Conic Sections and Engineering Curves Questions and Answers

Answer: A

The generating circle stays outside the directing circle throughout, so the arches curve away from the fixed centre. A hypocycloid is the corresponding curve for rolling inside, and its arches curve the other way entirely.

Answer: B

Rolling within the fixed circle turns the cusps inward, whereas rolling outside it throws them outward as an epicycloid. Where the rolling circle's radius is exactly half the fixed radius the curve degenerates into a diameter.

Answer: C

Correct answer: off the circumference of the rolling circle

A trochoid is the general curve traced by any point rigidly attached to the rolling circle. If the point lies on the circumference the curve is the cycloid, with cusps touching the line; inside it the curve is a curtate trochoid with rounded lows, and outside it a prolate trochoid with loops.

Answer: D

Correct answer: at a uniform rate while the radius rotates uniformly

Both motions are uniform, so the radius grows in proportion to the angle turned and successive turns of the spiral are evenly spaced. That even spacing is exactly the property a scroll cam needs to convert uniform rotation into uniform radial movement.

Answer: A

Correct answer: increases in geometric rather than arithmetic progression

In an Archimedean spiral equal angles add equal increments to the radius; in a logarithmic spiral they multiply it by a constant factor, so the turns spread further apart as the curve grows. The logarithmic form is self-similar under scaling, which is why it appears in shells and in the profiles of certain cams and cutters.

Answer: B

One revolution advances the point along the axis by the pitch, and developing the cylinder turns the helix into a straight line. Screw threads, helical springs and spiral staircases are all applications of the same curve.

Answer: C

Correct answer: one complete revolution

The pitch is the axial advance for one full turn, so the helix rises by its pitch each time it goes round. Together with the diameter it fixes the helix completely: the helix angle is the arctangent of the pitch divided by the circumference, and every property of a screw thread follows from those two dimensions.

Answer: D

Correct answer: an ellipse

An eccentricity between zero and one is an ellipse, so 0.6 lies comfortably inside that range. The nearer the value to zero the more nearly circular the curve, and at 0.6 the ellipse is noticeably elongated but far from the parabola, which is reached only at exactly one.

Answer: A

Correct answer: greater than one

Two branches require the cutting plane to meet both nappes of the cone, and that condition corresponds exactly to an eccentricity above one. At exactly one the curve is a parabola with a single unbounded branch, and below one it closes into an ellipse with no branch at all.

Answer: B

Correct answer: 100 mm

The latus rectum of a parabola is four times the distance from the vertex to the focus: \[ 4 \times 25 = 100\ \mathrm{mm}. \] It is the width of the curve measured at the focus, and it gives a quick check on a construction - the curve should pass 50 mm either side of the axis level with the focus.