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MFG-MECH15-01 · Lesson 3 · 28 clock hours · practice

Trade Math for Mechatronics Technicians

Objective. Students solve electrical, mechanical, and fluid-power calculation problems using trade formulas—Ohm's law, power, series/parallel resistance, synchronous speed, slip, torque, cylinder force, hydraulic speed, gear/pulley ratios, and 4–20 mA scaling—to within ±2% accuracy.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by a licensed engineer or certified journey-level tradesperson or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

VoTechWorks program approval and accreditation are still in progress. Credentials are issued only by the certifying bodies named in each program, after their own exams.

Why Trade Math Matters in Mechatronics

As a mechatronics technician you will size components, verify sensor readings, troubleshoot motors, and commission fluid-power circuits—all in real time on a production floor. Mistakes in calculation can damage equipment, injure workers, or cause costly downtime. Accuracy to within ±2% is the professional standard. This lesson builds fluency with the core formula families you will use every day and that appear on PMMI, Siemens SMSCP, and FANUC credential exams.

Safety note before you pick up a meter or touch a panel: Always follow OSHA 29 CFR 1910.147 lockout/tagout before working on energized systems, and apply NFPA 70E shock and arc-flash boundaries when voltage must be verified. Never bypass machine guarding (OSHA 1910.212). Math helps you understand what a circuit should do; proper PPE and LOTO keep you alive while you verify it.

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Ohm's Law and Power

Ohm's Law relates voltage (V, volts), current (I, amperes), and resistance (R, ohms):

Power (P, watts) formulas:

These are your starting point for every electrical calculation. Always check units: volts × amps = watts; amps² × ohms = watts.

Series and Parallel Resistance

In a series circuit, total resistance adds directly:

> R_total = R₁ + R₂ + R₃ + …

In a parallel circuit, use the reciprocal formula:

> 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

For exactly two parallel resistors the shortcut is:

> R_total = (R₁ × R₂) / (R₁ + R₂)

Current is the same everywhere in a series circuit; voltage splits. Voltage is the same across every branch in a parallel circuit; current splits.

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Synchronous Speed and Motor Slip

The synchronous speed of an AC induction motor depends on supply frequency (f, Hz) and the number of poles (P):

> Ns = 120 × f / P

Result is in RPM. A 4-pole motor on 60 Hz runs at 1,800 RPM synchronously. The rotor always turns slightly slower; that difference is slip:

> Slip (%) = (Ns − Nr) / Ns × 100

where Nr is actual rotor speed. Typical induction motors slip 2–5%.

Motor Torque

When horsepower and speed are known:

> T = hp × 5,252 / rpm

T is in pound-feet (lb·ft). This formula comes from the definition of horsepower (1 hp = 550 ft·lb/s) combined with rotational math. Use it to verify that a motor shaft can handle its mechanical load.

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Fluid Power: Pneumatic Cylinder Force

A pneumatic or hydraulic cylinder converts pressure into linear force:

> F = P × A

Always use the net area on the rod side (subtract rod cross-section area) when calculating retract force.

Hydraulic Actuator Speed

How fast a hydraulic cylinder extends depends on flow rate and bore area:

> v = (gpm × 231) / A

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Gear and Pulley Ratios

For a gear or pulley pair:

> Gear Ratio = N_driven / N_driver (teeth or pulley diameter)

> Speed_output = Speed_input / Gear Ratio

> Torque_output = Torque_input × Gear Ratio (ignoring losses)

A ratio greater than 1 (step-down) reduces speed and multiplies torque. A ratio less than 1 (step-up) increases speed and reduces torque.

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4–20 mA Signal Scaling

Industrial sensors transmit a current loop signal: 4 mA represents the zero/minimum of the range and 20 mA represents the full-scale maximum. To convert a measured current (I_mA) to an engineering value:

> Scaled Value = (I_mA − 4) / 16 × (Span) + Zero

Example: 12 mA on a 0–100 psi transmitter → (12 − 4)/16 × 100 + 0 = 50 psi.

