Chapter 27, Problem 8P

To determine

(a) To determine:

Gravitational potential energy of the person on the first step.

Answer to Problem 8P

Solution:

Gravitational potential energy of the person on the first step is $121.64\text{J}$.

Explanation of Solution

Given:

Length of the step of a flight stair, $h=20\text{cm}=\text{0}\text{.2}\text{m}$

Mass of the person, $m=62\text{kg}$

Formula used:

The gravitational potential energy is given by the formula

$PE=mgh$

Here,

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height of stair

Calculation:

The gravitational potential energy is given by the formula

$PE=mgh$

Plugging the values in the above equation

$\begin{array}{c}PE=62\times 9.81\times 0.2\\ =121.64\text{J}\end{array}$

To determine

(b) To determine:

Gravitational potential energy of the person on the second step.

Answer to Problem 8P

Solution:

Gravitational potential energy of the person on the second step is $243.288\text{J}$.

Explanation of Solution

Given:

Length of the step of a flight stair, $h=20\text{cm}=\text{0}\text{.2}\text{m}$

Mass of the person, $m=62\text{kg}$

Formula used:

The gravitational potential energy is given by the formula

$PE=mgh$

Here,

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height of stair

Calculation:

The gravitational potential energy is given by the formula

$PE=mgh$

On the second step of the stairs, the vertical distance of the person from the ground is

$\begin{array}{c}h\text{'}=h+h\\ =0.2+0.2\\ =0.4\text{m}\end{array}$

Plugging the values in the above equation

$\begin{array}{c}PE=62\times 9.81\times 0.4\\ =243.288\text{J}\end{array}$

To determine

(c) To determine:

Gravitational potential energy of the person on the third step.

Answer to Problem 8P

Solution:

Gravitational potential energy of the person on the third step is $364.93\text{J}$.

Explanation of Solution

Given:

Length of the step of a flight stair, $h=20\text{cm}=\text{0}\text{.2}\text{m}$

Mass of the person, $m=62\text{kg}$

Formula used:

The gravitational potential energy is given by the formula

$PE=mgh$

Here,

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height of stair

Calculation:

The gravitational potential energy is given by the formula

$PE=mgh$

On the third step of the stairs, the vertical distance of the person from the ground is

$\begin{array}{c}h\text{'}=h+h+h\\ =0.2+0.2+0.2\\ =0.6\text{m}\end{array}$

Plugging the values in the above equation

$\begin{array}{c}PE=62\times 9.81\times 0.6\\ =364.932\text{J}\end{array}$

To determine

(d) To determine:

Gravitational potential energy of the person on the n^th step.

Answer to Problem 8P

Solution:

Gravitational potential energy of the person on the n^th step is $121.64n\text{J}$.

Explanation of Solution

Given:

Length of the step of a flight stair, $h=20\text{cm}=\text{0}\text{.2}\text{m}$

Mass of the person, $m=62\text{kg}$

Formula used:

The gravitational potential energy is given by the formula

$PE=mgh$

Here,

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height of stair

Calculation:

The gravitational potential energy is given by the formula

$PE=mgh$

On the n^th step of the stairs, the vertical distance of the person from the ground is

$\begin{array}{c}h\text{'}=h+h+h+\mathrm{...}n\\ =nh\\ =n\times 0.2\text{m}\end{array}$

Plugging the values in the above equation

$\begin{array}{c}PE=62\times 9.81\times n\times 0.2\\ =121.64n\text{J}\end{array}$

To determine

(e) To determine:

Change in energy of the person when he descends from step 6 to step 2.

Answer to Problem 8P

Solution:

Change in energy of the person is $486.6\text{J}$.

Explanation of Solution

Given:

Length of the step of a flight stair, $h=20\text{cm}=\text{0}\text{.2}\text{m}$

Mass of the person, $m=62\text{kg}$

Formula used:

The gravitational potential energy is given by the formula

$PE=nmgh$

Here,

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height of stair

$n$ is the step number form the ground

Calculation:

The energy of the person on the $6^{\text{th}}$ step is calculated as

$\begin{array}{c}{E}_6=6\times 62\times 9.81\times 0.2\\ =729.8\text{J}\end{array}$ …… (1)

The energy of the person on the $2^{\text{nd}}$ step is calculated as

$\begin{array}{c}{E}_2=2\times 62\times 9.81\times 0.2\\ =243.2\text{J}\end{array}$ …… (2)

The change in energy is calculated as

$\begin{array}{c}\Delta E=E_6-E_2\\ =729.8-243.2\\ =486.6\text{J}\end{array}$