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If a scale model has 0.5x the original length in all directions, should its mass be 1/8th of the original mass?

DarkLightA
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  • probably yes, because the volume is proportional to the cube. – elcojon Jan 15 '13 at 21:28
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    Does the model use material of the same density? – Qmechanic Jan 15 '13 at 21:32
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    What is the purpose of your model? If it is just to look at, then the mass is irrelevant. If it is for some sort of test or simulation purposes, then it is a more complicated question. There's no a priori reason a model should have the same volumetric mass density. – user1631 Jan 15 '13 at 21:34
  • There is actually a lot of thinking that you have to do to answer this problem for any particular system. In some systems you have to use the same density, in others there are otehr variables (like Reynold's numbers) that need to be maittained). Sometimes there are competing requirements that limits how big a scaling you can use and still get representative results. Ship builders, for instance, don't scale down by more that about 3:1. – dmckee --- ex-moderator kitten Jan 15 '13 at 21:37
  • @user1631 Good question. I'm using a wood plank with wheels to simulate the effect of braking with the front wheel and the lift it causes on the back. I found the ratio of horizontal to vertical distance to be 1:1.76 for the center of mass of a bike to be replicated, but if I use 25 cm from the hub (Bike was 50cm), I'll end up having to use around 10kg of weight to get the right center of mass for both the rider and the bike, which will be a problem. – DarkLightA Jan 15 '13 at 21:40
  • @DarkLightA Ok, so you're adding weights to make sure it has the right center of mass, but the problem is that this will give a total mass higher than 1/8 of the mass of the original? – jkej Jan 15 '13 at 21:51
  • Well, the real problem is really that I'm unsure whether the construction I've built will be able to support 10kg. – DarkLightA Jan 15 '13 at 21:55
  • Would 1/8th the mass be correct in this situation though? – DarkLightA Jan 15 '13 at 21:56
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    By the way, if this is intended to mimic the behavior of a motorcycle (or a bicycle with articulated suspension) it will fail unless it also has a suspension with similar behavior. Weight transfer is heavily influenced by suspension. – Colin K Jan 15 '13 at 21:58
  • Good observation. However, it is for a bicycle. I guess I'll have to specify that suspension isn't taken into account. – DarkLightA Jan 15 '13 at 22:00
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    If you just want to know, for example, the max deceleration before the back wheels come up, you could scale the mass however you wanted, since inertial and graviational forces both scale the same way with m. You need to write down all the pertinent equations and see how things scale. – user1631 Jan 15 '13 at 22:09
  • @User1631 Thank you for the help! I'm planning to have a model hit a wall at a set velocity, and will measure the angle the back wheel bounces up to, changing the center of gravity in one direction. I'd like the torque produced to be the same as for a real bike, relative to the weight. Could you help me with how this would work out? Sorry, I'm a high schooler and currently haven't covered torque. – DarkLightA Jan 15 '13 at 22:31
  • Depends on whether you are building a fractal. – Mark Eichenlaub Jan 16 '13 at 19:23

2 Answers2

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Yes, assuming it has the same density. Volme scales as length to the power of three and mass is proportional to volume.

$$ \left(\frac{1}{2}\right)^3=\frac{1}{8} $$

jkej
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OK, now we have a better idea what you're trying to do. If we can assume elastic collisions, then answer should be independent of the mass of the bike (though it will depend on the relative distribution of the mass, i.e. center of gravity and moments of inertia). However you will need to think how to scale the velocity of your model.

One way to think of it: you want to make a coordinate transformation from your model system that replicates the physics of your target system. The length transformation from model to target system is $l_t = 2*l_m$. However, your model system has a gravitational acceleration of $g = 9.8 m/s^2$, so your simulated target system will have an effective gravitational acceleration of $g_t = 2*9.8 m/s^2$ which is not right. How do you fix this? You have to rescale the time between model and target systems: $t_t = \sqrt2*t_m$. This in turn means your models velocity will be related to the target system velocity by $v_m = l_m/t_m = (\sqrt2/2)*l_t/t_t=v_t/\sqrt2$, so you will want to reduce the velocity of your model accordingly.

user1631
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