Maharashtra SSC Workshop Science & Calculation ITI cheat sheet

A fraction a/b is proper when a<b, improper when a≥b; convert to a mixed number by dividing: 17/5 = 3 remainder 2 = 3 2/5.

To add or subtract fractions, take the LCM of the denominators: 3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 1 7/12.

Division of fractions: multiply by the reciprocal, a/b ÷ c/d = a/b × d/c; simplify by cancelling common factors before multiplying.

Decimal to fraction: place digits over the matching power of ten and reduce - 0.375 = 375/1000 = 3/8; a repeating decimal such as 0.333… equals 1/3.

Percentage means per hundred: x% = x/100. Find x% of N as (x/100)×N; express a/b as a percentage by (a/b)×100.

Percentage change = (new − old)/old × 100; a rise then equal fall does not cancel - 100 +10% −10% gives 99, not 100.

Ratio a:b compares like quantities in the same unit; the total of n parts splits as a/(a+b) and b/(a+b) of the amount.

To compare ratios, cross-multiply or convert to decimals: 3:4 vs 5:7 → 21 vs 20, so 3:4 is larger.

A proportion a:b = c:d satisfies ad = bc; use this to find an unknown term in a direct or inverse proportion.

Circle: circumference = 2πr = πd; area = πr² = πd²/4. Use π ≈ 3.1416 or 22/7 for quick workshop estimates.

Triangle: area = ½ × base × height; for sides a, b, c use Heron's formula with s = (a+b+c)/2 and area = √[s(s−a)(s−b)(s−c)].

Rectangle: area = l × b, perimeter = 2(l+b); parallelogram area = base × perpendicular height; trapezium area = ½(a+b)h.

Sector of a circle: arc length = (θ/360)×2πr; sector area = (θ/360)×πr², where θ is the central angle in degrees.

Cylinder: curved surface = 2πrh; total surface = 2πr(r+h); volume = πr²h.

Cone: slant height l = √(r²+h²); curved surface = πrl; total surface = πr(l+r); volume = ⅓πr²h.

Sphere: surface area = 4πr²; volume = (4/3)πr³. Hemisphere total surface = 3πr².

Prism and pyramid: prism volume = base area × height; pyramid volume = ⅓ × base area × height.

Cube: volume = a³, total surface = 6a², diagonal of a face = a√2, body diagonal = a√3.

Right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; tan θ = sin θ/cos θ.

Pythagoras: hypotenuse² = sum of squares of the other two sides; use it to find a slant height or a diagonal before applying ratios.

Standard values: sin 30° = 0.5, cos 30° = 0.866, tan 30° = 0.577; sin 45° = cos 45° = 0.707, tan 45° = 1; sin 60° = 0.866, tan 60° = 1.732.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ. Use these to find one ratio when another is known.

Taper and included angle: half-angle tan α = (D − d)/(2L); included angle = 2α.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A - used for non-right triangles such as dovetail and V-block layouts.

Angle of elevation and depression are measured from the horizontal; the same triangle ratios apply to heights and distances in the shop.

Chord and sagitta: for a circular arc, radius R = (c²/4 + h²)/(2h), where c is chord length and h is sagitta.

Convert degrees to radians by multiplying by π/180; 1 radian ≈ 57.3°, and 180° = π radians.

Density ρ = mass/volume; mass = ρ × V. Steel ≈ 7850 kg/m³ (7.85 g/cm³), aluminium ≈ 2700 kg/m³, copper ≈ 8900 kg/m³.

Weight is a force: W = m × g, with g ≈ 9.81 m/s². A 10 kg mass weighs about 98.1 N.

Work = force × distance moved in the direction of the force (W = F × s); unit is the joule (N·m).

Power = work/time = (force × distance)/time; unit is the watt. 1 kW = 1000 W; 1 metric horsepower ≈ 735.5 W.

Kinetic energy = ½mv²; potential energy = mgh. Energy is conserved and measured in joules.

