Chapter 10, Problem 66A

(a)

To determine

The force exerted by T.

Answer to Problem 66A

The force exerted by T is $300\text{N}$ .

Explanation of Solution

Given:

The weight of the Piano is $1200\text{N}$ .

The distance up to which the piano is raised is $5\text{m}$ .

The length of the rope pulled by T is $20\text{m}$ .

Formula used:

The efficiency of a machine is the ratio of output work to the input work.

The expression for the efficiency of the machine is

  $\begin{array}{c}\text{Efficiency}=\frac{W_{\text{output}}}{W_{\text{input}}}\\ =\frac{F_{\text{r}}{d}_{\text{r}}}{F_{\text{e}}{d}_{\text{e}}}\end{array}$

Here, $F_{\text{r}}$ and $F_{\text{e}}$ are the resistance force and the effort force, $d_{\text{r}}$ is the displacement of the load, and $d_{\text{e}}$ is the displacement of effort.

Calculation:

Consider T tries to lift the Piano up to a distance of $d_{\text{r}}=5\text{m}$ .

The resistance force exerted by the Piano is $F_{\text{r}}=1200\text{N}$ .

Consider the effort put by T in lifting the piano is denoted by $F_{\text{e}}$ .

The length of the rope pulled by T is $d_{\text{e}}=20\text{m}$ .

The efficiency of the machine is $1$ as the machine is ideal.

The force exerted by Tis,

  $\begin{array}{c}\text{Efficiency}=\frac{F_{\text{r}}{d}_{\text{r}}}{F_{\text{e}}{d}_{\text{e}}}\\ 1=\frac{\left(1200\text{N}\right)\left(5\text{m}\right)}{F_{\text{e}}\left(20\text{m}\right)}\\ F_{\text{e}}=300\text{N}\end{array}$

Conclusion:

Therefore, the force exerted by T is $F_{\text{e}}=300\text{N}$ .

(b)

To determine

The force utilized to balance the friction force.

Answer to Problem 66A

The force utilized to balance the frictional force is $f=40\text{N}$ .

Explanation of Solution

Given:

The weight of the Piano is $1200\text{N}$ .

The distance up to which the piano is raised is $5\text{m}$ .

The length of the rope pulled by T is $20\text{m}$ .

The actual effort force exert by T is $F_{\text{a}}=340\text{N}$ .

Formula used:

For an equilibrium condition, the total force acting on an object is equal to zero.

Calculation:

The actual effort force exerted by T in lifting the Piano is $F_{\text{a}}=340\text{N}$ .

From part (a), the effort exerted by the T in lifting the Piano is $F_{\text{e}}=300\text{N}$ .

The force $\left(f\right)$ utilized to balance the friction force is,

  $\begin{array}{l}f=F_{\text{a}}-F_{\text{e}}\\ f=340\text{N}-300\text{N}\\ f=40\text{N}\end{array}$

Conclusion:

Therefore, the force utilized to balance the frictional force is $f=40\text{N}$ .

(c)

To determine

The output work.

Answer to Problem 66A

The output work done is $6000\text{N}\cdot \text{m}$ .

Explanation of Solution

Given:

The weight of the Piano is $1200\text{N}$ .

The distance up to which the piano is raised is $5\text{m}$ .

The length of the rope pulled by T is $20\text{m}$ .

The actual effort force exert by T is $F_{\text{a}}=340\text{N}$ .

Formula used:

The output work $\left(W_{\text{output}}\right)$ done for a machine is the product of resistance force $\left(F_{\text{r}}\right)$ and the displacement of the load $\left(d_{\text{r}}\right)$ .

Calculation:

The displacement of the load is $d_{\text{r}}=5\text{m}$ .

The resistance force exerted by the Piano is $F_{\text{r}}=1200\text{N}$ .

The output work done for a machine as follows:

  $\begin{array}{c}{W}_{\text{output}}=F_{\text{r}}{d}_{\text{r}}\\ =\left(1200\text{N}\right)\times \left(5\text{m}\right)\\ =6000\text{N}\cdot \text{m}\end{array}$

Conclusion:

Therefore, the output work done is $6000\text{N}\cdot \text{m}$ .

(d)

To determine

The input work done.

Answer to Problem 66A

The input work done is $6000\text{N}\cdot \text{m}$ .

Explanation of Solution

Given:

The weight of the Piano is $1200\text{N}$ .

The distance up to which the piano is raised is $5\text{m}$ .

The length of the rope pulled by T is $20\text{m}$ .

The actual effort force exert by T is $F_{\text{a}}=340\text{N}$ .

Formula used:

The input work $\left(W_{\text{input}}\right)$ done for a machine is the product of effort force $\left(F_{\text{e}}\right)$ and the displacement of the effort $\left(d_{\text{e}}\right)$ .

Calculation:

The effort put by T in lifting the piano is $F_{\text{e}}=300\text{N}$ .

The length of the rope pulled by T is $d_{\text{e}}=20\text{m}$ .

The input work done for a machine as follows:

  $\begin{array}{c}{W}_{\text{input}}=F_{\text{e}}{d}_{\text{e}}\\ =\left(300\text{N}\right)\times \left(20\text{m}\right)\\ =6000\text{N}\cdot \text{m}\end{array}$

Conclusion:

Therefore, the input work done is $6000\text{N}\cdot \text{m}$ .

(e)

To determine

The mechanical advantage of the machine.

Answer to Problem 66A

The mechanical advantage of the machine is $4$ .

Explanation of Solution

Given:

The weight of the Piano is $1200\text{N}$ .

The distance up to which the piano is raised is $5\text{m}$ .

The length of the rope pulled by T is $20\text{m}$ .

The actual effort force exert by T is $F_{\text{a}}=340\text{N}$ .

Formula used:

The mechanical advantage of the machine $\left(MA\right)$ of a machine is the ratio of resistance force $\left(F_{\text{r}}\right)$ to the effort force $\left(F_{\text{e}}\right)$ .

Calculation:

The effort put by T in lifting the piano is $F_{\text{e}}=300\text{N}$ .

The resistance force exerted by the Piano is $F_{\text{r}}=1200\text{N}$ .

The mechanical advantage of the machine as follows:

  $\begin{array}{l}MA=\frac{F_{\text{r}}}{F_{\text{e}}}\\ MA=\frac{1200\text{N}}{300\text{N}}\\ MA=4\end{array}$

Conclusion:

Therefore, the mechanical advantage of the machine is $4$ .