Chapter 10, Problem 88A

To determine

The ramp used by person I in order to do least amount of work.

Answer to Problem 88A

Person I can choose any one of the two ramps as the work done by person I in moving the piano using the ramp 1 and ramp 2 are equal.

Explanation of Solution

Given:

The height of the platform is $h=2.0\text{m}$ .

Consider there are two ramp and person I can use one of these.

The length of the ramp 1 is $l_1=3.0\text{m}$ .

The length of the ramp 2 is $l_2=4.0\text{m}$ .

Consider the angle made by the ramp 1 with the horizontal is denoted by ${\theta }_1$ .

Consider the angle made by the ramp 2 with the horizontal is denoted by ${\theta }_2$ .

Formula used:

Consider an object is moved up the ramp, then the work done $\left(W\right)$ by the object is expressed as follows:

$W=\left(F_g\sin \theta \right)d$

Here, $F_g$ is the gravitational force acting on the object, $\theta$ is the angle made by the ramp with the horizontal, $d$ is the length of the ramp.

Calculation:

The two ramps are shown in Figure below.

Consider the work done by person I in moving the piano by using ramp is denoted by $W_1$ .

The work done $W_1$ is,

  $\begin{array}{l}{W}_1=\left(F_g{d}_1\sin {\theta }_1\right)\\ W_1=F_g\times 3\times \left(\frac{2}{3}\right)\\ W_1=2F_g\\ W_1=2mg\end{array}$

Consider the work done by person I in moving the piano by using ramp is denoted by $W_2$ .

The work done $W_2$ is,

  $\begin{array}{l}{W}_2=\left(F_g{d}_2\sin {\theta }_2\right)\\ W_2=F_g\times 4\times \left(\frac{2}{4}\right)\\ W_2=2F_g\\ W_2=2mg\end{array}$

Compare the work done $W_1$ and $W_2$ .The work done by person I in moving the piano to the platform using the ramp 1 is same as the work done by person I in moving the piano to the platform using ramp 2.Hence, person I can choose any one of the two ramps.

Conclusion:

Thus, person I can choose any one of the two ramps.