Chapter 10.2, Problem 28PP

a.

To determine

The ideal mechanical advantage of the given machine.

Answer to Problem 28PP

  $6.0$

Explanation of Solution

Given:

The radius of the crank, $r_e=45\text{cm}$

The radius of the drum, $r_r=7.5\text{cm}$

Number of revolutions undertaken by crank, $n=1$ to rotate the drum through one revolution.

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 in above formula, the ideal mechanical advantage of the given machine will be,

  $\begin{array}{c}IMA=\frac{n\times 2\pi r_e}{2\pi r_r}\\ =\frac{1\times 45}{7.5}\\ =6.0\end{array}$

Conclusion:

Thus, the ideal mechanical advantage of the machine is $6.0$ .

b.

To determine

The amount of force to be exerted on the handle of the crank to exert $750\text{}\text{N}$ of force on the rope.

Answer to Problem 28PP

  $1.67\times {10}^4\text{}\text{N}$

Explanation of Solution

Given:

The radius of the crank, $r_e=45\text{cm}$

The radius of the drum, $r_r=7.5\text{cm}$

Number of revolutions undertaken by crank, $n=1$ to rotate the drum through one revolution.

The force needed on the rope, $F_r=750\text{}\text{N}$

The efficiency, $e=75\%$

Formula used:

The efficiency of a machine is the ratio of real and idea mechanical advantages,

  $e=\frac{MA}{IMA}\times 100=\frac{F_r{d}_r}{F_e{d}_e}\times 100$

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

Calculation:

Using the given values in above formula, the force to be exerted on the handle of the crank will be,

  $\begin{array}{c}e=\frac{F_r\left(2\pi r_r\right)}{F_e\left(2\pi r_e\right)}\times 100\\ 0.75=\frac{\left(750\right)\left(7.5\right)}{F_e\left(45\right)}\times 100\\ F_e=1.67\times {10}^4\text{}\text{N}\end{array}$

Conclusion:

Thus, the amount of force to be exerted on the handle of the crank is $1.67\times {10}^4\text{}\text{N}$ .