Check: 4 mA should always give the minimum; 20 mA should always give the maximum. If your answer falls outside the sensor range, recheck your arithmetic.

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Working Safely with These Calculations

Before measuring current in a live circuit, confirm you are within NFPA 70E safe approach boundaries and wearing rated PPE (at minimum, safety glasses and insulated gloves rated for the voltage). Use a CAT-rated meter. For pneumatic and hydraulic systems, depressurize and lock out per OSHA 1910.147 before disconnecting fittings. Never exceed rated cylinder pressure—exceeding it can cause sudden, violent failure. On robot cells, ANSI/RIA R15.06 / ISO 10218 requires safeguarding to be verified before power is restored.

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Tips for the Quiz

1. Write the formula first, then substitute numbers—this prevents unit errors.

2. Carry one extra decimal place during calculation; round only the final answer.

3. For 4–20 mA problems, always sanity-check that your answer is between 0% and 100% of range.

4. Label every answer with its unit (psi, RPM, lbf, in/min, etc.)—unlabeled answers are incomplete.

Key terms8

Ohm's LawThe relationship V = IR, stating that voltage across a conductor equals the product of current and resistance; the foundation of all DC and AC circuit calculations.
Synchronous Speed (Ns)The theoretical no-load rotational speed of an AC induction motor's magnetic field, calculated as Ns = 120 × f / P, where f is frequency in Hz and P is the number of poles.
SlipThe percentage difference between synchronous speed and actual rotor speed in an induction motor; necessary for torque production, typically 2–5%.
Torque (T)Rotational force measured in pound-feet (lb·ft), calculated from horsepower and speed as T = hp × 5,252 / rpm.
Cylinder ForceThe linear push or pull force produced by a pneumatic or hydraulic cylinder, equal to pressure (psi) multiplied by piston area (in²), expressed in pounds-force (lbf).
4–20 mA SignalA standard industrial current-loop signal in which 4 mA represents the sensor's minimum value and 20 mA represents its maximum, used to transmit analog process measurements reliably over long cable runs.
Gear RatioThe ratio of driven-gear teeth to driver-gear teeth (or driven-pulley diameter to driver-pulley diameter); a ratio greater than 1 reduces output speed and multiplies torque.
Hydraulic Speed (v)The linear velocity of a hydraulic cylinder, calculated as v = (gpm × 231) / A, where 231 converts US gallons to cubic inches and A is piston area in square inches.

Worked examples4

1. A 480 V heater element draws 8 A. (a) What is its resistance? (b) What power does it consume?
  1. Write the formulas: R = V / I and P = V × I
  2. Substitute for resistance: R = 480 V / 8 A = 60 Ω
  3. Substitute for power: P = 480 V × 8 A = 3,840 W
  4. Check units: volts / amps = ohms ✓; volts × amps = watts ✓
Answer: R = 60 Ω; P = 3,840 W (3.84 kW)
2. Find the synchronous speed of a 6-pole AC motor operating on 60 Hz supply, then calculate slip if the rotor actually turns at 1,150 RPM.
  1. Apply the synchronous speed formula: Ns = 120 × f / P = 120 × 60 / 6 = 7,200 / 6 = 1,200 RPM
  2. Calculate slip: Slip (%) = (Ns − Nr) / Ns × 100 = (1,200 − 1,150) / 1,200 × 100
  3. Slip (%) = 50 / 1,200 × 100 = 4.17%
  4. Check: 4.17% is within the normal 2–5% range for an induction motor ✓
Answer: Ns = 1,200 RPM; Slip ≈ 4.17%
3. A pneumatic cylinder has a 2-inch bore diameter and is supplied at 90 psi. What extend force does it produce?
  1. Calculate piston area: A = π × d² / 4 = π × (2)² / 4 = π × 4 / 4 = π ≈ 3.1416 in²
  2. Apply the force formula: F = P × A = 90 psi × 3.1416 in²
  3. F = 282.7 lbf
  4. Check units: psi × in² = lbf ✓
Answer: F ≈ 283 lbf (to three significant figures)
4. A pressure transmitter is ranged 0–200 psi and outputs a 4–20 mA signal. The PLC reads 14 mA. What is the actual pressure?
  1. Write the scaling formula: Scaled Value = (I_mA − 4) / 16 × Span + Zero
  2. Identify values: I_mA = 14, Span = 200 − 0 = 200 psi, Zero = 0 psi
  3. Substitute: Scaled Value = (14 − 4) / 16 × 200 + 0 = 10 / 16 × 200
  4. Scaled Value = 0.625 × 200 = 125 psi
  5. Sanity check: 14 mA is between 4 and 20 mA; 125 psi is between 0 and 200 psi ✓
Answer: 125 psi

