# Does a scale model with 1/2 the linear length have 1/8 the mass?

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

-
probably yes, because the volume is proportional to the cube. – elcojon Jan 15 at 21:28
Does the model use material of the same density? – Qmechanic♦ Jan 15 at 21:32

1
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 at 21:34

1
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 at 21:58

1
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 at 22:09
show 8 more comments

## 2 Answers

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.

-

I think this is the answer to a different question. What bike, what velocity?? – Marcus Jan 17 at 14:27
Read the comment thread below the original question. – user1631 Jan 17 at 17:42

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

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

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9329211711883545, "perplexity_flag": "head"}