each part.

Step-by-Step Solution

Answer

(b) The tension in the rope above the Superhero is 1421 N.

(c) The tension in the rope in the lower circle is 539 N.

Step 1: Given data

• Superhero’s mass = 90.0 kg.
• Trusty Sidekick’s mass = 55.0 kg.

• The mass of the rope is negligible.

Step 2: (a) Draw the free body diagram showing the forces involved

Here, T’ is the tension in the upper rope above Superhero and T is the tension in the rope between the Superhero and Trusty Sidekick.

Step 3: (b) Determine the tension in the rope above the superhero

In the upper circle of the diagram, take the summation of forces in the vertical direction and equate it to zero as:

 

 $\begin{array}{c}\sum F_y=0\\ \mathbf{T}\mathbf{\text{'}}\mathbf{-}\mathbf{T}\mathbf{-}{\mathbf{w}}_{\mathbf{S}}\mathbf{=}\mathbf{0}\\ \mathbf{T}\mathbf{\text{'}}\mathbf{=}\mathbf{T}\mathbf{+}{\mathbf{w}}_{\mathbf{S}}\end{array}$

Here, w_S is the weight of the superhero.

 

Since the tension in the rope above Trusty Sidekick is due to its weight only, therefore

 

$\begin{array}{c}T\text{'}=w_T+w_S\\ T\text{'}=m_{T}g+m_{S}g\end{array}$

 

Substitute 90 kg for m_S and 55 kg for m_T in the above expression, and we get,

 

$\begin{array}{c}T\text{'}=90\text{\hspace{0.33em}kg}\times 9.8{\text{\hspace{0.33em}m/s}}^{\text{2}}+55\text{\hspace{0.33em}kg}\times 9.8{\text{\hspace{0.33em}m/s}}^{\text{2}}\\ =882\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2+539\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^{\text{2}}\\ =\left(882+539\right)\text{\hspace{0.33em}N}\\ =1421\text{\hspace{0.33em}N}\end{array}$

 

Hence, the tension in the rope above the Superhero is 1421 N.

Step 4: (c) Determine the tension in the rope between the Superhero and the Trusty Sidekick

In the lower circle of the diagram, take the summation of forces in the vertical direction and equate it to zero as:

 

$\begin{array}{c}\mathbf{T}\mathbf{-}{\mathbf{w}}_{\mathbf{T}}\mathbf{=}\mathbf{0}\\ \mathbf{T}\mathbf{=}{\mathbf{m}}_{\mathbf{T}}\mathbf{g}\end{array}$

 

Substitute 55 kg for m_T in the above expression, and we get,

 

$\begin{array}{c}T=55\text{\hspace{0.33em}kg}\times {\text{9.8 m/s}}^2\\ =539\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2\\ =539\text{\hspace{0.33em}N}\end{array}$

 

Hence, the tension in the rope in the lower circle is 539 N.