Why Is 60 rpm Not Equal to 1 Hz in PTC Mathcad Prime?

Scenario / Question

There is a built-in frequency unit "rpm" in PTC Mathcad Prime, which I assumed stands for revolutions per minute. But in Mathcad Prime 3.0 I find that 1 rpm evaluates to 6.283/min, or 0.105/sec, instead of 1/min, or 0.017/sec. I expected 60 rpm to equal 1/sec, that is 1 Hz, and that is not what Mathcad calculates. Am I missing something?

Overview

This is not a defect: in Mathcad Prime, rpm is an angular velocity, not an inverse-time frequency, and it is defined as 1 rpm = 2·pi/(60 sec) (0.105·rad/sec or 2·pi·rad/min). Because a radian is dimensionless, rpm still carries the dimension time to the power of minus one, so it looks like a frequency while differing from your intuitive "revolutions per minute" by the constant factor 2·pi. Mathcad Prime handles the same ambiguity for frequency by providing two units: Hz, defined as 1/sec, and Hza, the angular form, defined as 2·pi/sec (1·rev/sec).

Step-by-Step

Option 1: Use rpm as Mathcad intends, for angular velocity work

1. Treat rpm as angular speed: Read rpm as 2·pi radians per minute, that is 1 rpm = 2·pi/(60 sec), approximately 0.105 radians per second.
2. Multiply torque by shaft speed directly: Multiply torque in newton meters by shaft speed in rpm, or in Hza, and the result comes out correctly in watts, kilowatts, or horsepower with no manual 2·pi factor inserted.
3. Multiply rpm by radius for tip speed: Multiplying a rotational speed in rpm by a radius returns the correct tip speed, again without inserting 2·pi.
4. Pair rpm with Hza, not Hz: Use Hza, the Hertz unit expressed in terms of angular velocity, when an expression also contains rpm. Frequency units and angular units should not be mixed in one expression.

Option 2: Express plain revolutions per minute as an inverse time unit

1. Decide which quantity you actually need: If you want the count of full 360 degree rotations per minute, you need an inverse-time quantity, not Mathcad's angular rpm.
2. Use an inverse-time unit instead: Display or enter the value using 1/min or Hz, where 1 Hz = 1/sec, rather than the built-in rpm.
3. Apply the 2·pi factor when converting: The only difference between Mathcad's rpm and counted revolutions per minute is the constant factor 2·pi, since one revolution equals 360 degrees, or approximately 6.283 radians. Divide the angular value by 2·pi to obtain counted revolutions or multiply counted revolutions by 2·pi to obtain the angular value.

Key Notes & Limitations

• Under SI, revolution is not a unit at all; it is a semantic annotation, so rpm is not strictly an SI unit. The SI equivalent of rpm is 1/sec.
• In Mathcad Prime, 1 rpm = (2·pi)·(1/60)·(1/sec), which is why 60 rpm evaluates to 6.283 Hz rather than 1.000 Hz.
• Radians are dimensionless, so Mathcad does not display radian as a unit in the result. This is the main reason the output looks like a plain frequency.
• The tooltip on rpm in the Units dropdown under Frequency reads "revs per minute", which is accurate in Mathcad's terms, as Mathcad's "rev" is an angle equal to 360 degrees, or 6.283 radians, rather than a count of rotations.
• rpm is listed in the Units dropdown under Frequency
• The behavior described here is documented in PTC Mathcad Prime Help.

Related Questions (FAQ)

Q: Why does 60 rpm give 6.283 Hz instead of 1 Hz in Mathcad Prime?

A: Because Mathcad defines 1 rpm as 2·pi/(60 sec), an angular velocity. Multiplying by 60 gives 2·pi/sec, which is 6.283 in SI units of 1/sec. One full revolution per minute corresponds to a rotation of approximately 0.105 radians per second.

Q: What is the difference between Hz and Hza in Mathcad?

A: Hz is a frequency unit defined as 1/sec. Hza is the Hertz unit defined in terms of angular velocity, equal to 2·pi/sec. Use Hza in expressions that also contain rpm.

Q: Why is rpm defined as an angular velocity rather than a simple count?

A: So that common engineering products work as written. Torque in newton meters multiplied by shaft speed in rpm returns power directly in watts, kilowatts, or horsepower, and rpm multiplied by a radius returns tip speed correctly, with no need to insert a factor of 2·pi.

Q: How do I get the value I think of as revolutions per minute?

A: Work in inverse time units such as 1/min or Hz, and apply the constant factor 2·pi when moving between that value and Mathcad's angular rpm, since one revolution equals 360 degrees, or approximately 6.283 radians.

Links & Resources

• Frequency and Angular Velocity Units, PTC Mathcad Help (support.ptc.com)
• Frequency Conversion, related PTC Community thread on 60·rpm returning 6.283 Hz (community.ptc.com)
• RPM to m/s, related PTC Community thread on rpm differing from 1/min in Prime (community.ptc.com)
• rpm frequency unit?, original community thread (community.ptc.com)

This article is part of a new community content management program to bring more visibility to popular posts on the community.  It has been reviewed for accuracy by Andy McGough, Product Management Director.