Math Is Fun Forum

Filip

Member

Registered: 2006-02-24

Posts: 3

THE hardest question on earth (gr 10)

... or at least for a grade 10. This is what the question states.
A semi-circle of radius 8cm, rocks back and forth along a line. The
distance between the line on which the semi-cicrle sits and a line
above the circle is 12cm. As it rocks without slipping, the
semi-circle touches the line above at two points. (When the
semi-circle hits the line above, it immediately rocks back i nthe
other direction.) The distance between the two points in millimetres
is ________.

The semicircle does not rotate around its centre of gravity but rater its center shifts as teh circle rocks form left to right. Yes i know you can figure it out by using cycloids but its grade 10.. not calculus.

Lets see is someone can proove me wrong ("THE hardest question on earth")

Filip

Member

Registered: 2006-02-24

Posts: 3

Re: THE hardest question on earth (gr 10)

Heres a sketch

Messed up a little point A nad B are teh same point

Last edited by Filip (2006-02-24 09:01:21)

ryos

Member

Registered: 2005-08-04

Posts: 394

Re: THE hardest question on earth (gr 10)

You can find the angle between point B, the center, and point A with the inverse sin function: arcsin 1/2. In radians, this comes out to π/6. This is the amount of the circle that rolls to one side before the point hits the top. You can convert this from radians to distance by dividing by the total number of radians in a circle (2π) and multiplying by the circumference of the circle (2π8). This comes out to 4π/3, which is the horizontal distance the semicircle travels.

The diameter of the semicircle is 16 cm. 4π/3 cm are cut out of one side, and 4π/3 are cut out of the other. So, you can find the distance between the two points thusly:
16 - 2(4π/3) = 7.6224.

If you need more explanation I can post diagrams and such; this was kind of a hurried post.

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El que pega primero pega dos veces.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: THE hardest question on earth (gr 10)

So ryos has shown that you can figure out certain points on a cycloid by
working from the center of rotation and adding in the horizontal
movement that the center moves.

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igloo myrtilles fourmis

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: THE hardest question on earth (gr 10)

Yes, exactly.
Yuo can have equation for cycloid using rotation+translation.

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IPBLE:  Increasing Performance By Lowering Expectations.

blah

Guest

Re: THE hardest question on earth (gr 10)

u guys are sooo smart!!!

JUNBONE

Guest

Re: THE hardest question on earth (gr 10)

i think your answer is wrong.. i have my calculation and check it with autocad.. and found out that the answer is 6.6080, try to check also your answer using drawings coz it will verify your answer.

JUNBONE

Guest

Re: THE hardest question on earth (gr 10)

your right that the angle of displacement is 30deg from sinØ=4/8 and Ø=30deg
but dont forget that the semicircle is moving and we need to get its linear displacement. lets say X=to its linear displacement where X=2πr(360deg/30deg)=2π8(360deg/30deg)=4.1888
then we can create a triangle where the two sides are give which is 8cm and 12.1888cm(8+4.1888) and one angle between the two which is 30deg. now using the cosine law which state b²=c²+a² - 2ca cos(B).. therefore the answer is 6.60862...

if u need some diagram.. email me.. junbone102@hotmail.com