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If I have a cart that has a force sensor on it and then I have the available supplies: Motion sensor, computer to read the motion and force sensors, weights and weight holder, wood blocks (tilt the track) and also some springs, then how could I go about getting the mass of the cart without a scale or spring scale?

The computer outputs a graph of the position, velocity, acceleration, and the force.

I'm guessing there are multiple ways to go about determining the mass of the cart, but I'm not sure what all of them could be, so I'll write about my idea, but it would be nice if someone can help confirm I'm doing it correctly and if they have any other ideas.

My first idea is to tilt the track (with the wood blocks) and then start recording the data of $x(t)\;v(t)\;a(t)\;\;and\;\; F(t)(not \;useful\;here)$ so that I'm collecting data that I can then get values from. So, start with the track tilted and the cart being held at zero velocity and then release it. I think that it doesn't matter if I let the cart come to rest or if I stop recording data while it is still rolling (shouldn't make a difference in my calculations). Then I can use these equations to find acceleration: $$v_f^2=v_i^2+2a(x_f-x_i)\;\;\;or\;\;\;x_f^2=x_i^2+v_it+{1\over2}at^2$$ I can only use the second if I keep track of time. So, at this point, I may be able to use Newton's second law $F=ma$ but I'm not sure what to do with the force if I want to solve for m. I'm stuck at this point for this idea.

For my second idea, I can just pull on the force sensor hook and measure the force and acceleration but I'm not sure what to do with friction here. If I ignore friction (I would like to be able to justify doing that) then it should be simple to calculate the mass. I could also use the spring here, but that just complicates everything at this step.

If anyone can answer my above questions I would be very grateful, sorry for them being so easy I just can't get a hang of physics for some reason, it doesn't come naturally to me. Thank you!

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Do your second experiment, where you apply a known force on flat ground, measure the acceleration, and apply $F=ma$. If you’re worried about friction, measure it by bringing the cart to a known velocity and observing it decelerate. If the friction forces end up being much smaller than the force you applied in your first measurement, then don’t worry about it. If not, then you’ll have to characterize the friction more carefully and disentangle them.

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  • $\begingroup$ Yeah, that is the easy way. I'm assuming that friction won't be a problem since the cart starts rolling even if the track is just slightly higher on one end. Can you think of any other ways (maybe obvious?) that I could go about measuring the mass in my given situation? I'm still not completely familiar with all the equations so I can't see any obvious methods. Thanks for your help! $\endgroup$
    – JustHeavy
    Commented Feb 25, 2018 at 2:06
  • $\begingroup$ Well, you could hang the cart by the force sensor, measure the force, and get the mass from F=ma using a=9.8 m/s^2, the gravitational acceleration. But really, you’re not going to be able to avoid using F=ma because that’s where the mass comes in to the equations. Ideally your computer could just spit out the acceleration as a function of force, and that would give you mass trivially. But you could also get this info indirectly by, e.g., applying a known force for a known amount of time and then measuring the final velocity. $\endgroup$
    – Gilbert
    Commented Feb 25, 2018 at 2:16
  • $\begingroup$ Oh, I didn't think about hanging the cart by the force sensor, that would work, but I only have springs I'm able to use for this experiment, although I don't see why I couldn't just use my fingers to grab it. Thanks again! $\endgroup$
    – JustHeavy
    Commented Feb 25, 2018 at 2:24

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