Mechanical advantage = load/effort; velocity ratio = distance moved by effort/distance moved by load; efficiency = MA/VR × 100%.

Torque = force × perpendicular radius (T = F × r), measured in N·m; it differs from work because no displacement along the force occurs.

Pressure = force/area, in N/m² (pascal); 1 bar = 100 kPa ≈ 1.02 kgf/cm².

Moment of a force = force × perpendicular distance from the pivot; for balance, clockwise moments = anticlockwise moments.

Cutting speed V = πDN/1000 (m/min) when D is in mm and N in rpm; rearrange to N = 1000V/(πD).

Turning time per pass = L/(f × N), where L is the length of cut in mm and f is feed in mm/rev; add approach and overtravel to L.

Milling: table feed = fz × z × N (mm/min), where fz is feed per tooth and z the number of teeth.

Drilling time = (depth + 0.3D)/(f × N) for a standard 118° drill point; the 0.3D term is the point allowance.

Tapping: spindle speed N = 1000V/(πD) using the tap's major diameter; feed rate equals pitch (mm/rev).

Carbide tools tolerate roughly 3–6 times the cutting speed of HSS on the same material; always check the tool and work material chart.

For a given surface speed, doubling the diameter halves the rpm - smaller work or tools run faster.

Depth of cut = (D₁ − D₂)/2 for a turning pass reducing diameter from D₁ to D₂; it is measured radially, not on diameter.

Unit check: V in m/min, D in mm, N in rev/min - the factor 1000 converts mm to m.

Gear ratio = driven teeth/driver teeth = speed of driver/speed of driven; for a train, multiply each stage's ratio.

Module m = pitch circle diameter/number of teeth (m = PCD/z); circular pitch p = πm; addendum = m, dedendum = 1.25m (standard).

Two meshing gears must have the same module and pressure angle; centre distance = m(z₁ + z₂)/2.

Metric thread: pitch is the axial distance between crests; lead = pitch × number of starts. For single-start, lead = pitch.

Thread depth for ISO metric ≈ 0.6134 × pitch; tap drill size ≈ major diameter − pitch.

Taper: taper ratio = (D − d)/L; half-angle tan α = (D − d)/(2L); included angle = 2α.

Morse tapers are self-holding because of their small included angle (roughly 2–3°); they transmit torque by friction.

Clearance fit: hole is always larger than shaft (e.g. H8/c7). Interference fit: shaft always larger (e.g. H7/r6). Transition fit may give either.

Fit designation: capital letter = hole, small letter = shaft; H is the basic hole whose lower deviation is zero.

Heat is energy in transit, measured in joules; temperature is the degree of hotness, measured in °C, K or °F.

Convert: °F = (9/5)°C + 32; °C = (5/9)(°F − 32); K = °C + 273.15.

Sensible heat Q = m × c × ΔT; specific heat of water ≈ 4.18 kJ/kg·K, steel ≈ 0.49 kJ/kg·K.

Latent heat of fusion of ice ≈ 334 kJ/kg; of vaporisation of water ≈ 2260 kJ/kg - absorbed at constant temperature.

Thermal expansion: ΔL = L × α × ΔT; steel α ≈ 11–12 × 10⁻⁶ per K.

Ohm's law: V = I × R; power P = V × I = I²R = V²/R.

Series: R total = R₁ + R₂ + …; current is the same, voltage divides. Parallel: 1/R = 1/R₁ + 1/R₂ + …; voltage is the same, current divides.

Electrical energy = P × t; in kWh, P(kW) × t(h). One unit of electricity = 1 kWh = 3.6 MJ.

Conductors have low resistance (copper, aluminium); insulators have high resistance (rubber, mica, porcelain).

Ferrous metals contain iron: cast iron, mild steel, high-carbon steel, alloy steels. Non-ferrous: copper, aluminium, brass, bronze, zinc, lead.

Mild steel: about 0.15–0.3% carbon, tensile strength ≈ 400–500 N/mm², weldable and machinable - the general workshop default.

