How to measure mass for an object of changing mass?

[AEGIS]

Hi,

I'm doing a high school physics project that involves a small toy car tied to a bottle filled with pressurized air. By removing a certain thumb tack on the bottle, an orifice is exposed which expels a stream of air that acts as a force of thrust.

I'm going to measure AND theoretically calculate simple motion variables such as acceleration, frictional force, etc...

What I was struggling with was coming up with a procedure that accurately measures (not theoretically calculates) the mass of the bottle attached to the car over a period of time as air is released. I was considering simply laying the car + bottle upside down on top of an electronic scale and recording the weight as air was expelled, but it occurred to me that an electronic scale might not be the most accurate at measuring change over time for a high level of samples per second.

I was wondering if there was any better way to measure the mass of the car + bottle as it changed due to the release of air-- I considered hanging it from a force meter and recording data as the air was released, but this was problematic because the air stream made the system twist and turn every which way and gave inconsistent results. I also considered securing the system with strings, but was worried that this could potentially skew the readings of the force meter.

Thanks in advance for any suggestions offered.

EDIT: I forgot to mention this, but I also would like to measure the force of friction over time as the system releases air. (m*g*μ_k). This one is more problematic, though... I had no idea on how I could measure it as the system was moving using lab equipment, unless I came up with some sort of complicated setup, or revamped the procedure entirely (ex. if I took trials on a angled ramp, then used the tan(θ) of the ramp to roughly estimate the coefficient of friction. The only reason I'm against this is because my teacher advised against using tan(θ) as an approximation to μ when dealing with "rolling" friction between wheels and a surface.)

 

[Andy Resnick]

<snip>Thanks in advance for any suggestions offered.

A first step is to estimate the magnitude of what you are measuring- do you need to measure grams, milligrams, micrograms? How accurately and precisely do you need to measure this quantity? What are the sources of variation (in your case, one source is how repeatably you pressurize the bottle).

It's also worth thinking about exactly what you are trying to measure- for example, since the mass of the toy doesn't changes, you don't need to bother weighing it.

I can think of a few approaches, but I'll let you work on it for a bit.

 

[AEGIS]

A first step is to estimate the magnitude of what you are measuring- do you need to measure grams, milligrams, micrograms? How accurately and precisely do you need to measure this quantity? What are the sources of variation (in your case, one source is how repeatably you pressurize the bottle).

It's also worth thinking about exactly what you are trying to measure- for example, since the mass of the toy doesn't changes, you don't need to bother weighing it.

I can think of a few approaches, but I'll let you work on it for a bit.

I used PV = nRT as an approximation to test the various pressures I'm observing (30-60 psi, at 10 psi intervals) assuming a 2 L bottle (the largest size I'm using), a temperature of 294.261 K (approx. 70 deg. F, which is the temperature I'll be running tests under), and the conversion factor 1 mol air = roughly 28.97 g. Using this, I found that each pressure of air adds only a few additional grams to the total mass. It would make most sense to measure in grams then--the percent error I'll have to aim for will have to be relatively low, given that the difference between the initial masses for each of the pressure is so small. This might actually be problematic given the inaccuracy of the pump we're using to pump air into the bottle--I may have to find a more accurate pump in its place.

I'll definitely only measure the bottle of air's mass, since the car's mass is fixed. Would hanging it from one of the hooked force meters that automatically records graphical data on a program work with the bottle spinning due to the release of air? Alternatively, is there a better way to keep the bottle secure without affecting the data measurements?

EDIT: Also still trying to figure out a way to measure (NOT calculate) the force of friction acting on the system at various times. It's mostly problematic because the method I'm familiar with (pulling a force meter attached to the car via a string) doesn't work due to the unpredictability of the wheels rolling. Every time I try to pull the car ends up 'catching up' with my pull and loosening the tension on it from the string/force meter as a result, if that makes sense.)

 

[Andy Resnick]

I used PV = nRT as an approximation to test the various pressures I'm observing (30-60 psi, at 10 psi intervals) assuming a 2 L bottle (the largest size I'm using), a temperature of 294.261 K (approx. 70 deg. F, which is the temperature I'll be running tests under), and the conversion factor 1 mol air = roughly 28.97 g.

I like the direction you are going, but let me pause you here. Significant digits is not something invented to irritate students- the concept is critically important for any measurement. 294.261 K is in no way equivalent to 'approx. 70 deg. F', and 1 mol air is not equivalent to 28.97 g. Based on your comments, I think you are going to have, at best, 10% precision in your measurement- there's no point in trying to measure something to more than 2 decimals. This simplifies your task considerably.

The reason I like your direction is that you are (correctly) setting an equivalence between 'mass of air' and 'pressure in bottle', because measuring the pressure is going to be a lot easier that measuring the mass. Rather than PV= nRT, converting pressure (or volume) to mass is best done through measured data, for example:

https://www.google.com/url?sa=t&rct...qMcqZ65Le48pLy29g&sig2=53QHfwLCNn-Q4jMucxkjIA

Now your problem of measuring a time-varying mass is equivalent to measuring a time-varying *pressure*. Note the pressure changes independently of the 'cargo', and there are many ways to measure pressure.

