Question

Construct a hypocycloid, rolling circle 50 mm diameter and directing circle

175 mm diameter. Draw a tangent to it at a point 50 mm from the centre

of the directing circle.​

Answer 1

Answer:

Given Data:- Diameter of rolling circle Φ = 50 mm

Normal and tangent point = 30 mm above straight line.

Assumption – Circle is rolling towards right side (Show the direction)

Procedure:-

1) Draw a circle of 50 mm diameter.

2) Draw horizontal and vertical axis and mark centre of the circle say C.

3) Divide the circle into 12 equal parts and name each division 1, 2, 3, ….12 as per shown in fig.

4) Draw a straight horizontal line of length πD from point P at the contact surface of circle and ground .

5) Divide the line into 12 equal parts 1’, 2’, 3’…..12’ (same no. as that of circle).

6) Draw again a circle of 50 mm diameter at πD distance with centre C’.

7) Draw horizontal and vertical axis for 2nd circle.

8) Draw horizontal lines from points (1,11) (2,10) (3,9) (4,8) (5,7) and 6 up to vertical axis of 2nd circle .

9) Draw vertical lines from point 1’, 2’, 3’,…. 12’ up to horizontal axis and name it C1, C2, C3….C12 respectively.

10) Taking C1 as centre and 25 mm radius (radius of rolling circle) cut the horizontal line passing through point on the circle near point P. Mark that point P1.

11) Repeat the same procedure up to C12 and accordingly marks points up to P12.

12) Draw smooth curve passing through all 12 points (P1, P2, ….. P12) and name the curve.

13) Mark a point M on the curve at a distance of 30 mm from horizontal line.

14) Taking M as centre and 25 mm radius (radius of rolling circle) cut the horizontal axis and mark that point Q.

15) Draw perpendicular from Q on horizontal and mark it as N.

16) Draw a line passing through M and N (NMN is normal).

17) Draw a line perpendicular to normal from point M (TMT is required tangent).

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Answer 2

Answer:

To construct a hypocycloid having a rolling circle of 50mm and directing circle of 175mm diameter. Following below are the construction steps:

1. To determine, use the formula r/R * 360° = 20/100 * 360° = 72°.
2. Draw an arc PQ that subtends an angle = 72o at any point O with a radius (R) of 100mm.
3. P should act as the generating point. PC = r = 20 mm = rolling circle radius on OP mark.
4. Draw the rolling circle with C as the center and r as the radius (20 mm).
5. Given that it is expected that the rolling circle rolls in a clockwise direction (CW), divide the rolling circle into 12 equal sections as 1, 2, 3, etc.
6. In issue 6, repeat steps 5 through 10 to create a smooth Hypocycloid curve. At a given point M, the Tangent and Normal are.
7. Using r=CP and M as the radius At the N-point, cut the locus of the center.
8. JOIN ON and extend it to S, where it will cross the base circle.
9. Join the Normal, MS.
10. Draw a line parallel to MS at M to obtain the necessary Tangent.

Explanation:

A hypocycloid is a special type of curve that is formed by tracing a point on a smaller circle as it rolls inside a larger circle. The curve is defined by the equation:

$x = (R + r) cos(t) - r cos((R + r) / r * t)\\y = (R + r) sin(t) - r sin((R + r) / r * t)$

where R is the radius of the larger circle,

r is the radius of the smaller circle,

and t is the angle between the x-axis and the line connecting the center of the smaller circle to the point on the smaller circle that is in contact with the larger circle.

The number of cusps (or loops) on the hypocycloid depends on the ratio of the radii of the two circles. For example, a hypocycloid with R / r = 3 will have 3 cusps.

Hypocycloids have various applications in physics, engineering, and mathematics. For example, they are used in the design of gears and cam mechanisms, and in the study of fluid dynamics and wave propagation.

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