Comprehensive Notes on Pulleys, Mechanical Advantage, and Simple Machines

Pulley System and Mechanical Advantage

Numerical Problems and Solutions

Question 8: Block and tackle system.
- (a) How many strands support the load?
- (b) Indicate tension in each strand with arrows.
- (c) What is the system's mechanical advantage?
- (d) If the load is pulled up by $1 m$ , how far does the effort end move?
- (e) Effort needed to lift a $100 N$ load?
- Answers: (a) 4 strands, (c) $M.A. = 4$ , (d) Effort moves $4 m$ , (e) Effort needed is $25 N$ .
Question 9: Block and tackle system with velocity ratio 3.
- Diagram needed: Label points of application and directions of load and effort.
- Man exerts $200 kgf$ pull. Efficiency is 60%.
  - (a) Maximum load he can raise?
  - (b) If the effort end moves $60 cm$ , how far does the load move?
- Answers: (a) $360 kgf$ , (b) $20 cm$ .
Question 10: Given 4 pulleys and 3 strings, achieve a mechanical advantage of 8.
- Diagram needed: Show load, effort, and tension directions.
- Assumptions to obtain required mechanical advantage?
- Hint: Three movable pulleys with one fixed pulley.

Single vs. Movable Pulleys

Question 10: When is a single pulley used with a mechanical advantage greater than 1? How to change the force direction without changing the mechanical advantage? Illustrate with a diagram.
Question 11: Velocity ratio of a single movable pulley?
- Effect of friction in the pulley bearing on velocity ratio?
- Answer: Velocity Ratio = 2, No effect from friction.
Question 11: In a single movable pulley, if the effort moves up by a distance $x$ , how much is the load raised?
- Answer: $x/2$
Question 12: Diagram of two pulleys, one fixed and one movable. Mark all forces.
- Ideal mechanical advantage of the system?
- How can it be achieved?

Pulley Arrangement and Tension

Question 13: Pulley arrangement.
- (a) Name pulleys A and B.
- (b) Mark tension direction on each string strand.
- (c) Purpose of pulley B?
- (d) If tension is $T$ , deduce the relation between:
  - (i) $T$ and $E$
  - (ii) $E$ and $L$
- (e) What is the velocity ratio of the arrangement?
- (f) Assuming 100% efficiency, what is the mechanical advantage?
- Answers: (a) A - movable, B - fixed, (d) (i) $T = E$ , (ii) $E = L/2$ , (e) 2, (f) 2
Question 14: Four differences between single fixed and single movable pulleys.
Question 15: Arrangement of three pulleys A, B, and C.
- (a) Name the pulleys A, B, and C.
- (b) Mark the directions of load $L$ , effort $E$ , and tensions $T1$ and $T2$ .
- (c) How are magnitudes of $L$ and $E$ related to tension $T_1$ ?
- (d) Calculate the mechanical advantage and velocity ratio.
- (e) Assumptions made in parts (c) and (d)?
- Answers: (a) A and B - movable pulleys, C - fixed pulley, (c) $L = 4T1$ , $E = T1$ , (d) $M.A. = 4$ , $V.R. = 4$ , (e) Weightless pulleys A and B, no friction.

Combination Pulleys

Question 16: Diagram of three movable and one fixed pulley to lift a load.
- Show load, effort, and tension directions.
- Find mechanical advantage, velocity ratio, and efficiency (ideal situation).
- Answers: (i) $2^3$ (ii) $2^3$ (iii) 1.
Question 17: Block and tackle system diagram with velocity ratio of 5.
- Indicate application points and directions of load and effort. Mark tension.
Question 18: Reasons for the following:
- (a) Single fixed pulley: the velocity ratio > mechanical advantage.
- (b) Movable pulley efficiency < 100%.
- (c) Block and tackle: mechanical advantage increases with number of pulleys.
- (d) Lower block of block and tackle: negligible weight.
Question 19: Name a machine used to:
- (a) Multiply force
- (b) Multiply speed
- (c) Change force direction
- Answers: (a) movable pulley (b) class III lever (c) single fixed pulley
Question 20: True or false:
- (a) Single fixed pulley: the velocity ratio > 1.
- (b) Single movable pulley: the velocity ratio is always 2.
- (c) Combination of $n$ movable pulleys with a fixed pulley: velocity ratio is always $2^n$ .
- (d) Block and tackle: velocity ratio = number of tackle strands supporting load.
- Answers: (a) F (b) T (c) T (d) T