EDIT: Also still trying to figure out a way to measure (NOT calculate) the force of friction acting on the system at various times. It's mostly problematic because the method I'm familiar with (pulling a force meter attached to the car via a string) doesn't work due to the unpredictability of the wheels rolling. Every time I try to pull the car ends up 'catching up' with my pull and loosening the tension on it from the string/force meter as a result, if that makes sense.)

I'm not being pedantic when I say that force is not measurable. You basically want to know how much energy stored in compressed air is being lost to dissipative processes, or equivalently, how much heat is produced during the process. These types of calorimetric measurements are difficult and require very specialized instrumentation. It may be easier to characterize the process in terms of power loss- the rate at which mechanical energy is lost to dissipation.

 

[AEGIS]

I'm not being pedantic when I say that force is not measurable. You basically want to know how much energy stored in compressed air is being lost to dissipative processes, or equivalently, how much heat is produced during the process. These types of calorimetric measurements are difficult and require very specialized instrumentation. It may be easier to characterize the process in terms of power loss- the rate at which mechanical energy is lost to dissipation.

I'm not extremely experienced with the topic, but this is what I understand this to mean. Unfortunately, my teacher does ultimately want me to come up with an experimental value for frictional force, but my understanding is that I can determine it indirectly using other observed values using a process such as the below:

1. Calculate mechanical energy (ME), which in this case is equivalent to kinetic energy (no potential energy for an object along a flat 'ground level'), or 0.5mv². Both mass and velocity can definitely be determined, either from direct measurement or using other values (ex. using pressure to determine mass at a particular point in time)
2. The change in kinetic energy over a given time period can be used to determine the acceleration of the system, by determining the change in velocity over a particular period of time. I know that this acceleration is just the sum of the forces in the x direction divided by the system mass (from F = ma), which in this case is just the force of thrust and the force of friction. Alternatively, I have a friend who may be able to get me a wireless lab accelerometer which would enable me to effectively skip steps 1-2 and measure acceleration directly.
3. From there, I can use the equations describing rolling friction and the force of thrust (mass flow rate * velocity of the air stream) to determine what would essentially be the measured/actual values of frictional force and thrust force, as compared to theoretical/calculated ones.
4. I may have to either take measurements at regular intervals or find an equation that describes the rate of change of the frictional and thrust forces, given that they will change as the system mass does.

Please correct me if I'm interpreting this incorrectly--I have not had much opportunity to experiment in lab with mechanical/kinetic energy conceptually. This makes a lot of sense to me... this then means that I would be able to compare the 'measured' force of friction determined using other measured values to the calculated force of friction given ideal experimental conditions, right? (one main goal of the project is to determine the % error between measured and calculated values of certain variables.)

Thank you for all of the insight that you've given me thus far. This has been very enlightening for me.

 

[Dr. Courtney]

What percent of the total mass is the mass of the air that gets expelled?

How small do you think the change in mass needs to be to safely be neglected?

 

[Andy Resnick]

I'm not extremely experienced with the topic, but this is what I understand this to mean. Unfortunately, my teacher does ultimately want me to come up with an experimental value for frictional force, but my understanding is that I can determine it indirectly using other observed values using a process such as the below: <snip>

(one main goal of the project is to determine the % error between measured and calculated values of certain variables.)

How about this: you start off with a certain amount of potential energy by compressing the air. Given a measurement of the pressure and knowing the temperature and volume, you can calculate this energy. Then you set up your car and remove the tack to allow air to escape. What happens next? Lots of things, but in essence: 1) The potential energy of compressed air is converted into kinetic energy of moving air and car/bottle and 2) frictional/viscous forces act on the wheels/axles/outflow/etc to dissipate the kinetic energy.

Note that it's possible the car never moves- if you like, the escaping air doesn't produce enough force to overcome friction. Perhaps the car moves at a (near) constant velocity: the sum of forces = 0; the frictional force = thrust force. Perhaps the car accelerates, so thrust > friction. Maybe all three happen during the course of an experimental run.

This is why I think it would be easier to think in terms of power: rate of change of energy. You can estimate the rate of change of pressure (and mass), it's probably linear for most of the time, especially given 10% precision. Then you know the rate at which potential energy is lost. (say 0.1 J/min). The motion of the car+bottle tells you the rate of change of kinetic energy (say constant speed, so 0 J/min), so the deficit is the rate at which potential energy is lost to dissipation and over the duration of 'experiment', you have lost (say) 0.03 J (20 second duration). Note, I included kinetic energy carried away by the exhaust as dissipated.

I like the idea of attaching an accelerometer to the car: iPhones have apps that can output the sensor (SensorLog is one,AccelMeter is another). Not sure if that will work as iPones are (relatively) heavy and I don't know the sampling rate/device precision, but you get the idea. Then you have directly measured the rate of change of kinetic energy of the car.

Now your teacher says "ok, but I told you to calculate a force". In my example above, the thrust force = friction force, but how do you convert that 0.03J into a force? You need to know the total distance that the car traveled (say 2 meters); the calculated average force is then 0.015 J/m = 0.015 N.

