Q92P - University Physics with Modern Physics

Question

For a physics lab experiment, four classmates run up the stairs from the basement to the top floor of their physics building—a vertical distance of 16.0 m. The classmates and their masses are: Tatiana, 50.2 kg; Bill, 68.2 kg; Ricardo, 81.8 kg; and Melanie, 59.1 kg. The time it takes each of them is shown in Fig. P6.92. (a) considering only the work done against gravity, which person had the largest average power output? The smallest? (b) Chang is very fit and has mass 62.3 kg. If his average power output is 1.00 hp, how many seconds does it take him to run up the stairs?

Step-by-Step Solution

Answer

1. Ricardo has the largest average power output, and Melanie has the smallest average power output.
2. The required time is 13.1 s.

Step 1: Identify the given data

The vertical height is $h=16 m$.

Mass of Tatiana is $m_T=50.2 kg$.

Mass of Bill is $m_B=68.2 \mathrm{kg}$.

Mass of Ricardo is $m_R=81.8 \mathrm{kg}$.

Mass of Melanie is $m_M=59.1 \mathrm{kg}$.

 

From the diagram,

Time taken by Tatiana is $T_T=31 \mathrm{s}$.

Time taken by Bill is $T_B=50 \mathrm{s}$.

Time taken by Ricardo is $T_R=45 \mathrm{s}$.

Time taken by Melanie is $T_M=53 \mathrm{s}$.

Step 2: Concept/Significance of power, and work done

The average amount of work done or energy converted per unit time is known as average power output that is given by,

${\mathit{P}}_{\mathbf{a}\mathbf{v}\mathbf{g}}\mathbf{=}\frac{\mathbf{W}}{\mathbf{T}}$       ...........(1) 

Here, W is work done, and T is time.

 

The work done against the gravity to reach height h is given by,

$\mathit{W}\mathbf{=}\mathit{m}\mathit{g}\mathit{h}$       ..........(2) 

Here, m is mass, and g is acceleration due to gravity.

Step 3: Determine the person that has person largest and smallest average power output (a)

From equations (1) and (2),

                                 $P_{\mathrm{avg}}=\frac{mgh}{\mathrm{T}}$      ..........(3)        

Therefore, the required time is 3.64 hr.

 

Find the average power output for Tatiana as follows.

                                    $$\begin{gathered} P_{{\mathrm{avg}}_{\mathrm{T}}}=\frac{m_{T}gh}{{\mathrm{T}}_{\mathrm{T}}} \\ =\frac{\left(50.2 kg\right)\left(9.8 m/s^2\right)\left(16 m\right)}{31 s} \\ =253.9 \mathrm{watt} \end{gathered}$$

 

Find the average power output for Bill as follows.

                                   $$\begin{gathered} P_{{\mathrm{avg}}_B}=\frac{m_{B}gh}{{\mathrm{T}}_B} \\ =\frac{\left(68.2 kg\right)\left(9.8 m/s^2\right)\left(16 m\right)}{50 s} \\ =213.89 \mathrm{watt} \end{gathered}$$

 

Find the average power output for Ricardo as follows.

                                         $$\begin{gathered} P_{{\mathrm{avg}}_R}=\frac{m_{R}gh}{{\mathrm{T}}_R} \\ =\frac{\left(81.8 kg\right)\left(9.8 m/s^2\right)\left(16 m\right)}{45 s} \\ =285 \mathrm{watt} \end{gathered}$$

 

Find the average power output for Melanie as follows.

                                    $$\begin{gathered} P_{{\mathrm{avg}}_M}=\frac{m_{M}gh}{{\mathrm{T}}_M} \\ =\frac{\left(59.1 kg\right)\left(9.8 m/s^2\right)\left(16 m\right)}{53 s} \\ =175.85 \mathrm{watt} \end{gathered}$$

 

From the above calculations, Ricardo has the largest average power output of $P_{{\mathrm{avg}}_R}=285 \mathrm{watt}$, and Melanie has the smallest average power output of $P_{{\mathrm{avg}}_M}=175.85 \mathrm{watt}$.

 

Therefore, Ricardo has the largest average power output, and Melanie has the smallest average power output

Step 4: Determine the time taken by the Chang to run up the stairs (b)

It is given that Chang is very fit, and has a mass of $m_C=62.3 \mathrm{kg}$and his average power output is $P_{av{g}_C}=1 \mathrm{hp}$ or $P_{{\mathrm{avg}}_C}=746 \mathrm{watt}$.

 

From equation (3),

                                       $$\begin{gathered} T_C=\frac{m_{C}gh}{P_{av{g}_C}} \\ =\frac{\left(62.3 \mathrm{kg}\right)\left(9.8 \mathrm{m}/{\mathrm{s}}^2\right)\left(16 \mathrm{m}\right)}{746 \mathrm{watt}} \\ =13.1 s \end{gathered}$$      

 

Therefore, it will take 13.1 s to run up the stairs.