Carbon content raises hardness and tensile strength but lowers ductility and weldability; above about 0.6% C the steel is hard and brittle.

Cast iron: 2–4% carbon, excellent compressive strength and damping, brittle in tension; grey cast iron is used for machine beds.

Brass = copper + zinc; bronze = copper + tin. Bronze has better wear resistance and is used for bearings and bushes.

Aluminium: density ≈ 2700 kg/m³ (about one-third of steel), good conductivity, forms a protective oxide layer; soft, so it is alloyed for structural use.

Hardness tests: Brinell (ball, HB), Vickers (pyramid, HV), Rockwell (cone or ball, HRC/HRB).

Heat treatments: annealing softens and relieves stress; normalising refines grain; hardening (quench) increases hardness; tempering restores toughness after hardening.

Case hardening adds carbon to the surface of low-carbon steel, giving a hard skin with a tough core.

Lever: load × load arm = effort × effort arm. Class I has the fulcrum between load and effort; Class II has the load between; Class III has the effort between.

Pulley systems: a single fixed pulley gives MA = 1; a movable pulley gives MA = 2; a block and tackle with n rope parts supports MA ≈ n.

Inclined plane: MA = length of incline/height; screw jack: MA = 2πr/pitch, where r is the handle radius.

Efficiency = (output work/input work) × 100 = (MA/VR) × 100; friction always makes efficiency below 100%.

Newton's laws: a body stays at rest or in uniform motion unless a net force acts; F = ma; every action has an equal and opposite reaction.

Friction force F = μN, where μ is the coefficient of friction and N the normal reaction; it opposes relative sliding.

Threads: a bolt and nut is a helical inclined plane; a right-hand thread tightens clockwise viewed from the head.

Fastener types: through bolt, tap bolt, stud, machine screw, self-tapping screw, rivet, key, pin and circlip - each resists a specific load type.

A key transmits torque between shaft and hub; a circlip locates parts axially; a rivet gives a permanent joint.

Current is measured in amperes (A) with an ammeter in series; voltage in volts (V) with a voltmeter in parallel.

A multimeter selects function and range; for an unknown value, start on the highest range and step down.

A clamp meter reads current without breaking the circuit by sensing the magnetic field around a conductor.

A Vernier caliper: least count = 1 main-scale division − 1 Vernier division. For 0.02 mm, 49 main divisions equal 50 Vernier divisions.

A micrometer: least count = pitch/number of thimble divisions, typically 0.5 mm/50 = 0.01 mm.

A dial gauge (comparator) measures small variations and runout; it must be zeroed against a standard.

Magnetism: like poles repel, unlike attract; a magnetic field exists around a current-carrying conductor (right-hand grip rule).

Electromagnetic induction: a changing magnetic flux through a coil induces an emf (Faraday); a rotating coil in a magnetic field generates AC.

A fuse and an MCB protect a circuit by opening it on excess current; earthing connects the metal body to earth for safety.

Personal protective equipment: safety goggles for grinding and chipping, gloves for handling rough stock (not for rotating spindles), safety shoes, ear protection above about 85 dB.

Machine guards must be in place before starting; never remove a guard or reach across a rotating part.

Before any maintenance or tool change, isolate the machine and apply lockout/tagout; verify that the spindle has stopped.

A file cuts on the forward stroke only; a file without a handle must never be used because the tang can pierce the hand.

A hacksaw blade is fitted with teeth pointing forward; about 40–50 strokes per minute on mild steel.

A chisel has a cutting angle of about 60° for mild steel and 70° for cast iron; a sharper 30–40° edge suits softer metals.

Fire classes: A solid combustibles (water), B flammable liquids (foam, CO₂), C gases (dry powder), D metals (special dry powder), E electrical (CO₂, never water).

Waste segregation: metal swarf, oily rags, coolant and electronic waste go to separate streams; used oil is never poured into drains.

5S practice: sort, set in order, shine, standardise, sustain - housekeeping that prevents accidents and improves quality.

