Experimentally determining if g depends on mass

[Raze]

Homework Statement

I am designing an experiment where I will measure a value proportional to g, and determining if g then depends on the mass of the object falling. The particular problem explored is my choice, but I really like this one.

I plan on using a long ramp and either some cylindrical or spherical weights (of the same size and shape, different weights). I will do several runs with different weights and graph the data. A straight line indicates that g remains constant regardless of weight.

Homework Equations

x = mgt²

The Attempt at a Solution

I will use the above equation (since initial velocity is zero) to graph a line of my data, where x is position, m is some unknown proportionality (due to the objects moving on a ramp, friction and other things), g is the acceleration due to gravity, t is time. With this set up, I will have a line with a slope of mgt going through the origin. I am NOT trying to find out what m or g is. Rather, what my experiment needs to show is that the slope is constant within my uncertainty (although if I can minimize all the factors affecting the motion, maybe I can get a good value for g).

The main problems:

Before I propose this to my instructor, I need to find out if things like moment of inertia can be neglected, or what dimensions of my ramp and size of weights need to be in order for such things to be negligible. I will also be waxing up my ramp to minimize friction, and since that should be constant I don't anticipate much a problem from that (although will it be constant? isn't friction proportional to normal force, which is proportional to mg? All that matters is that I minimize it, though).

Other potential ideas: What if I put flat weights on little cars? Would this help? If not that, what shapes would be best to minimize friction and energy from the rotation of the body?

So, how would I go about determining what the dimensions of my tools need to be in order to make these potential contaminates negligible? Are there any other factors that would effect my results that I am missing? Any help whatsoever would be appreciated.

 

[Simon Bridge]

The first step is to define your problem.
You need to carefully define what you mean by "g".
You appear to mean the rate of acceleration of a falling object, as measured at a particular place inside your lab/classroom.
Notice how carefully worded this is? This value may vary with geographic location and/or altitude - will your experiment be affected by this?

Also carefully word the question you are investigating:
"How does the value of g depend on the mass of the falling object?"

The wording you used could have been read: "How does g depend on the mass of the Earth?"

Almost all of science is about asking careful questions ;)

Some notes:

if you plan to roll objects down your ramp, then the shape and mass distribution of the object is important - i.e. you generally cannot discount the moment of inertia.

Your questions are good ones and you should devise preliminary experiments to investigate their impact - i.e. can you neglect air friction? Hopefully the masses will be going slowly enough that air-resistance is not important ... but you can test this quite quickly, so do so - it'll be worth marks.

 

[Raze]

The first step is to define your problem.
You need to carefully define what you mean by "g".
You appear to mean the rate of acceleration of a falling object, as measured at a particular place inside your lab/classroom.
Notice how carefully worded this is? This value may vary with geographic location and/or altitude - will your experiment be affected by this?

Also carefully word the question you are investigating:
"How does the value of g depend on the mass of the falling object?"

The wording you used could have been read: "How does g depend on the mass of the Earth?"

Almost all of science is about asking careful questions ;)

Some notes:

if you plan to roll objects down your ramp, then the shape and mass distribution of the object is important - i.e. you generally cannot discount the moment of inertia.

Your questions are good ones and you should devise preliminary experiments to investigate their impact - i.e. can you neglect air friction? Hopefully the masses will be going slowly enough that air-resistance is not important ... but you can test this quite quickly, so do so - it'll be worth marks.

Thanks for the input. How about this: We are trying to determine if the rate of acceleration of an object falling down an incline near the surface of the Earth (around 1000 feet above see level) depends upon the mass of the falling object in question (I believe that is a clearer way of putting it), and by extension, if the rate of acceleration in a straight drop depends upon the mass of the falling object (we can figure this by doing several angles of incline, including near straight down, although the more the incline the higher the uncertainty in time might be).

In any case, I will certainly do some preliminary experiments as soon as I can get to the store and buy some weights and make a ramp (hopefully this weekend). I'm hoping air-resistance is negligible (it should be, as long as the objects are small), and that the effect from the moment of inertia isn't too big.





