Put these eccentricities in order of increasing openness of the curve.
1. 0.5 2. 1.0 3. 0 4. 2.0
Answer: A
Correct answer: 3, 1, 2, 4
Eccentricity measures openness, so the order is simply the numerical one: 0 gives a circle, 0.5 an ellipse, 1.0 a parabola and 2.0 a hyperbola. The sequence runs from a closed curve of no elongation, through progressively longer ellipses, to the open curves beyond.
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Consider the following statements about conics.
1. A parabola has eccentricity exactly one 2. The four-centre method gives a true ellipse 3. A rectangular hyperbola has perpendicular asymptotes
Answer: B
Correct answer: 1 and 3 only are correct
Statements 1 and 3 are correct: the parabola has eccentricity exactly one, and a rectangular hyperbola is defined by its perpendicular asymptotes. Statement 2 is the one to watch - the four-centre method builds the figure from four circular arcs, and no combination of circular arcs is a true ellipse.
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Which of the following is NOT a conic section?
Answer: C
Correct answer: a cycloid
A conic section is a curve obtained by cutting a cone with a plane, which gives the circle, ellipse, parabola and hyperbola and nothing else. A cycloid is a roulette, traced by a point on a rolling circle, and no plane section of a cone will produce one.
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Three of the following describe an ellipse. The exception is
Answer: D
Correct answer: each point equidistant from focus and directrix
Equal distances from focus and directrix mean an eccentricity of exactly one, which is the parabola. An ellipse has eccentricity below one, so its focal distance is always less than its directrix distance. The other three statements all describe an ellipse correctly.
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A draughtsman needs a curve for a gear tooth flank that transmits motion at a constant velocity ratio. The appropriate curve is
Answer: A
The involute's normal at the point of contact always passes through the pitch point, which is exactly what keeps the velocity ratio constant. It is also tolerant of small errors in centre distance, which cycloidal teeth are not.
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A scroll cam is required to convert uniform rotation into uniform radial movement. The profile needed is
Answer: B
Correct answer: an Archimedean spiral
Uniform rotation must produce uniform radial movement, which requires the radius to grow in proportion to the angle - precisely the Archimedean spiral. A logarithmic spiral would accelerate the follower as it went, and neither a cycloid nor an involute has the required constant relation between angle and radius.
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Assertion: the four conics can all be drawn by the eccentricity method. Reason: each is the locus of a point whose distances from a focus and a directrix bear a constant ratio.
Answer: C
Correct answer: both the assertion and the reason are true and the reason explains the assertion
Both are true and the reason is why the method is general. The focus-directrix definition characterises all four conics, differing only in the value of the ratio, so one construction serves them all: fix the focus, the directrix and the eccentricity, and the same steps produce a circle, an ellipse, a parabola or a hyperbola as the value dictates.
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Assertion: the four-centre construction produces a geometrically exact ellipse. Reason: it is built from arcs struck from four centres on the axes.
Answer: D
The arcs are indeed struck as described, but circular arcs cannot reproduce a curve of continuously varying radius. The construction is an approximation adopted for speed, adequate for isometric circles and not for precise work.
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Comparing an Archimedean with a logarithmic spiral, the Archimedean form has turns that are
Answer: A
Correct answer: evenly spaced radially
Because the radius grows in proportion to the angle, each complete turn adds the same amount and successive turns lie the same distance apart. In a logarithmic spiral each turn multiplies the radius by a constant factor, so the spacing widens continuously as the curve grows outward.
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None of the following is generated by rolling one curve on another except
Answer: B
Correct answer: an epicycloid
An epicycloid is traced by a point on a circle rolling on the outside of another circle, so it belongs with the cycloid and the hypocycloid as a roulette. The other three are conic sections, obtained by cutting a cone with a plane, and no rolling motion produces them.
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