A fraction a/b is proper when a<b, improper when a≥b; convert to a mixed number by dividing: 17/5 = 3 remainder 2 = 3 2/5.

To add or subtract fractions, take the LCM of the denominators: 3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 1 7/12.

Division of fractions: multiply by the reciprocal, a/b ÷ c/d = a/b × d/c; simplify by cancelling common factors before multiplying.

Decimal to fraction: place digits over the matching power of ten and reduce - 0.375 = 375/1000 = 3/8; a repeating decimal such as 0.333… equals 1/3.

Percentage means per hundred: x% = x/100. Find x% of N as (x/100)×N; express a/b as a percentage by (a/b)×100.

Percentage change = (new − old)/old × 100; a rise then equal fall does not cancel - 100 +10% −10% gives 99, not 100.

Ratio a:b compares like quantities in the same unit; the total of n parts splits as a/(a+b) and b/(a+b) of the amount.

To compare ratios, cross-multiply or convert to decimals: 3:4 vs 5:7 → 21 vs 20, so 3:4 is larger.

A proportion a:b = c:d satisfies ad = bc; use this to find an unknown term in a direct or inverse proportion.

Circle: circumference = 2πr = πd; area = πr² = πd²/4. Use π ≈ 3.1416 or 22/7 for quick workshop estimates.

Triangle: area = ½ × base × height; for sides a, b, c use Heron's formula with s = (a+b+c)/2 and area = √[s(s−a)(s−b)(s−c)].

Rectangle: area = l × b, perimeter = 2(l+b); parallelogram area = base × perpendicular height; trapezium area = ½(a+b)h.

Sector of a circle: arc length = (θ/360)×2πr; sector area = (θ/360)×πr², where θ is the central angle in degrees.

Cylinder: curved surface = 2πrh; total surface = 2πr(r+h); volume = πr²h.

Cone: slant height l = √(r²+h²); curved surface = πrl; total surface = πr(l+r); volume = ⅓πr²h.

Sphere: surface area = 4πr²; volume = (4/3)πr³. Hemisphere total surface = 3πr².

Prism and pyramid: prism volume = base area × height; pyramid volume = ⅓ × base area × height.

Cube: volume = a³, total surface = 6a², diagonal of a face = a√2, body diagonal = a√3.

Right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; tan θ = sin θ/cos θ.

Pythagoras: hypotenuse² = sum of squares of the other two sides; use it to find a slant height or a diagonal before applying ratios.

Standard values: sin 30° = 0.5, cos 30° = 0.866, tan 30° = 0.577; sin 45° = cos 45° = 0.707, tan 45° = 1; sin 60° = 0.866, tan 60° = 1.732.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ. Use these to find one ratio when another is known.

Taper and included angle: half-angle tan α = (D − d)/(2L); included angle = 2α.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A - used for non-right triangles such as dovetail and V-block layouts.

Angle of elevation and depression are measured from the horizontal; the same triangle ratios apply to heights and distances in the shop.

Chord and sagitta: for a circular arc, radius R = (c²/4 + h²)/(2h), where c is chord length and h is sagitta.

Convert degrees to radians by multiplying by π/180; 1 radian ≈ 57.3°, and 180° = π radians.

Density ρ = mass/volume; mass = ρ × V. Steel ≈ 7850 kg/m³ (7.85 g/cm³), aluminium ≈ 2700 kg/m³, copper ≈ 8900 kg/m³.

Weight is a force: W = m × g, with g ≈ 9.81 m/s². A 10 kg mass weighs about 98.1 N.

Work = force × distance moved in the direction of the force (W = F × s); unit is the joule (N·m).

Power = work/time = (force × distance)/time; unit is the watt. 1 kW = 1000 W; 1 metric horsepower ≈ 735.5 W.

Kinetic energy = ½mv²; potential energy = mgh. Energy is conserved and measured in joules.

