Chapter 10, Problem 72A

a.

To determine

The force parallel to the ramp $\left(F_A\right)$ required.

Answer to Problem 72A

  $61.25\text{N}$

Explanation of Solution

Given:

The height of the ramp, $h=4.5\text{m}$

The length of the ramp, $l=18\text{m}$

The mass of the box, $m=25\text{kg}$

Formula used:

Work-energy theorem,

  $W=\Delta E$

Where $\Delta E$ is the change in the energy.

And, the work done is given by,

  $W=Fd\cos \theta$

Where $F$ is the force done, $d$ is the distance, and $\theta$ is the angle between the distance and force.

Calculation:

Then, combining above two formulas,

  $\begin{array}{c}{F}_{A}l\cos 0°=mgh\\ F_{A}l=mgh\end{array}$

Using the given values in above,

  $\begin{array}{c}{F}_{A}l=mgh\\ F_A\left(18\right)=\left(25\right)\left(9.8\right)\left(4.5\right)\\ F_A=61.25\text{N}\end{array}$

Conclusion:

Thus, the needed force $F_A$ is $61.25\text{N}$ .

b.

To determine

The IMA of the ramp.

Answer to Problem 72A

  $4$

Explanation of Solution

Given:

The height of the ramp, $h=d_r=4.5\text{m}$

The length of the ramp, $l=d_e=18\text{m}$

The mass of the box, $m=25\text{kg}$

Formula used:

The idea mechanical advantage is the ratio of the distances moved,

  $IMA=\frac{d_e}{d_r}$

Where $d_e$ is the displacement by the effort force, and $d_r$ is the displacement of the load.

Calculation:

Using the given values, IMA of the ramp can be calculated as,

  $\begin{array}{c}IMA=\frac{d_e}{d_r}\\ =\frac{18}{4.5}\\ =4\end{array}$

Conclusion:

Thus, the IMA of the ramp is $4$ .

c.

To determine

The actual MA and the efficiency of the ramp if a $75\text{}\text{N}$ parallel force is needed.

Answer to Problem 72A

  $3.267$ and $81.675\%$ , respectively.

Explanation of Solution

Given:

The mass of the box, $m=25\text{kg}$

The actual force needed, $F_e=75\text{N}$

From part (a), $IMA=4$

Formula used:

The mechanical advantage of a system is given by,

  $MA=\frac{F_r}{F_e}$

Where $F_r$ is the resistance force, and $F_e$ is the effort force.

And, the efficiency of a machine is the ratio of mechanical advantage and ideal mechanical advantages,

  $e=\frac{MA}{IMA}\times 100$

Calculation:

The weight of the box will act as a resistance force, thus

  $F_r=mg$

Using the given values, the actual MA can be calculated as,

  $\begin{array}{c}MA=\frac{F_r}{F_e}\\ =\frac{mg}{75}\\ =\frac{\left(25\right)\left(9.8\right)}{75}\\ =3.267\end{array}$

And, then the efficiency of the ramp will be,

  $\begin{array}{c}e=\frac{MA}{IMA}\times 100\\ =\left(\frac{3.267}{4}\times 100\right)\%\\ =81.675\%\end{array}$

Conclusion:

Thus, the actual MA and the efficiency of the ramp is $3.267$ and $81.675\%$ , respectively.