# What are the reasons behind the different period in vertical and horizontal circular motion?

Why does hand-spinning a small mass in a vertical circular motion produces different period if the same mass is spun in a horizontal circular motion given that the radius, slotted mass are kept the same?

I did the following 2 experiments and collected their period with the same150g slotted mass, 1m radius. where I hand spin a mass (red) that is attached to the same string with the slotted mass at the other end. I spin the mass by grabbing the hollow tube (blue). Same applies to horizontal.  Now the period that I collected is $$0.8s$$ for horizontal circular motion and $$0.7s$$ for vertical circular motion. So my question is why do they produce a different or similar period? Is there a mathematical reason to explain why this happens? What are the reasons/physics concepts behind the different period?

• How do you control that you apply the same amount of torque in both experiments? The angular velocity/period is not defined by only the radius and the mass. It depends on the amount of energy you put into the system. Apr 15 '20 at 12:35
• $0.7\text{ s}$ and $0.8\text{ s}$ is not a huge difference. Are you sure the $0.1\text{ s}$ is statistically significant?
– Gert
Apr 15 '20 at 12:57
• I've no great interest in the problem but will note this. Your 'horizontal' circle cannot be perfectly horizontal because of gravity. Gravity means that the vector $\vec{r}$ has an angle to the horizontal, ALWAYS.
– Gert
Apr 15 '20 at 13:52
• the vertical circular motion will not be homogenous, it will usually be slower at the top and faster on the bottom, so you compare different things, if you find any type at home you can do the experiment there and see, that your weight got up and down if you spin vertically, but stays ar the same height, if you rotate fairly constant horizontally. Apr 15 '20 at 14:42
• @Gert : And, something more stable and less prone to error when it comes to spinning it than your very imprecise hand. I would suggest a small electric motor, suitably mounted, and connected through a rheostat. Apr 19 '20 at 2:05

Probably you horizontal and vertical frequencies are similar, so the the centripetal forces are similar, but since one is a constant frequency, the other is not, so why should the effect be the same. In the vertical case the velocity at the top can be very small, the velocity at the bottom you can calculate from $$v_t^2/2+4gr=v_b^2$$ For the horizontal motion you should not measure the string length, but the distance of the circling mass from your tube.

• trula thanks for replying. What do you mean by constant frequency? Also this does not really answer the question. Yes I get that velocity on top is < velocity bottom. So what though? I could not see the correlation between different velocities at different times and the difference in period? Apr 17 '20 at 12:08
• the frequency of the horizontal circle is determined by the centrifugal force the weight at he end gives. Ao you have $m\omega^2*r=F$ . The time for the vertical loop is a more complicated problem to calculate, but why should it be the same?. Apr 17 '20 at 17:39
• isnt v the tangential velocity? Apr 18 '20 at 7:41

Gravity for sure has a play in this Mathematically when we have to find tension in a string during vertical motion we write T-Mg=Mv^2/r I.e acceleration at every point is changing so time period for such motion is difficult to determine Case 2 In horizontal circle tension solely is responsible for acceleration T=mv^2/r In this time can be calculated by T= 2 pi/ omega

• maybe you explain why you expect the same frequency for two different experiments? Apr 18 '20 at 17:09

If you wanna go, in a given time, from a point A to a point B, then A and B will be the farthest apart if you travel directly to B with constant velocity. Just so, if you travel with constant velocity on a circle (like in the horizontal trajectory), then, for a given time, you will have traveled the biggest angle possible. If you travel sometimes faster, sometimes slower (like in the vertical circular trajectory), then the distance traveled in that same time will be less and so the vertical trajectory takes more time. So your measurements are not accurate.

Maybe this begs the question of why traveling with constant speed in a given time takes you the farthest. but similar questions about this have been asked here. This question and its answers shine a light on this topic. It's Fermat's Principle. principle.