Mechanical advantage = load/effort; velocity ratio = distance moved by effort/distance moved by load; efficiency = MA/VR × 100%.

Torque = force × perpendicular radius (T = F × r), measured in N·m; it differs from work because no displacement along the force occurs.

Pressure = force/area, in N/m² (pascal); 1 bar = 100 kPa ≈ 1.02 kgf/cm².

Moment of a force = force × perpendicular distance from the pivot; for balance, clockwise moments = anticlockwise moments.

Cutting speed V = πDN/1000 (m/min) when D is in mm and N in rpm; rearrange to N = 1000V/(πD).

Turning time per pass = L/(f × N), where L is the length of cut in mm and f is feed in mm/rev; add approach and overtravel to L.

Milling: table feed = fz × z × N (mm/min), where fz is feed per tooth and z the number of teeth.

Drilling time = (depth + 0.3D)/(f × N) for a standard 118° drill point; the 0.3D term is the point allowance.

Tapping: spindle speed N = 1000V/(πD) using the tap's major diameter; feed rate equals pitch (mm/rev).

Carbide tools tolerate roughly 3–6 times the cutting speed of HSS on the same material; always check the tool and work material chart.

For a given surface speed, doubling the diameter halves the rpm - smaller work or tools run faster.

Depth of cut = (D₁ − D₂)/2 for a turning pass reducing diameter from D₁ to D₂; it is measured radially, not on diameter.

Unit check: V in m/min, D in mm, N in rev/min - the factor 1000 converts mm to m.

Gear ratio = driven teeth/driver teeth = speed of driver/speed of driven; for a train, multiply each stage's ratio.

Module m = pitch circle diameter/number of teeth (m = PCD/z); circular pitch p = πm; addendum = m, dedendum = 1.25m (standard).

Two meshing gears must have the same module and pressure angle; centre distance = m(z₁ + z₂)/2.

Metric thread: pitch is the axial distance between crests; lead = pitch × number of starts. For single-start, lead = pitch.

Thread depth for ISO metric ≈ 0.6134 × pitch; tap drill size ≈ major diameter − pitch.

Taper: taper ratio = (D − d)/L; half-angle tan α = (D − d)/(2L); included angle = 2α.

Morse tapers are self-holding because of their small included angle (roughly 2–3°); they transmit torque by friction.

Clearance fit: hole is always larger than shaft (e.g. H8/c7). Interference fit: shaft always larger (e.g. H7/r6). Transition fit may give either.

Fit designation: capital letter = hole, small letter = shaft; H is the basic hole whose lower deviation is zero.

Heat is energy in transit, measured in joules; temperature is the degree of hotness, measured in °C, K or °F.

Convert: °F = (9/5)°C + 32; °C = (5/9)(°F − 32); K = °C + 273.15.

Sensible heat Q = m × c × ΔT; specific heat of water ≈ 4.18 kJ/kg·K, steel ≈ 0.49 kJ/kg·K.

Latent heat of fusion of ice ≈ 334 kJ/kg; of vaporisation of water ≈ 2260 kJ/kg - absorbed at constant temperature.

Thermal expansion: ΔL = L × α × ΔT; steel α ≈ 11–12 × 10⁻⁶ per K.

Ohm's law: V = I × R; power P = V × I = I²R = V²/R.

Series: R total = R₁ + R₂ + …; current is the same, voltage divides. Parallel: 1/R = 1/R₁ + 1/R₂ + …; voltage is the same, current divides.

Electrical energy = P × t; in kWh, P(kW) × t(h). One unit of electricity = 1 kWh = 3.6 MJ.

Conductors have low resistance (copper, aluminium); insulators have high resistance (rubber, mica, porcelain).

Ferrous metals contain iron: cast iron, mild steel, high-carbon steel, alloy steels. Non-ferrous: copper, aluminium, brass, bronze, zinc, lead.

Mild steel: about 0.15–0.3% carbon, tensile strength ≈ 400–500 N/mm², weldable and machinable - the general workshop default.

