Machines Notes

Machines

Syllabus Overview

• The chapter covers machines as force multipliers, including concepts like load, effort, mechanical advantage, velocity ratio, and efficiency.
• It also discusses simple machines like levers and pulley systems.

3.1 Machines

• Machines make tasks easier by:
  • Lifting heavy loads with less effort (force multiplier).
    • Examples: jack, bar, spade, pulleys, wheelbarrow.
  • Changing the point of application of effort.
    • Example: bicycle pedals transferring effort to the rear wheel.
  • Changing the direction of effort.
    • Example: using a pulley to lift a bucket of water by applying downward effort.
  • Obtaining a gain in speed (greater movement of load with smaller effort).
    • Examples: scissors, knife.
• Definition of a machine:
  • A device that overcomes a large resistive force (load) by applying a small force (effort) or obtains a gain in speed.

3.2 Technical Terms Related to a Machine

• Load (L):
  • The resistive force to be overcome by a machine.
• Effort (E):
  • The force applied to the machine to overcome the load.
• Mechanical Advantage (M.A.):
  • The ratio of load to effort.
  • $M.A. = \frac{Load (L)}{Effort (E)}$ (3.1)
  • It has no unit.
  • If M.A. > 1: Effort needed is less than the load.
  • If M.A. < 1: Effort needed is greater than the load, gain in speed.
  • If M.A. = 1: Generally used to change the direction of the effort with no gain in force or speed.
• Velocity Ratio (V.R.):
  • The ratio of the velocity of effort to the velocity of load.
  • $V.R. = \frac{Velocity of effort (V<em>E)}{Velocity of load (V</em>L)}$
  • If $d<em>E$ and $d</em>L$ are the distances moved by the effort and the load respectively in the same time t, then:
    • $V<em>L = \frac{d</em>L}{t}$
    • $V<em>E = \frac{d</em>E}{t}$
  • $V.R. = \frac{\frac{d<em>E}{t}}{\frac{d</em>L}{t}} = \frac{d<em>E}{d</em>L}$ (3.2)
  • $V.R. = \frac{d<em>E}{d</em>L}$
  • It has no unit.
  • A machine cannot be used as a force multiplier and speed multiplier simultaneously.
  • If V.R. > 1: Machine acts as a force multiplier.
  • If V.R. < 1: Machine gives gain in speed.
  • If V.R. = 1: Machine changes the direction of effort.
• Work Input:
  • Work done on the machine by the effort.
  • $Work input = work done by the effort$ (3.4)
• Work Output:
  • Work done by the machine on the load.
  • $Work output = work done on the load$ (3.5)
• Efficiency $(\eta)$
  • The ratio of work output to work input.
  • $\text{Efficiency } \eta = \frac{\text{Work output (}W<em>{\text{output}})}{\text{Work input (}W</em>{\text{input}})}$
  • $\text{Efficiency } \eta = \frac{\text{Work output (}W<em>{\text{output}})}{\text{Work input (}W</em>{\text{input}})} \times 100\%$
  • It has no unit.

3.3 Principle of a Machine

• Ideal Machine:
  • No energy loss during operation.
  • $Output energy = Input energy$
  • Efficiency is 100%.
• Actual Machine:
  • Energy loss occurs due to:
    • Moving parts not weightless or smooth.
    • String not perfectly elastic.
    • Parts not perfectly rigid.
  • The most prominent loss is in overcoming friction, which appears as heat.
• When energy is supplied to a machine by applying effort, it overcomes the load by doing some useful work on it.
• If a machine is 80% efficient, 80% of the total energy supplied is obtained as useful energy, the rest of the 20% is lost due to friction which appears as heat energy.

3.4 Relationship between Efficiency $(\eta)$, Mechanical Advantage (M.A.) and Velocity Ratio (V.R.)

• Suppose a machine overcomes a load L by the application of an effort E, in time t. Let the displacement of effort be $d<em>E$ and the displacement of load be $d</em>L$
• Work input = Effort x displacement of effort = $E \times d_E$
• Work output = Load x displacement of load = $L \times d_L$
• $\text{Efficiency } \eta = \frac{\text{work output}}{\text{work input}}$
• $\eta = \frac{L \times d<em>L}{E \times d</em>E} = \frac{L}{E} \times \frac{d<em>L}{d</em>E}$
• $\frac{L}{E} = M.A.$
• $\frac{d<em>E}{d</em>L} = V.R.$
• $\eta = \frac{M.A.}{V.R.}$ (3.8)
• $M.A. = V.R. \times \eta$
• For an ideal machine, M.A. = V.R.
• In actual practice, M.A. < V.R. because n < 1

3.5 Levers

• Simplest kind of machines.
• A rigid, straight (or bent) bar that turns about a fixed axis (fulcrum).
• Principle of moments:
  • At equilibrium, the moment of load about the fulcrum equals the moment of effort about the fulcrum.
  • $Load \times load :arm = Effort \times effort :arm$
  • $L \times FB = E \times FA$
  • $\frac{L}{E} = \frac{FA}{FB}$ (3.9)
  • $M.A = \frac{Effort : arm : FA}{Load : arm : FB}$ (3.10)
• The mechanical advantage of a lever is equal to the ratio of the length of its effort arm to the length of its load arm.
• If effort arm = load arm, M.A. = 1.
• If effort arm > load arm, M.A. > 1.
• If effort arm < load arm, M.A. < 1.
• M.A. can be increased by increasing effort arm or decreasing load arm (shifting the fulcrum towards the load).

3.6 Kinds of Levers

• Three types of levers based on the relative positions of effort, load, and fulcrum.

(1) Class I Levers

• Fulcrum is between the effort and the load.
• Effort and load are on opposite sides of the fulcrum, acting in the same direction while producing rotation in opposite sense.
• Can act as a force multiplier.
• Examples: seesaw, scissors, crowbar, handle of water pump, claw hammer, pliers, spoon used to open lid, spade, catapult, nodding of the human head.
• M.A. can be greater than 1, equal to 1, or less than 1.

(2) Class II Levers

• Fulcrum and effort are at the two ends of the lever, and the load is between them.
• Effort arm is always longer than the load arm.
• M.A. is always greater than 1.
• Always acts as a force multiplier.
• Examples: nutcracker, bottle opener, wheelbarrow, lemon crusher, paper cutter, mango cutter, bar used to lift a load, door, raising the weight of the human body on toes.

(3) Class III Levers

• Fulcrum and load are at the two ends, and the effort is between them.
• Effort arm is always smaller than the load arm.
• M.A. is always less than 1.
• Gain in speed.
• Examples: sugar tongs, forearm lifting a load (or action of the biceps muscle), fire tongs, foot treadle, knife, spade used to lift coal (or soil), fishing rod.

3.7 Examples of Each Class of Levers as Found in the Human Body

• Muscles exert force (effort) by contraction.

(1) Class I Lever in the Action of Nodding of Head

• Fulcrum at the spine, load at the front part of the head, and effort at the rear part.
• This is an example of class I lever.

(2) Class II Lever in Raising the Weight of the Body on Toes

• Fulcrum at the toes, load (weight of the body) in the middle, and effort by muscles at the other end.
• This is an example of class II lever.

(3) Class III Lever in Raising a Load by Forearm

• Elbow joint acts as the fulcrum, effort from biceps in the middle, and load on the palm at the other end.
• This is an example of class III lever.