There are other ways to do this- use the rocket equation to calculate the acceleration, etc. etc. My approach is different and grounded in the fact that energy is measurable but force is not. There is no such thing as measured/actual values of force. Thinking about forces as measurable quantities leads to other problems as well- say you observe that while air is escaping, the car initially accelerates, reaches a constant velocity for some time, and then the velocity slowly decreases back to zero. It's possible to conceptually treat this process in terms of an initial large (net) force, a period of zero net force, and a period of small negative force. But that model only describes the net force, which you can calculate knowing the mass (whether assumed constant or not) and measured accelerations. It doesn't tell you anything about the component forces and how they may change in time. Dissipative forces, in particular, are often called 'passive' forces because the magnitude of the force can vary in response to other applied forces- as you jump off the ground, the normal force between you and the ground varies even though the acceleration of your feet never exceeds zero until your feet have left the ground. Using the force law F ≤ μN is another example of a passive force. Air resistance varies with velocity.

As for the main goal- compare measured and calculated values of certain variables, I'm not trying to over-complicate your situation but trying to educate you on experimental science and principles of measurement. The goal sounds simple, and in fact most introductory labs use a phrase like this. The problem is that it's a misleading thing to say. As you can see, using good measurement principles makes this experiment somewhat cumbersome, which is why intro labs often blur the distinction between measurable data (position, velocity, acceleration, mass) and conceptual tools (forces) as the analysis becomes much simpler. And in the end, you may choose to go this route. Just remember that forces cannot be measured.

 

[russ_watters]

What percent of the total mass is the mass of the air that gets expelled?

How small do you think the change in mass needs to be to safely be neglected?

Pre-requisite question: why do you want to know the mass of the air? What does it do for you/allow you to calculate?

Then you can move on to the above questions.

 

[Dr. Courtney]

Pre-requisite question: why do you want to know the mass of the air? What does it do for you/allow you to calculate?

Then you can move on to the above questions.

Since mass is not constant, Newton's second law is not F = m dv/dt, but rather

F = dp/dt = m dv/dt + v dm/dt. Knowing the mass of the air and the time span of the propulsion event allows estimating the dm/dt term.

If the dm/dt term is small enough, the changing mass can be neglected. In most interesting rocket propulsion problem, the changing mass CANNOT be neglected, but it is worth considering, and students should give it some consideration.

 

[russ_watters]

I wasn't asking you!

 

[AEGIS]

How about this: you start off with a certain amount of potential energy by compressing the air. Given a measurement of the pressure and knowing the temperature and volume, you can calculate this energy.

This is a bit of a digression from the original topic (and perhaps might best be answered on the engineering forum), but I thought I might as well as it here as well since we're already on the issue.

I looked into determining the potential energy of a compressed gas in a cylinder and found the information below:

Since the process is quick, we’ll consider it to be adiabatic. Then, PV^g=K, where I am writing ‘g’ for gamma. For air, g=7/5. The value of K can be obtained by putting in the initial values of P and V, which you had given.

Work done= integral (P-Pa)dV between V1 and V2, where V1 is the volume of the compression chamber and V2 is (V1+volume of barrel), and Pa is the outside pressure.

After integrating, W=K[V2^(-g+1)-V1^(-g+1)]/(-g+1) – Pa(V2-V1).

Thankfully I've already taken calculus, but I was a bit confused as to what the 'compression chamber' and 'barrel' of my setup are (this seemed to apply more to guns, but the thread that this post originated from here seemed to have a setup very similar to my own.) Since this is not strictly physics this may not necessarily be in your field of expertise, but do you know if the above scenario/equation does in fact apply to my setup? If it does, what are the 'compression chamber' and 'barrel' of the empty bottle with the orifice (which seems to be a single chamber)?

Sorry if I'm misunderstanding something regarding determining the initial potential energy of the container with the compressed air given P, V, and T, but I don't believe I've yet learned how to determine this.

 

[Andy Resnick]

Sorry if I'm misunderstanding something regarding determining the initial potential energy of the container with the compressed air given P, V, and T, but I don't believe I've yet learned how to determine this.

Not at all, and I hope I haven't created unnecessary work/confusion on your end. The work required to compress a gas is given here:

https://en.wikipedia.org/wiki/Compressed_air_energy_storage

Using pV = nRT = constant, the isothermal work done to compress the air W = p_iV_iln(p_i/p_f). The initial pressure is ambient air, the volume is constant, etc. etc. If you want to use pV^γ= constant, you may be able to work out the integral yourself. Again, this may be getting you too far off-track.

Let's review the main idea: pressurize the tank, set up the car and remove the tack. Time the process ('t') and measure the distance the car traveled ('d'). Then calculate how much energy you started with (p_iV_iln(p_i/p_f) is as good anything), determine the rate at which energy was lost (W/t) and the corresponding frictional force (W/d). Anything else- using an accelerometer, etc.- is likely a distraction.

Again- in this case, measuring the pressure is a *lot* easier than measuring the mass. You can convert between the two with tables (I gave you a reference earlier) if your teacher insists on reporting a rate of mass loss.