The issue is that we have not covered rotational motion in my class, and I'm not sure if I'm allowed to use much from that (due to the fact that the rest of the class will be somewhat lost during the presentation and I'm not so sure my group partners will be able to keep up).


Regarding minimizing moment of inertia: It seems to be dependent on r², so I imagine the smaller the weight the smaller this effect. I will try this myself with really tiny weights and a long ramp and see if my data gives me something reasonable. I know I need the rotational energy to be really small compared to the transnational energy.


HOWEVER, something just occurred to me: since I = mr², if I also used really light masses, that could also help to bring that effect down.






If I can't get this to be reasonable, I'll have to go to my plan B, which is just to see if g depends upon the height from which the ball is released. I find that less interesting, so hopefully I can find some values that would minimize the effect from the spatial extent of the weights.



What about this? I suppose first I'll need to figure out what my uncertainty is going to be, and then do some calculations to see what specific values I'll need for

mr² + mv² ≃ mv²


within my uncertainty (is that correct?).


I'm guessing this might be quite impossible, due to both parts depending on m. Maybe I'll need really tiny sized weights and a very high incline (but then this will add to the uncertainty in my time measurement). I suppose I'll see when I do some preliminary stuff.

 

[nasu]

Homework Statement

x = (mgt)t


I will use the above equation (since initial velocity is zero) to graph a line of my data, where x is position, m is some unknown proportionality (due to the objects moving on a ramp, friction and other things), g is the acceleration due to gravity, t is time. With this set up, I will have a line with a slope of mgt going through the origin. I am NOT trying to find out what m or g is. Rather, what my experiment needs to show is that the slope is constant within my uncertainty (although if I can minimize all the factors affecting the motion, maybe I can get a good value for g).

Are you saying that you will plot the distance or position versus time and you expect to obtain a straight line?
Or I misunderstood, maybe.

You need to plot x versus t^2 and then you have a line whose slope will be proportional to g.

 

[Raze]

Are you saying that you will plot the distance or position versus time and you expect to obtain a straight line?
Or I misunderstood, maybe.

You need to plot x versus t^2 and then you have a line whose slope will be proportional to g.

Yes, that's what I'm saying. x = (some stuff) t^2. Wrote that wrong.

 

[Simon Bridge]

You are tracking uncertainty?
You'll want to do each trial about 10 times (or more) so you can work out the standard deviation on the times ... to get the mean time, and the uncertainty on the mean.

note:

$mr^2 + mv^2 \approx mv^2$

... your units don't match there.
You will need to be more careful with your equations if you want to make sure the rotation does not significantly impact on your experiment. The fastest and surest way is to just test it out empirically. Nature knows more physics than you do.

also note: $mr^2$ is the moment of inertia only for a particular geometry.
The same mass and radius in a hoop will roll down the incline in a different amount of time to a uniform solid ball. If your rolling masses are small, small diameter, and light, then you may get away with it ... or you can try just putting them on a cart like you thought.

It would be a nice finesse to use your experiment to measure g though.

 

[Raze]

Just an update on what we did (and thanks again for the suggestions) since you all gave great suggestions:

(1) Instead of measuring g, we decided to measure the coefficient of friction at the end of the ramp, using a completely inelastic collision at the end of the ramp (using a tennis ball filled with hot glue and a empty box with a velcro strip, which we filled with varying masses). We did this by deriving an expression between the distance the new mass traveled and the coefficient of friction (using conservation of energy, conservation of momentum, and Newton's second law).

(2) Regarding friction down the ramp, we derived an expression for acceleration neglecting friction, and it was found to be (5/7)*g*sin(theta), using torque and moment of inertia, etc. We then used kinematics to find the initial velocity just before collision. However, we in the end also used conservation of energy for this and it was much much easier. But in any case, the absolute value of the acceleration would be the above minus Force of friction/mass, but it turns out for the angle of our ramp the friction force/mass is much smaller than the acceleration without friction with our ball. But again, this wasn't even the most interesting part of our experiment.

