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Archimedean Spiral – How to draw Archimedean Spiral

Archimedean Spiral

Archimedean Spiral: If a line rotates in a plane about one of its ends and if at the same time, a point moves along the line continuously in one direction, the curve traced out by the moving point is called a spiral.

The point about which the line rotates is called a pole. Each complete revolution of the curve is termed the Convolution. A spiral may make any number of convolutions before reaching the pole. Archimedean Spiral:  It is a curve generated by a point moving uniformly along a straight line, while the line swings or rotates about a fixed point with uniform angular velocity.



Example 1 – Archimedean Spiral

A point P is 120 mm away from the fixed point pole 0.A point P moves towards pole 0 and reaches the position Q in one convolution where OQ is 24 mm. The point P moves in such a way that its movement towards fixed point 0, being uniform with its movement around fixed point pole O. Draw the curve traced out by the point P. Name the curve.

Archimedean Spiral 1

Procedure:

  1. With pole O as centre and radii equal to R1 = 120 mm and R2 = 24 mm draw two circles.
  2. Divide 360 degree at pole O into twelve equal parts and draw radial lines OP, O1’, O2’…, O12’.
  3. Divide (R1 – R2) = 120 – 24 = 96 mm, length also into same twelve equal parts as shown in the figure.
  4. Initial position of point is P.
  5. Point P 1 is obtained by cutting radial line 0 – I ‘ by arc with centre 0 and having radius as 0 – 11 length.
  6. Similarly, points P2, P3, .. etc. are obtained.
  7. Observe that point P 12 is coincident with Q.
  8. Thus, point P starts moving towards pole 0 and reaches the point Q in one convolution and hence the curve obtained by joining P, P 1 , P2 ,… P 12 in proper sequence is an Archimedean Spiral.

Example 2 – Archimedean Spiral

A line OA, 105 mm long is rotated in the plane of the paper clockwise, about the pivot point 0 with uniform angular velocity, while a bead is free to slide along the line outwards from the centre of rotation with uniform linear velocity. If the centre of the bead moves from its initial point P (which is at a distance of 25 mm from 0) to P’ (which is at a distance of 97 mm from 0) during one complete revolution of the link, draw to suitable scale, the locus of the point P. Also draw a tangent and a normal to the curve at a point M on the curve which is at a distance of 76 mm away from the centre O.

Archimedean Spiral How to draw Archimedean Spiral

Procedure:-

  1. With pole 0 as centre and radii equal to R = 1O5 mm and R2  = 25 mm, draw two circles.
  2. Divide 360° at pole 0 into twelve equal parts and draw radial lines OA, OA1, …OA12 as shown.
  3. Divide the length P’P i.e. 97 – 25 = 72 mm length also into same twelve equal parts as shown in the figure.
  4. Initial position of point is at P and end position of point P is P’.
  5. Point P1 is obtained by cutting radial line OA1 by arc with centre O and having radius as O – 1 length.
  6. Similarly, points P2, P3, … etc. are obtained .
  7. Observe that points P12 is coincident with P’.
  8. Thus, bead at P starts moving towards P’ in one complete revolution of the link and hence, the curve obtained by joining P, P1, P2, .. P12 in proper sequence to get a curve called Archemedian Spiral.
  9. To get Tangent and Normal to the curve at point M, first mark the position of point M on the curve by OM = 76mm.

Next, obtain the constant of the curve as mentioned below:-

Constant of the curve = C = (Difference in length of any two radius vectors / Angle between the corresponding radius vector in radians) = (OP’ – OP9)/(π/2) = (97-79)/1.57 = 11.465 mm

Therefore, C = OQ = 11.465 mm.

The normal to an Archemedian spiral at point M is the hypotenuse QM of the right angled triangle QOM as shown in the figure. The tangent T-S is drawn perpendicular to the line QM at point M.



 

Example 3 – Archimedean Spiral

A straight line AB of 60 mm length rotates clockwise about its end A for one complete revolution and during this period, a point P moves along the straight line from A to B and returns back to the point A. If rotary motion of the straight line about point A and linear motion of point P along AB are both uniform, draw the path of the point P. Name the curve. Draw a normal and a tangent to the curve at a point 35 mm from the point A. 

Archimedean Spiral drawing spiral

Procedure:-

  1. Follow the procedure explained for the construction of archemedian spiral, except divide the distance AB into 6 equal parts and circle into 12 equal parts.
  2. Point P starts moving from A and reaches at B in 1800 of revolution and returns back to A in further 180˚ of revolution and hence the required curve is P, P1, P2, ..P6, P7, ..P12 as shown. The curve is an archemedian spiral.    ·
  3. Tangent and normal at point M is obtained with usual procedure.

Example 4 – Archimedean Spiral

A thin square plane ABCD of 50 mm length of each side is standing on its corner C with diagonal AC vertical. The plate is rotated about the diagonal AC for one complete revolution. During this period, a point P moves from A to B assuming uniform motions trace the paths of point P in front and top views. Name the curve traced by the point P.

Archimedean Spiral 4

Procedure:-

  1. Draw plan ABCD and elevation A’B’C’D’ of a square plate ABCD as shown.
  2. Mark the diagonal of square plate as line A’C’ in elevation and point view AC in plan.
  3. When the place is rotated about the diagonal AC through one complete revolution, the point P moves from A to B in elevation represented by path A’, A 1′,A2′, ..A 12′ and in plan by path A, A1, A2, ..A 12 as shown in the figure.
  4. Curve traced by the point P in the plan is a spiral.

Example 5 – Archimedean Spiral

A rectangular door ABCD has its vertical edge AB 2 m long and a horizontal edge BC 0.8 m long. It is rotated about the hinged vertical edge AB as the axis and at the same time, a fly X moves from point C towards D and another fly Y moves from A towards D. By the time the door rotates through 180 degree, both the flies reach point D. Using suitable scale trace the paths of the flies in elevation and plan, if the motion of the flies and the door are uniform. Name the curves traced out by the flies. Assume the door to be parallel to the V.P. in the initial position and the thickness of the door equal to that of your line.

Archimedean Spiral 5

Procedure:-

  1. Draw a plan ABCD and elevation A’B’ of a rectangle ABCD as shown.
  2. When the rectangular door ABCD is rotated about the hinged vertical edge AB as the axis through 180 degree, the fly X moves from C towards D represented in elevation by path X’, X 1’, X2′, .. X6′ and the another fly Y moves from A towards D represented in plan by path Y, Y1, Y2, …Y6 as shown in the figure.
  3. Curve traced by the fly X and fly Y in the plan and elevation are spirals.