Worked Examples-Calculation Part 2 - Simple Machines

Considered Formula;

$$\text {i)}\quad V.R = \frac {2\pi R}{P} \quad \Rightarrow For~Screw~and~Jack$$ $$\qquad \text {Where}~~V.R \to Velocity~ratio,\quad R \to Radius \\\qquad P \to Pitch$$ $$\text {ii)}\quad V.R = \frac {R}{r}\quad \Rightarrow For~Wheel~and~Axle$$ $$\qquad \text {Where}~~V.R \to Velocity~ratio,\quad R \to Radius~of~the~wheel \\\qquad r \to Radius~of~the~axle$$ $$\text {iii)}\quad V.R = \frac {R^2}{r^2}\quad \Rightarrow For~Hydraulic~press$$ $$\qquad \text {Where}~~V.R \to Velocity~ratio,\quad R \to Radius~of~large~piston \\\qquad r \to Radius~of~large~piston$$

Example 1

A screw jack has 5 threads per centimeter, if the length of the turning lever is 20 cm. find the velocity ratio (ϖ = 3.14)

Solution

Data given

Radius (R)= 20 cm

$$ \text {Pitch (P)} = 20~cm= \Bigl ( \frac {1}{5} \Bigr )~cm = 0.2~cm$$

Velocity ratio (V.R) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {2\pi R}{P}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {2\times 3.14 \times 20}{0.2}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {125.6 \times 10}{0.2 \times 10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {1256}{2}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 628$$

The Velocity ratio = 628

Example 2

A wheel and axle with an efficiency of 90% is to be raised a load of 10000 N. the radius of the wheel is 40 cm while radius of the axle is 5 cm. find

a)    Velocity ratio,

b)    Mechanical advantage and

c)     Effort

Solution

Data given

Radius of the wheel (R) = 40 cm

Radius of the axle (r) = 5 cm

Load (L) = 10000 N

Efficiency (ε) = 90%

a)        Velocity ratio (V.R) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {R}{r}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {40}{5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 8$$

          The velocity ratio = 8

b)        Mechanical advantage (M.A) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \epsilon = \frac {M.A}{V.R} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~90 \% = \frac {M.A}{8} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {90 \% \times 8}{100 \%}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {720}{100}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 7.2$$

          The Mechanical advantage = 7.2

c)        Effort (E) = ?

$$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {L}{E}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~7.2 = \frac {10000}{E}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~E= \frac {10000}{7.2}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {10000 \times 10}{7.2 \times 10}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {100000}{72}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 1389~N$$

                   The Effort = 1389 N

Example 3

The diagram below shows a hydraulic press being used to lift a container weighting 100000 N

Radii of the effort and load piston are 20 cm and 5 m respectively, if the efficiency of the hydraulic press is 90%. Determine

a)  Velocity ratio,

b)  Mechanical advantage and

c)   Minimum Effort

c)   The distance the container raised through if the effort piston pushed through 1 m

Solution

Data given

Radius of large piston (R) = 5 m = (5 x 100) cm = 500 cm

Radius of small piston (r) = 20 cm

Load (L) = 100000 N

Efficiency (ε) = 90%

a)        Velocity ratio (V.R) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~V.R = \frac {R^2}{r^2}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {500^2}{20^2}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {250000}{400}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 625$$

          The velocity ratio = 625

b)        Mechanical advantage (M.A) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \epsilon = \frac {M.A}{V.R} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~90 \% = \frac {M.A}{625} \times 100 \%$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {90 \% \times 625}{100 \%}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac {56250}{100}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 562.5$$

          The Mechanical advantage = 562.5

c)        Effort (E) = ?

$$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~M.A = \frac {L}{E}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~562.5 = \frac {100000}{E}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~E= \frac {100000}{562.5}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {10000 \times 10}{562.5 \times 10}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1000000}{5625}$$ $$ \qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 178~N$$

                   The Effort = 178 N

b)        Distance moved by load (Ld) =?
    Distance moved by effort (Ed) = 1 m

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~ V.R = \frac {Ed}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~625= \frac {1}{Ld}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ld = \frac {1}{625}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 0.0016~m$$

          The distance raised by the container = 0.0016 m