Multiple Choice Questions

Question 1: Single fixed pulley is used because it:
- (a) Has mechanical advantage > 1
- (b) Has velocity ratio < 1
- (c) Gives 100% efficiency
- (d) Helps apply effort in a convenient direction
- Answer: (d)
Question 2: Mechanical advantage of an ideal single movable pulley:
- (a) 1 (b) 2 (c) < 2 (d) < 1
- Answer: (b)
Question 3: A movable pulley is used as:
- (a) a force multiplier (b) a speed multiplier (c) a device to change the direction of effort (d) an energy multiplier
- Answer: (a)

Numerical Problems

Problem 1: Woman draws water using a fixed pulley; mass of bucket + water = $6 kg$ . The applied force = $70 N$ . Calculate mechanical advantage. (Take $g = 10 m/s^2$ )
- Answer: 0.857
Problem 2: Fixed pulley driven by $100 kg$ mass falling at $8.0 m$ in $4.0 s$ . Lifts a load of $75.0 kgf$ .
- (a) Power input to the pulley (gravity on $1 kg$ is $10 N$ ).
- (b) Efficiency of the pulley.
- (c) Height to which load is raised in $4.0 s$ .
- Answers: (a) $2000 W$ , (b) 0.75, (c) $8.0 m$
Problem 3: Block and tackle with 3 pulleys; a load of $75 kgf$ is raised with an effort of $25 kgf$ .
- Find mechanical advantage, velocity ratio, and efficiency.
- Answers: (i) 3, (ii) 3, (iii) 100%
Problem 4: Block and tackle with 5 pulleys. Effort of $1000 N$ needed to raise load of $4500 N$ .
- Calculate mechanical advantage, velocity ratio, and efficiency.
- Answers: (a) 4.5 (b) 5 (c) 90%
Problem 5: Tackle to lift load applying downward force (diagram related).
- (a) Mark load and effort direction.
- (b) If load raised by $1 m$ , through what distance does the effort move?
- (c) Number of tackle strands supporting the load.
- (d) Mechanical advantage of the system?
- Answers: (b) $5 m$ , (c) 5, (d) 5
Problem 6: Pulley system: velocity ratio 3, efficiency 80%. Calculate:
- (a) Mechanical advantage
- (b) Effort needed to raise a load of $300 N$
- Answers: (a) 2.4, (b) $125 N$
Problem 7: System of four pulleys, upper two fixed, lower two movable.
- (a) Draw string around pulleys, show application point and effort direction.
- (b) What is the system velocity ratio?
- (c) How are load and effort related?
- (d) What assumption do you make in arriving at your answer in part (c)?
- Answers: (b) 4, (c) $load = 4 \times effort$ . Assumptions: (1) No friction, (2) Weight of lower block negligible, (3) Effort applied downwards

Effect of Pulley Weight

Equation $(3.27)$ : $V.R. = n$
Let $w$ = total weight of lower block + pulleys.
In balanced position: $E = T$ and $L + w = nT$
$L = nT - w = nE - w$
Equation $(3.26)$ : $M.A. = \frac{L}{E} = n - \frac{w}{E}$
Mechanical advantage is less than ideal value $n$ .
Velocity ratio remains $n$ .
Equation $(3.28)$ : $\eta = 1 - \frac{w}{nE}$
Efficiency is reduced due to weight of lower block; more weight, less efficiency. Pulleys should be light, and friction minimized with lubricants.

Example: Fixed Pulley

Lifting a $400 N$ load through $5 m$ in $10 s$ with $480 N$ effort.
- (a) What is the velocity ratio?
- (b) What is the mechanical advantage?
- (c) Calculate the efficiency.
- (d) Why is the efficiency not 100%?
- (e) What is the energy gained by the load in $10 s$ ?
- (f) How much power was developed by the boy in raising the load?
- (g) Justification for using the pulley if the effort is greater than the load?
Answers:
- (a) $V.R. = 1$
- (b) $M.A. = \frac{400}{480} = 0.833$
- (c) $\eta = \frac{0.833}{1} = 0.833 \approx 83.3\%$
- (d) Energy wasted overcoming friction.
- (e) Energy gained = $400 \times 5 = 2000 J$
- (f) Power developed = $\frac{480 \times 5}{10} = 240 W$
- (g) Changes direction of force, allowing user to use own weight.

