Conic Sections and Engineering Curves Questions and Answers

Answer: C

Correct answer: 0.8

With a semi-major axis of 50 and a semi-minor of 30, \[ c = \sqrt{50^2 - 30^2} = \sqrt{1600} = 40, \] \[ e = \frac{c}{a} = \frac{40}{50} = 0.8. \] The 3-4-5 triangle appears throughout ellipse work; here it gives the focal distance without a calculator.

Answer: D

Correct answer: 80 mm

The foci lie at \( \pm c \) from the centre, so the distance between them is \[ 2c = 2 \times 40 = 80\ \mathrm{mm}. \] With the major axis 100 mm, the foci sit 10 mm inside each vertex, which is the measurement used to set them out on the drawing.

Answer: A

Correct answer: 94.2 mm

One arch spans the circumference of the rolling circle, since the circle turns exactly once between cusps: \[ \pi d = \pi \times 30 = 94.2\ \mathrm{mm}. \] Using the radius instead gives 47.1 mm, and the factor of two is the commonest error in setting out a cycloid.

Answer: B

Correct answer: 30 mm

The tracing point rises from the line to the top of the circle and back, so the height of the arch is the full diameter of the rolling circle, 30 mm. Base and height together, \( \pi d \) by \( d \), fix the arch completely and are the two dimensions to lay out first.

Answer: C

Correct answer: 14.3 degrees

The helix angle is measured from the plane perpendicular to the axis: \[ \tan\theta = \frac{\text{pitch}}{\pi d} = \frac{40}{\pi \times 50} = \frac{40}{157.1} = 0.2546, \] \[ \theta = 14.3^\circ. \] Taking the complement gives 75.7 degrees, which is the angle to the axis and is offered as a distractor.

Answer: D

Correct answer: 120 mm

The pitch is the rise for one turn, so three turns rise \[ 3 \times 40 = 120\ \mathrm{mm}. \] Developed on the flat the helix becomes a straight line, the hypotenuse of a triangle whose vertical side is that 120 mm and whose horizontal side is three circumferences - which is how a helical development is set out.

Answer: A

Correct answer: 5 mm

One convolution is 360 degrees, which is twelve intervals of 30, and the radius grows from 10 to 70 mm over it: \[ \frac{70 - 10}{12} = \frac{60}{12} = 5\ \mathrm{mm}. \] Because the growth is uniform with angle, the same 5 mm is added at every step, which is what makes the construction so simple.

Answer: B

With the radius ratio at one half, the tracing point sweeps back and forth along a diameter of the fixed circle instead of describing arches. A quarter ratio gives four cusps and the astroid rather than a straight line.

Answer: C

Correct answer: 1-Q, 2-R, 3-P

Three cutting planes and three sections. Perpendicular to the axis gives a circle. Parallel to a generator gives a parabola, the boundary case that never closes. And through the apex gives a degenerate section, in the general case two straight lines crossing at the apex rather than a curve at all.

Answer: D

Correct answer: 1-Q, 2-R, 3-P

Three curves generated three different ways. An involute is traced by the end of a taut string unwound from a shape, which is why it is the profile used for gear teeth. A cycloid comes from a circle rolling along a straight line, and an epicycloid from a circle rolling outside another circle.