Linear to Rotational Motion Conversion Calculator

Posted by Robbie Dickson on

Linear to Rotational Motion Conversion Calculator + Formula, Examples & Applications

You've picked a motor. You know the force and speed you need at the output. But the motor spins — and your load moves in a straight line. The lead screw bridges that gap, and getting the math right is the difference between a drivetrain that works and one that stalls or overshoots. This calculator converts between linear speed and RPM, and between linear force and shaft torque, using lead screw pitch as the conversion factor. Below you'll find the formulas, worked examples, and practical guidance to spec your next actuator drivetrain with confidence.

What Is Linear to Rotational Motion Conversion?

It's the math that translates a motor's spinning output — RPM and torque — into the straight-line speed and push/pull force delivered by a lead screw. The screw's pitch determines the exchange rate.

Simple Explanation

Think of a lead screw like a ramp wrapped around a cylinder. Every time the screw completes 1 full turn, the nut riding on it moves forward by a fixed distance — that's the pitch. A coarse thread is like a steep ramp: fast travel, but harder to push heavy loads. A fine thread is a gentle ramp: slower, but you get a big mechanical advantage. This calculator handles both directions — rotational to linear and linear to rotational — so you can work from whichever side of the problem you're starting on.

Motor Shaft RPM Acme Lead Screw P (in/rev) 1 rev = P inches Nut v (in/sec) F (lbs) Speed Conversion RPM = (v × 60) / P Speed Conversion v = (RPM × P) / 60 Force → Torque T = (F × P) / (2π × η) Torque → Force F = (T × 2π × η) / P

Linear to Rotational Motion Conversion Calculator

Linear travel per one full screw revolution. Acme standard is typically 0.1–0.5 in/rev.
80% for Acme thread. 90–95% for ball screw.

🎥 Video — Linear to Rotational Motion Conversion Calculator

How to Use This Calculator

This calculator covers 4 conversion directions. Pick your mode, enter your known values, and get the answer you need to spec your drivetrain.

  1. Select your conversion mode. Choose from the dropdown: Linear Speed → RPM, RPM → Linear Speed, Linear Force → Shaft Torque, or Shaft Torque → Linear Force.
  2. Enter your lead screw pitch. This is the linear travel per revolution of the screw — check your screw manufacturer's datasheet. Common Acme values range from 0.1 to 0.5 in/rev.
  3. Enter your known value. Depending on the mode, this will be linear speed, RPM, linear force, or shaft torque. For force/torque modes, also enter the screw's efficiency.
  4. Click "= Calculate." Your results appear instantly with both imperial and metric units where applicable.
  5. Use "Try Example" to see it in action. The button loads real-world default values for whichever mode you've selected — a fast way to understand the math before entering your own numbers.

Linear to Rotational Motion Conversion Formulas

Speed Conversions

RPM = (Linear Speed × 60) / Pitch
Linear Speed (in/sec) = (RPM × Pitch) / 60

Force / Torque Conversions

Torque (lb-in) = (Linear Force × Pitch) / (2π × Efficiency)
Linear Force (lbs) = (Torque × 2π × Efficiency) / Pitch
Symbol Variable Unit
P Lead Screw Pitch inches/rev
v Linear Speed inches/sec
RPM Rotational Speed revolutions per minute
F Linear Force lbs
T Shaft Torque lb-in
η Lead Screw Efficiency decimal (e.g., 0.80)

Simple Example

Problem: You need 100 lbs of linear push force from an Acme lead screw with a 0.2 in/rev pitch and 80% efficiency. What shaft torque does your motor need to deliver?

Formula: Torque = (Force × Pitch) / (2π × Efficiency)

Calculation:
Torque = (100 × 0.2) / (2 × 3.14159 × 0.80)
Torque = 20 / 5.02655
Torque = 3.979 lb-in (≈ 0.4496 Nm)

What this means: Your gearmotor needs to deliver roughly 4 lb-in of continuous torque at the screw shaft to produce 100 lbs of linear force. That's a very achievable number for a small DC gearmotor — which is exactly why lead screws are so effective at converting modest motor torque into substantial push/pull force.

Engineering Applications

Why Pitch Is Everything

Lead screw pitch is the fundamental conversion factor between rotational and linear motion. It defines how far the nut travels per revolution of the screw — and it sets the trade-off between speed and force for your entire drivetrain. A coarser pitch (larger number, like 0.5 in/rev) gives you faster linear speed but less mechanical advantage. A finer pitch (smaller number, like 0.1 in/rev) delivers more force at the expense of speed. There's no free lunch here — you're just choosing where on the speed-force trade-off curve you want to operate.

The 300 RPM / 0.2 Pitch Reference Point

Here's a design shortcut worth memorizing: at 0.2 in/rev pitch and 300 RPM, your linear speed comes out to exactly 1 in/sec. We use this as a baseline reference when sizing FIRGELLI actuators because it's a realistic operating point for many DC gearmotors. If you need 2 in/sec, you either double the pitch to 0.4 in/rev or double the RPM to 600. Both have consequences — higher pitch reduces your force capability, and higher RPM demands a faster (often more expensive) motor.