Combination Pulleys Example

Pulley combination with two pulleys $P1$ and $P2$ to lift load $W$ .
- (a) State the kind of pulleys $P1$ and $P2$ .
- (b) State the function of pulley $P_1$ .
- (c) If free end $C$ moves through $x$ , by what distance is $W$ raised?
- (d) What effort $E$ is needed to raise $W = 20 kgf$ (neglect pulley weight and friction)?
Answers:
- (a) $P1$ movable, $P2$ fixed.
- (b) $P_1$ changes effort direction.
- (c) $W$ rises by $x/2$ .
- (d) $E = 10 kgf$

Block and Tackle with Four Pulleys

Block and tackle has two pulleys in each block, tackle tied to the hook of the lower block, and effort applied upwards.
- (a) Draw a diagram and calculate mechanical advantage.
- (b) If the load moves up a distance $x$ , by what distance will the free end of the string move up?
Answers:
- (a) Diagram, $M.A. = 5$
- (b) $5x$
Note: Effort applied upwards means that mechanical advantage and velocity ratio are more than the number of pulleys used.
It equals the number of strands supporting the load.

Pulley System Example with Velocity Ratio Or 4

Lifts a load of $150 kgf$ through $20 m$ , effort required is $50 kgf$ .
- (a) Distance moved by the effort.
- (b) Work done by the effort.
- (c) Mechanical advantage.
- (d) Efficiency of the pulley system.
- (e) Total number of pulleys and pulleys in each block.
Given $V.R. = 4$ , $L = 150 kgf$ , $d_L = 20 m$ , $E = 50 kgf = 500 N$ .
Answers:
- (a) $d_E = 80 m$
- (b) $W = 40000 J$
- (c) $M.A. = 3$
- (d) $75 \%$ .
- (e) Four total: two in each block

Exercise Questions

What is a fixed pulley? State one use.
What is the ideal mechanical advantage of a single fixed pulley? Can it be used as a force multiplier?
Name the pulley with no gain in mechanical advantage. Why is it used?
What is the velocity ratio of a single fixed pulley?
In a single fixed pulley, if the effort moves down $x$ , how high is the load raised?
What is a single movable pulley? What is its mechanical advantage in the ideal case?
Name the type of single pulley with an ideal mechanical advantage of 2. Draw a labeled diagram.
Give two reasons why the efficiency of a single movable pulley is not 100%.
In which direction should the force be applied for a single movable pulley?

Pulleys and Mechanical Advantage Combination

Derivation of Mechanical Advantage:
- Effort: $E = T_1$ (i)
- Load Segments Supported: Load $L$ supported by two segments over Pulley $A$ , so $2T1 = L$ or $T1 = L/2$ (ii)
- Similarly, Tensions: Supporting strings over Pulley $B$ implies: $2T2 = T1$ or $T2 = T1/2$ (III) through $C$ : $2T3 = T2$ or $T3 = T2/2$ (iv)
- Using (iv): $L = 2^3 \times T_3$ (v)
- Mechanical Advantage:
 \begin{align} M.A. &= \frac{Load \; L}{Effort \; E} = \frac{2^3 \times T3}{T3} = 2^3
 \end{align} (vii)
Generalization: For $n$ movable pulleys, the mechanical advantage is $2^n$ . (3.19)
Velocity Ratio:
- If the load attached to pulley $A$ moves a distance $x$ , i.e., $d_L = x$ , pulleys increase as $2 \times x \simeq dx$ , the end of the string passing mover the fixed pulley $D$ as $2 \times 2^2x = 2^3x$ , or $dE = 2^3x$
- Velocity Ratio Formula:
  \begin{align} V.R. &= \frac{dE}{dL} = \frac{2^3x}{x} = 2^3 \end{align} (viii)
Generalization: If there are $n$ movable pulleys connected to a fixed pulleys, then

\begin{align} V.R = 2^n \end{align}

(3.20)

Efficiency:
\begin{align} \frac{M.A.}{V.R.} \times 100\% = \frac{2^3}{2^3}= 1 \simeq 100 \% \end{align} (3.21)
Note: The efficiency of this arrangement is 100% only in an ideal situation. In actual practice both(1) the weight of the pulleys and string, and(2) the friction between the bearings of the pulleys, reduce the mechanical advantage and so the efficiency becomes less than 100%.

BLOCK AND TACKLE SYSTEM EXPLANATION

Pulley systems where two blocks of pulleys are used. One block (upper) having several pulleys is attached to a rigid support (i.e., fixed) and the other block (lower) having several pulleys is movable. This arrangement is called the block and tackle system.

The number of pulleys used in the movable lower block is either equal to or one less than the number of pulleys in the fixed upper block.
A strong inextensible string (or rope) of negligible mass passes around all pulleys. One end attaches to the hook if the number of pulleys in the upper block is more. If not, then in both blocks the pulley number is equal to apply the effort in the downward direction
Thus in a block and tackle system, the effort gets multiplied n times, where n is the total number of pulleys in the system. It therefore acts as a force multiplier.
Mechanical advantage : In Fig. 3.25, the tension in the five segments of string supports the load L. Therefore, L = 5T and E = T and M.A.= number n