Round & Round, It Goes
A coin of diameter 1 cm rolls from point B along the (solid) arc of a big circle and reaches point A after revolving 8 full rounds.
If the (red) line segment A B is 24 cm long, how many rounds in total will this coin need to revolve to complete one full circle (from B back to B)?
The answer is 48.
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1 solution
In the second to last step, what are you doing? It seems like you're saying
c o s ( x ) = 1 - y -> s i n ( x ) = y
but I don't think that's an identity
@Worranat Pakornrat Is there any way we can "solve" the equation?
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Yes, there is. By using sine rule for triangle ABP, where P is the point on the big arc, we will obtain sin R 4 π = R 1 2 .
Combining with your results and using sin 2 x = 2 sin x cos x formula, we will have:
2 R 2 4 2 = 2 1 − R 2 1 2 2 ( R 1 2 )
Then solving for R, we will get R=24. ;)
How many rounds does a coin radius 1 take around another circle of radius 5 ?
Its 6, not 5..
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That's when you lie it flat on the ground, and the center is shifted outwards. But in this case, it's rolling, and its center is along the circular arc.
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It wasn't a question but a statement which meant to question the answer being 48..
I mean by the above analogy the answer should be 49 as the circle is rolling on circle.... Not 48
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@Vishal Yadav – @Worranat Pakornrat I am inclined to agree. The rotations of the coin here include not just the distance moved along the perimeter, but the revolution that it would have made.
For example, if we had the coin rotate about a point, to me it would have made 1 revolution, even though the "circumference distance travelled" is 0.
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@Calvin Lin – The picture makes it clear that the coin is perpendicular to the page. The added revolution only applies if the coin is parallel.
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@Alex Li – Cool..but the question must be made clearer. Can I say that the coin revolves around a sphere(along a particular circle, if sphere is supposed to made of infinite circles with varying radii from R to 0.) And if it is so then are points A and B in 3d ( lying in plane passing through the the centre at the intersection of sphere and the plane)?
This is simply hard to imagine..:(
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@Vishal Yadav – I think the question and the picture made it quite clear that it's rolling on the ground. By making it revolving around the sphere is simply not as practical and can add extra revolution to the answer, making the question more complicated.
@Calvin Lin – I agree with Alex. The added revolution around the point is necessary if the center is off the point, but we just spin around itself, there's no need to add any extra round. In the question, the center of the coin is along the circumference, so the distance travelled is the distance of revolutions, no need to add extra one.
R × θ = n o . o f r e v o l u t i o n s × d i s t a n c e c o v e r e d i n a s i n g l e r e v o l u t i o n . . [ θ = r a d i u s a r c
R × θ = 8 × 2 π r
Is there other formula which is used ?
The circumference of the coin is π and the arc length is 8 π . Let us assume that the arc subtended angle θ at the centre of the circle of radius (say) r and centre O .
Applying the cosine rule to Δ O A B we get cos θ = 2 r 2 2 r 2 − 2 4 2
Also we have r × θ cos θ = 8 π = cos r 8 π Equating the above two equations we get cos r 8 π sin r 8 π = 1 − 2 r 2 2 4 2 = 2 r 2 2 4 2
We can now see that r = 2 4 satisify's the above expression. Hence the number of revolutions required will be π 2 π × 2 4 = 4 8 .