Where Does 2π Come From?

The 2π factor in the torque/force formulas isn't arbitrary — it comes directly from the relationship between rotational and linear work. One revolution of the screw equals 2π radians of rotation and produces exactly "pitch" inches of linear travel. When you equate rotational work (torque × 2π) to linear work (force × pitch), the 2π falls naturally out of the physics. Understanding this helps you sanity-check your numbers: if your torque result seems too high or too low, you've probably got a units mistake or forgot the 2π.

Efficiency Losses Are Real

At 80% Acme screw efficiency, 20% of your motor's input torque goes straight to friction and heat — it never becomes useful linear force. This matters a lot when you're sizing a motor. If you calculate the "ideal" torque and buy a motor rated at exactly that number, you'll come up short in the real world. Always use realistic efficiency values: 80% for standard Acme threads, 90–95% for ball screws. And if your screw is dirty, misaligned, or running without proper lubrication, actual efficiency can drop well below these numbers.

Practical Motor Sizing

The most common use of this calculator is working backwards from a force requirement. You know you need, say, 100 lbs of push force with a 0.2 in/rev Acme screw. Plug those numbers in and you get roughly 4 lb-in of required shaft torque. Now you can shop for a gearmotor that delivers at least 4 lb-in continuously — with margin for startup loads, temperature derating, and wear over time. That's how you spec the right gearmotor without guessing.

Advanced Example

Scenario: You're designing a custom positioning stage that needs to deliver 250 lbs of push force at a linear speed of 0.5 in/sec. You're choosing between an Acme screw (80% efficiency) and a ball screw (93% efficiency), both with 0.25 in/rev pitch. What motor specs do you need for each option?

Step 1 — Required RPM (same for both screws)

RPM = (v × 60) / Pitch = (0.5 × 60) / 0.25 = 120 RPM

Step 2 — Required Torque with Acme Screw (η = 0.80)

Torque = (250 × 0.25) / (2π × 0.80)
Torque = 62.5 / 5.0265
Torque = 12.43 lb-in (1.40 Nm)

Step 3 — Required Torque with Ball Screw (η = 0.93)

Torque = (250 × 0.25) / (2π × 0.93)
Torque = 62.5 / 5.843
Torque = 10.70 lb-in (1.21 Nm)

Design Interpretation

The ball screw reduces your torque requirement by about 14% — from 12.43 to 10.70 lb-in. That's meaningful because it might let you use a smaller, lighter, cheaper gearmotor. But ball screws cost 3–5× more than Acme screws and aren't self-locking. For a positioning stage that needs to hold position when the motor stops, the Acme screw's self-locking property might be worth the extra torque. Both options need a motor rated for at least 120 RPM continuous — a fairly gentle speed that most 12V or 24V DC gearmotors handle easily.

Frequently Asked Questions

What's the difference between pitch and lead on a screw? +

For single-start screws — which is most of what you'll encounter — pitch and lead are the same thing. Pitch is the distance between adjacent threads. Lead is the distance the nut advances per revolution. On a multi-start screw, lead = pitch × number of starts. This calculator uses "pitch" in the lead sense: linear travel per revolution.

Can I use this calculator with metric lead screws? +

Yes — just convert your metric pitch to inches first. Divide your pitch in mm by 25.4 to get inches/rev. For example, a 5 mm pitch screw is 0.197 in/rev. Enter that, and all the formulas work exactly the same way. The RPM-to-linear-speed mode also outputs mm/sec for convenience.

Why doesn't the speed conversion use efficiency? +

Efficiency affects force and torque — it represents energy lost to friction. But the kinematic relationship between RPM and linear speed is purely geometric. One revolution always moves the nut exactly "pitch" inches regardless of friction. Speed doesn't change with efficiency; only the force you can deliver does.

What efficiency should I use if I don't know my screw type? +

Default to 80% — that covers most standard Acme thread screws, which is what you'll find inside the vast majority of linear actuators including ours. If you're using a ball screw, bump it to 90–93%. If your screw is old, dirty, or running without lubrication, drop to 60–70% to be safe. When in doubt, be conservative.

Does this calculator account for backdrive and self-locking? +

No — this calculator handles the forward direction only: motor torque driving the screw to create linear force. Self-locking (whether the load can backdrive the screw) depends on the screw's helix angle and friction coefficient. As a rule of thumb, Acme screws with efficiency below 50% are self-locking. Most standard Acme screws are self-locking; ball screws are not.

Can I use this for rack and pinion or belt drive systems? +

The speed formulas work for any mechanism where you know the linear distance per revolution — for a rack and pinion, that's the pinion circumference (π × diameter). The torque formulas apply too, but the efficiency values will be different. Belt drives typically run 95–98% efficient, and rack-and-pinion systems around 90–95%. Adjust accordingly.

About the Author

Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations

Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.

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