Name: Lever: Principle of Moments and Mechanical Advantage
Uploaded: 2019-07-19
Duration: 6 min 29 s
Description: Engineers use a lever to magnify the force applied to an object. On what principle a lever works, and how to calculate the mechanical advantage of a lever? L...

Question

Draw a diagram of a combination of three movable pulleys with a fixed pulley showing the directions of load, effort, and tension in each strand. Find (i) mechanical advantage, (ii) velocity ratio, and (iii) efficiency of the combination in an ideal situation.

Accepted Answer

Diagram :

(i) In equilibrium,
Effort $E = T_{3} - - - (1)$
Tension $T_{1}$ in the string passing over the pulley A is given as $2 T_{1} = L$ .
$T_{1} = \frac{L}{2} - - - - - (2)$
Tension $T_{2}$ in the string passing over the pulley B is given as
$2 T_{2} = T_{1} T_{2} = \frac{T_{1}}{2}$
Substituting the value of $T_{1}$ from equation $2$ ,
$T_{2} = \frac{L}{2^{2}} - - - - - (3)$
Tension $T_{3}$ in the string passing over the pulley C is given as
$2 T_{3} = T_{2} T_{3} = \frac{T_{2}}{2}$
Substituting the value of $T_{2}$ from equation $3$ ,
$T_{3} = \frac{L}{2^{3}} - - - - - (4)$
In equilibrium, $T_{3} = E$
From the above equation
$E = \frac{L}{2^{3}} - - - - - - (5)$
We know that mechanical advantage, $M . A = \frac{L}{E}$
$M . A = \frac{L}{(\frac{L}{2^{3}})} = 2^{3}$

(ii) As we know, one end of each string passing over a movable pulley is fixed, so the free end of the string moves up twice the distance moved by the movable pulley.
If the load $L$ attached to the pulley A moves a distance $d$ ,
then $d_{L} = d$ .
Now, the string connected to the axle of pulley B, moves up by a distance $2 d$ . Then the string connected to the axle of the pulley C, moves up by a distance $2^{2} d$ .
Then the end of the string passing over the fixed pulley D, moves up by a distance $2^{3} d$ .
Hence, $d_{E} = 2^{3} d$ . As we know, velocity ratio is given by,
$V . R = \frac{Distance moved by effort}{Distance moved by load} = \frac{d_{E}}{d_{L}}$
Therefore, by substituting the values in the above formula we get, $V . R = \frac{2^{3} \times d}{d} = 2^{3}$ .

(iii) The efficiency of the combination is given by,
$η = \frac{M . A}{V . R} = \frac{2^{3}}{2^{3}} = 1$ .

Draw a diagram of a combination of three movable pulleys with a fixed pulley showing the directions of load, effort, and tension in each strand. Find (i) mechanical advantage, (ii) velocity ratio, and (iii) efficiency of the combination in an ideal situation.

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