Carbon content raises hardness and tensile strength but lowers ductility and weldability; above about 0.6% C the steel is hard and brittle.

Cast iron: 2–4% carbon, excellent compressive strength and damping, brittle in tension; grey cast iron is used for machine beds.

Brass = copper + zinc; bronze = copper + tin. Bronze has better wear resistance and is used for bearings and bushes.

Aluminium: density ≈ 2700 kg/m³ (about one-third of steel), good conductivity, forms a protective oxide layer; soft, so it is alloyed for structural use.

Hardness tests: Brinell (ball, HB), Vickers (pyramid, HV), Rockwell (cone or ball, HRC/HRB).

Heat treatments: annealing softens and relieves stress; normalising refines grain; hardening (quench) increases hardness; tempering restores toughness after hardening.

Case hardening adds carbon to the surface of low-carbon steel, giving a hard skin with a tough core.

Lever: load × load arm = effort × effort arm. Class I has the fulcrum between load and effort; Class II has the load between; Class III has the effort between.

Pulley systems: a single fixed pulley gives MA = 1; a movable pulley gives MA = 2; a block and tackle with n rope parts supports MA ≈ n.

Inclined plane: MA = length of incline/height; screw jack: MA = 2πr/pitch, where r is the handle radius.

Efficiency = (output work/input work) × 100 = (MA/VR) × 100; friction always makes efficiency below 100%.

Newton's laws: a body stays at rest or in uniform motion unless a net force acts; F = ma; every action has an equal and opposite reaction.

Friction force F = μN, where μ is the coefficient of friction and N the normal reaction; it opposes relative sliding.

Threads: a bolt and nut is a helical inclined plane; a right-hand thread tightens clockwise viewed from the head.

Fastener types: through bolt, tap bolt, stud, machine screw, self-tapping screw, rivet, key, pin and circlip - each resists a specific load type.

A key transmits torque between shaft and hub; a circlip locates parts axially; a rivet gives a permanent joint.

Current is measured in amperes (A) with an ammeter in series; voltage in volts (V) with a voltmeter in parallel.

A multimeter selects function and range; for an unknown value, start on the highest range and step down.

A clamp meter reads current without breaking the circuit by sensing the magnetic field around a conductor.

A Vernier caliper: least count = 1 main-scale division − 1 Vernier division. For 0.02 mm, 49 main divisions equal 50 Vernier divisions.

A micrometer: least count = pitch/number of thimble divisions, typically 0.5 mm/50 = 0.01 mm.

A dial gauge (comparator) measures small variations and runout; it must be zeroed against a standard.

Magnetism: like poles repel, unlike attract; a magnetic field exists around a current-carrying conductor (right-hand grip rule).

Electromagnetic induction: a changing magnetic flux through a coil induces an emf (Faraday); a rotating coil in a magnetic field generates AC.

A fuse and an MCB protect a circuit by opening it on excess current; earthing connects the metal body to earth for safety.

Personal protective equipment: safety goggles for grinding and chipping, gloves for handling rough stock (not for rotating spindles), safety shoes, ear protection above about 85 dB.

Machine guards must be in place before starting; never remove a guard or reach across a rotating part.

Before any maintenance or tool change, isolate the machine and apply lockout/tagout; verify that the spindle has stopped.

A file cuts on the forward stroke only; a file without a handle must never be used because the tang can pierce the hand.

A hacksaw blade is fitted with teeth pointing forward; about 40–50 strokes per minute on mild steel.

A chisel has a cutting angle of about 60° for mild steel and 70° for cast iron; a sharper 30–40° edge suits softer metals.

Fire classes: A solid combustibles (water), B flammable liquids (foam, CO₂), C gases (dry powder), D metals (special dry powder), E electrical (CO₂, never water).

Waste segregation: metal swarf, oily rags, coolant and electronic waste go to separate streams; used oil is never poured into drains.

5S practice: sort, set in order, shine, standardise, sustain - housekeeping that prevents accidents and improves quality.