Worked Examples-Calculation Part 1 - Simple Machines

Considered Formula;

$$\text {i)}\quad M.A = \frac {L}{E}$$ $$\qquad \text {Where}~~M.A \to Mechanical~advantage,\quad L \to Load \\\qquad E \to Effort$$ $$\text {ii)}\quad V.R = \frac {Ed}{Ld}$$ $$\qquad \text {Where}~~V.R \to Velocity~ratio,\quad Ed \to Effort~distance \\\qquad Ld \to Load~distance$$ $$\text {iii)}\quad \epsilon = \frac {M.A}{V.R} \times 100%$$ $$\qquad \text {Where}~~\epsilon \to Efficiency,\quad M.A \to Mechanical~advantage \\\qquad V.R \to Velocity~ratio$$

Example 1

A simple machine raises a load of 100N by using a force 50N. Calculate the mechanical advantage.

Solution

Data given

Load (L) = 100 N

Effort (E) = 50 m

Mechanical advantage (M.A) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {L}{E}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {100}{50}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 2$$

The mechanical advantage = 2

Example 2

A force of 20N raises a load of 100kg. Calculate mechanical advantage of the machine.

Solution

Data given

$$\text {Load (L) = Weight} = m \times g = 100 \times 10~N = 1000~N$$

Effort (E) = 20 N

Mechanical advantage (M.A) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {L}{E}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {1000}{20}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 50$$

The mechanical advantage = 50

Example 3

When a machine pressed by effort moved down a distance of 100 cm, while the load is raised through 25 cm at the same time. Find the velocity ratio.

Solution

Data given

Effort distance (Ed) = 100 cm

Load distance (Ld) = 25 cm

Velocity ratio (V.R) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {Ed}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {100}{25}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 4$$

The velocity ratio = 4

Example 4

In a certain machine a force of 10N moves down a distance of 0.05 m in order to raise a load of 100N through a height of 0.5cm calculated the velocity ratio (V.R) of the machine.

Solution

Data given

Effort distance (Ed) = 0.05 m = (0.05 x 100) cm = 5 cm

Load distance (Ld) = 0.5 cm

Velocity ratio (V.R) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {Ed}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {5 \times 10}{0.5 \times 10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {50}{5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 10$$

The velocity ratio = 10

Example 5

In a certain machine a force of 10N moves down a distance of 5 cm in order to raise a load of 80N through a height of 0.5cm, calculated the

i)        M.A

ii)       V.R

iii)     Efficiency of the machine.

Solution

Data given

Effort (E) = 10 N

Load (L) = 80 N

Effort distance (Ed) = 5 cm

Load distance (Ld) = 0.5 cm

Velocity ratio (V.R) =?

i)        Mechanical advantage (M.A) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A= \frac {L}{E}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {80}{10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 8$$

          The mechanical advantage = 8

ii)        Velocity ratio (V.R) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {Ed}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {5}{0.5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {5 \times 10}{0.5 \times 10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {50}{5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 10$$

          The velocity ratio = 10

iii)        Efficiency (ε) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~\epsilon = \frac {M.A}{V.R} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {8}{10} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 80%$$

          The efficiency of the machine = 80%

Example 6

When an effort of 50N is applied through 6 m on a machine, whose efficiency is 75%, such that load is overcome through 1.5 m. calculate the

i)        Velocity ratio

ii)       Mechanical advantage

iii)     Load of which the machine overcome

Solution

Data given

Effort (E) = 50 N

Effort distance (Ed) = 6 cm

Load distance (Ld) = 1.5 cm

Efficiency (ε) = 75%

Mechanical advantage (M.A) = ?

Velocity ratio (V.R) = ?

ii)        Velocity ratio (V.R) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {Ed}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {6}{1.5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {6 \times 10}{1.5 \times 10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {60}{15}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 4$$

          The velocity ratio = 4

iii)        Mechanical advantage (M.A) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\epsilon = \frac {M.A}{V.R} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~75\% = \frac {M.A}{4} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~300\% = 100% \times M.A$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {300 \%}{100 \%}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 3$$

          The Mechanical advantage = 3

iii)        Load (L) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A= \frac {L}{E}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3= \frac {L}{50}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L= 3 \times 50$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 150$$

          The load of which the machine overcome = 150 N