Chapter 19, Problem 1P

(a)

To determine

The energy equivalent in joules for one jelly doughnut.

Answer to Problem 1P

The energy equivalent in joules for one jelly doughnutis $2.26\times {10}^6\text{J}$.

Explanation of Solution

Given information:

Mass of the woman is $55.0\text{kg}$ and energy in one doughnut is $540\text{Calorie}\left(540\text{kcal}\right)$.

The conversion of $\text{kcal}$ into joules is,

$1\text{kcal}=4.186\times {10}^3\text{J}$

Therefore, the conversion of $540\text{kcal}$ into joules is,

$\begin{array}{c}540\text{kcal}=540\text{kcal}\times \frac{4.186\times {10}^3\text{J}}{1\text{kcal}}\\ =2.26\times {10}^6\text{J}\end{array}$

Conclusion:

Therefore, the energy equivalent in joules for one jelly doughnut is $2.26\times {10}^6\text{J}$.

(b)

To determine

The number of steps woman must climb to change the required potential energy.

Answer to Problem 1P

The number of steps woman must climb to change the required potential energy is $2.8\times {10}^4\text{steps}$.

Explanation of Solution

Given information:

Mass of the woman is $55.0\text{kg}$ and energy in one doughnut is $540\text{Calorie}\left(540\text{kcal}\right)$. The height of a single stair is $15.0\text{cm}$.

Formula to calculate the change in potential energy to climb one step is,

$\Delta U=mgh$

• $h$ is the height of one step.
• $U$ is the gravitational potential energy.
• $m$ is the mass of the woman.
• $g$ is the acceleration due to gravity.

Formula to calculate the total change in potential energy is,

$\Delta U_{\text{total}}=nmgh$

• $n$ is the number of steps.

Rearrange the above equation for $n$.

$n=\frac{\Delta U_{\text{total}}}{mgh}$

Substitute $2.26\times {10}^6\text{J}$ for $\Delta U_{\text{total}}$, $55.0\text{kg}$ for $m$, $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ and $15.0\text{cm}$ for $h$ to find $n$.

$\begin{array}{c}n=\frac{2.26\times {10}^6\text{J}}{55.0\text{kg}\times 9.8\text{m}/{\text{s}}^{\text{2}}\times \left(15.0\text{cm}\times \frac{{10}^{-2}\text{m}}{1\text{cm}}\right)}\\ =2.8\times {10}^4\text{steps}\end{array}$

Conclusion:

Therefore, the number of steps woman must climb to change the required potential energy is $2.8\times {10}^4\text{steps}$.

(c)

To determine

The number of steps woman must climb to work off her breakfast.

Answer to Problem 1P

The number of steps woman must climb to work off her breakfast is $6.99\times {10}^3\text{steps}$.

Explanation of Solution

Given information:

Mass of the woman is $55.0\text{kg}$ and energy in one doughnut is $540\text{Calorie}\left(540\text{kcal}\right)$. The height of a single stair is $15.0\text{cm}$. The efficiency of the human body to convert the energy into mechanical work is $25\%$.

Formula to calculate the mechanical work equivalent of potential energy is,

$W=25\%\times \Delta U_{\text{total}}$

• $W$ is the mechanical work equivalent of potential energy.

Formula to calculate the number of steps must climb to work off her breakfast is,

$n=\frac{W}{mgh}$

Substitute $25\%\times \Delta U_{\text{total}}$ for $W$ in the above equation.

$n=\frac{25\%\times \Delta U_{\text{total}}}{mgh}$

Substitute $2.26\times {10}^6\text{J}$ for $\Delta U_{\text{total}}$, $55.0\text{kg}$ for $m$, $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ and $15.0\text{cm}$ for $h$ to find $n$.

$\begin{array}{c}n=\frac{\frac{25}{100}\times 2.26\times {10}^6\text{J}}{55.0\text{kg}\times 9.8\text{m}/{\text{s}}^{\text{2}}\times \left(15.0\text{cm}\times \frac{{10}^{-2}\text{m}}{1\text{cm}}\right)}\\ =6.99\times {10}^3\text{steps}\end{array}$

Conclusion:

Therefore, the number of steps woman must climb to work off her breakfast is $6.99\times {10}^3\text{steps}$.