The math in this lesson3

Learners compute synchronous speed and percent slip for a 6-pole 60 Hz motor.
7.RP.3G-MG.2A-CED.4
Brush up in the math halls at newSchoolWorks.com:
In class: Work through a structured problem set in pairs: 30 problems progressing from unit conversions and Ohm's law (V = IR, P = IV, P = I²R) through series/parallel circuits, synchronous speed (Ns = 120f/P), motor torque (T = hp × 5252/rpm), pneumatic cylinder force (F = P × A), hydraulic speed (v = gpm × 231/A), and gear-ratio calculations, with peer checking at each section before moving forward. · Use a scientific calculator and a 4–20 mA scaling chart to convert raw sensor current signals to engineering units (e.g., 12 mA on a 0–100 psi transmitter), completing ten scaling exercises and confirming answers against a provided answer key. · Participate in a 'formula relay': teams of three rotate through six problem stations posted around the room, each station focusing on one formula family; teams record solutions on a shared whiteboard and defend one answer verbally to the instructor before the group rotates.
Materials: Scientific calculator; trade math workbook (instructor-developed, referencing standard formulas from verified research brief); 4–20 mA scaling reference chart; whiteboard and markers for relay stations; answer key for self-check.
Check: Timed 20-problem written quiz at lesson close; students must achieve ≥80% (16/20 correct within ±2% tolerance on numerical answers); any student below 80% completes a targeted remediation worksheet and retakes an alternate quiz within one week.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

Industrial Electricity / PMMI Industrial Electricity 1
1. A 120 V control circuit has a single resistor that limits current to 0.5 A. What is the resistance of that resistor?
Industrial Electricity / PMMI Industrial Electricity 1
2. Two resistors, 12 Ω and 6 Ω, are connected in parallel. What is the total resistance of the combination?
Motor and Motor Controls / PMMI Motor and Motor Controls
3. A 4-pole induction motor is connected to a 60 Hz supply. What is its synchronous speed?
Motor and Motor Controls / PMMI Motor and Motor Controls
4. An AC induction motor has a synchronous speed of 1,800 RPM and runs at 1,746 RPM under full load. What is the percent slip?
Motor and Motor Controls / PMMI Motor and Motor Controls
5. A motor produces 5 hp at 1,750 RPM. What is its output torque in pound-feet?
Fluid Power / PMMI Fluid Power
6. A pneumatic cylinder has a 3-inch bore diameter and is supplied at 80 psi. Approximately what extend force does it produce? (Use π ≈ 3.1416.)
Fluid Power / PMMI Fluid Power
7. A hydraulic cylinder with a 4 in² piston area is supplied at 3 gpm. What is the cylinder's extension speed in inches per minute?
Safety / OSHA Standards
8. A flow transmitter is ranged 0–50 gpm and outputs 4–20 mA. The PLC reads 20 mA. Which action is most appropriate before a technician opens the pipeline to investigate the reading?
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