(3) We used conservation of energy, including the moment of inertia for a sphere (which turned out to be a really good approximation since we basically made the tennis ball solid by filling it with hot glue), then conservation of momentum after collision, and Newton's second law for the motion of the combined masses (which we used to get an expression for friction). With conservation of energy we used that and obtained an expression for friction to get an expression for work, which then related the coefficient of friction with the displacement of the combined mass after collision.

This in the end worked out pretty well. We were able to make good predictions about how far the masses would move using our obtained value for the coefficient of friction, so I can only assume that all the approximations were reasonable.

Of course, I can only imagine there is an easier way to get this relationship, because we basically had to do 4 separate calculations (2 conservation of energy, 1 conservation of momentum, one Newton's 2nd law + a free body diagram).

Anyway, thanks for all the help. From what I can tell, our professor likes it, and the physics actually worked (we were quite excited about that), which makes it a complete success.

 

[Simon Bridge]

You are doing well - the retarding forces would normally be measured by comparing with the actual acceleration with that predicted by a model that excludes retarding forces.

The geometry of the experiment is not clear to me - you are rolling a tennis ball down a ramp or sliding a box or what?

If rolling - then how does the concept of a "coefficient of friction" come into it?
Tennis balls are usually furry rolling requires using energy to bend the fur strands flat(ish)... is that "friction"?
i.e. is is reasonable to expect rolling resistance due to deformation of the ball to be proportional to a the normal force?

Did you try the experiment at different slope angles?

For a box sliding a distance $d$ down a slope $\theta$, the gravitational PE lost falling height $h=mgd\sin\theta$ goes into kinetic energy and work $W=Fd$ against friction $F=\mu N$:$$mgd\sin\theta = \frac{1}{2}mv^2 + \mu mgd\cos\theta$$
... notice how the mass cancels out?

Ideally you want to collect lots of values of $v$ and $\theta$ and devise some sort of graphical method to find $\mu$.

 

[Raze]

Thanks for the response Simon.

Anyway, the set up was, we rolled the ball down the ramp to give it some velocity, and then at the bottom it collided with a box, and the two slide together along a flat surface. What we were trying to measure was the coefficient of friction of the flat surface.I personally wanted to do something more with the ramp, but I was outvoted. :) However, I liked being able to use so many calculations in the report, so in the end I guess it worked out for the best. In any case, I don't think that the friction from the tennis ball rolling down the slope would be proportional to the normal force anyway, at least not in the way we've considered so far in the class. Or what I mean is, it doesn't make much sense to me for the relation between normal force and friction to be completely independent of the shape of the object in question.

In our class we've basically just approximated everything as a point mass, but to me that makes zero sense. Wouldn't the AREA of the bottom of the object play a major roll in friction? But I suppose the math would be too sophisticated for this class. On the other hand, it must be reasonable to approximate extended objects as point masses with this because as I said earlier, we were getting solid predictions for how far the ball+cube slide along the table.

Thanks again for your input!

 

[Simon Bridge]

In any case, I don't think that the friction from the tennis ball rolling down the slope would be proportional to the normal force anyway, at least not in the way we've considered so far in the class. Or what I mean is, it doesn't make much sense to me for the relation between normal force and friction to be completely independent of the shape of the object in question.

What does or does not make sense to you makes no difference to the physics - have you conducted an experiment to see if the friction depends on the normal force alone?

The fact you are getting solid prediction for how far the ball+block slides should be telling you that the area of the bottom of the object does not play a big role in determining the amount of friction. This is generally what students find at college level when this is investigated experimentally ... the shape of the object does not matter, the area of contact does not matter.

Also see:
http://hyperphysics.phy-astr.gsu.edu/hbase/frict3.html
... but you should see it for yourself experimentally.