I wrote this paper in response to a declaration made by Dynamics Professor that "there is no difference between a lbm and lbf." The discussions from the students that followed exposed a huge concept error that seems to stem from the misuse of the above statement. It has some comedic relief, so it makes it more bearable ;) Enjoy!
The lbm-lbf Relationship: Why it Matters
by Kevin McConnell
Is there really a difference between a pound-mass and a pound-force? Many people might even ask, “What the hell is a pound-mass?” Well, you can point the finger at your sixth-grade physics teacher (or anyone else who may have misled you) for the confusion that surrounds this simple question. But don’t worry, it’s never too late to learn something new (and something undeniably important).
Here’s something to mull over: let’s say that you step on a scale and it reads “150.” The readout of the scale may even provide you with units of “lbs.” Well, a scale measures the amount of force that an object exerts so we can assume that the units then are lbf (pound-force). And your physics teacher told you that there is no difference between a pound-mass and a pound-force so that must mean that your body is composed of 150 pounds of mass as well, right? What your physics teacher DIDN’T tell you are the hidden assumptions that must be true for that relationship to exist. There is something so fundamentally wrong with the statement, “pounds-mass and pounds-force are the same thing!”
First off, pounds-mass is a unit of mass, and pounds-force is a unit of force (wait… WHAT?!). Newton’s second law of motion tells us that net force is equated by the product of mass and acceleration. So, we can see that a relationship exists between mass and force, but we would NEVER say, “mass and force are the same thing!”
Let’s say that I took the same scale from above on a trip to mars; what would the scale read there? Would you be surprised if the scale readout as “57 lbs?” Or what if I brought the scale to Jupiter and it told me I weighed “380 lbs?” Is the scale correct? Absolutely! As we learned previously, the scale is measuring the amount of force that you exert due to gravity (acceleration). And we know that gravity on these planets differs because of a difference in their size and mass.
Note that your mass DOES NOT change from planet to planet; only the amount of force exerted by your mass.
So why do we keep hearing that there is no difference between pounds-mass and pounds force? Because English units were created such that 1 lbm exerts 1 lbf here on Earth! And without further ado, here is the relationship that makes it happen:
1 lbf = 32.174 lbm ft/s^2
So, the statement that people are trying to say should sound something more like “on earth, pounds-mass subject to gravity IS pounds-force!” To further illustrate this point, lets use newtons second law to calculate the force exerted by a 1 lbm object here on earth:
Force=mass x acceleration
let acceleration=g=32.174 ft/s^2 (this is Earth' s gravitational constant)
F=m x g= 1 lbm x (32.174 ft/s^2) = 32.174 (lbm ft)/s^2
But we can’t really conceptualize the units lbm-ft /s2, so we use the relationship from above to convert it to pound-force (lbf):
F= 32.174 lbm-ft/s^2 x (1 lbf / 32.174 lbm ft/s^2) = 1 lbf
We have just proved that 1 lbm exerts 1 lbf here on Earth! If this is new to you, you should drink a beer tonight to celebrate a breakthrough in your understanding! Let’s go one step further to demonstrate why the scale would read differently on Mars and Jupiter
‘NOTHER KEY CONCEPT
The relationship (eq. 1) from above DOES NOT change if you’re on a different planet just because the gravity changes; this wouldn’t make sense and you’ll see why
Force = mass x acceleration
let acceleration = g = 12.176 ft/s^2 (this is the gravitational constant on Mars)
let mass = m = 150 lbm
F = m x g = 150 lbm x 12.176 ft/s^2 = 1826.4 (lbm ft)/s^2
Once again, lets convert this quantity from lbm-ft /s2, to something we know (lbf) by using the relationship illustrated above:
F=(1826.4 lbm ft/s^2) x (1 lbf / 32.174 lbm ft/s^2) = 56.8 lbf
Even though I imagine that you now have a firm grasp on this concept, let’s try it out on Jupiter to really send it the point home:
Force = mass x acceleration
let acceleration = g = 81.336 ft/s^2 (this is the gravitational constant on Jupiter)
let mass=m=150 lbm
F = m x g = 150 lbm) x 81.336 ft/s^2 x (1 lbf / 32.174 lbm ft/s^2 )=379.2 lbf
Now you’ve seen it and you can say that you understand it! So, let’s highlight the crucial points to everything that we just went over:
pounds-mass (lbm) and pounds-force (lbf) are NOT the same
an object’s mass is constant from place to place (i.e. from Earth to
Mars) but the force that it exerts IS different
The following relationship is key to understand the link between lbm
1 lbf=32.174 lbm ft/s^2
Arm yourself with this knowledge so that you can fight the good fight: the next time you hear someone say that pound-mass and pound-force are the same thing, you can confidently say they are not!