Locus - Pocus

Geometry Level 4

An uniform Semi-circular disc is sliding on a smooth wall and smooth horizontal Surface.

Then Find Locus Of point 'P' which is fixed with respect to disc ?

∙ \bullet Is this is physics or a maths Question ?

Ellipse straight line circle Parabola

Deepanshu Gupta by Deepanshu Gupta , 25, India

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2 solutions

Let us the wall and horizontal surface intersection point designate by O. It is easily seen that all four points A,O, B and P are equally spaced from the middle point of the diameter AB and hence all lie on the same circumference with the radius R=AB/2. Then angles POB and PAB are equal (as subtending the same arc PB) and every instant point P belongs to the line making the same angle with the horizontal surface.

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There also exists a simple physical solution. Let us construct two orthogonal lines - one from the point A (perpendicular to the wall and hence to the instant velocity of the pont A), and the second - from the point B (perpendicular to the horisontal surface and the instant velocity of the point B). There intersection -point Q - will be the point of instant rotation of the semi-circle. Four points P,B,Q, and O (O - the intersection of the wall and the horisontal surface) will lie on the same curcumference with the diameter QO. The instant velocity of the point P will be perpendicular to the line QP, which means that every moment the point P velocity will direct to the point O and its trajectory will form a part of straight line.

Сергей Кротов - 6 years, 6 months ago

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Yeah Nice :) I solved This question in a Math's Book and Then i tried To solve it By Using concept's of physics (i.e Instantaneous Centre of Rotation (ICR)) and Succeed , So I posted This question on brilliant.org by modifying it ! And You did it by Both method !

Deepanshu Gupta - 6 years, 5 months ago

https://brilliant.org/problems/find-the-areaonly-your-logic-can-help-you/?group=3UHxOzwinQpA&ref_id=384997

PLEASE TRY TO DO THIS AWSOME PROBLEM TOO..post a solution if you get......................i am waiting for an awesome solution that i made while creating this problem

Yash Sharma - 6 years, 3 months ago

The question clearly states that the point is fixed to the disc( forget the diagram for some time). Take A A or B B to be the point mentioned in the question( now look at the diagram). Clearly the locus is a straight line .

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