The minute hand of a wall clock measures 14 cm from its tip to the axis about wh

Question

The minute hand of a wall clock measures14 cmfrom its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the(a)magnitude and(b)angle from a quarter after the hour to half past, the(c)magnitude and(d)angle for the next half hour, and the(e)magnitude and(f)angle for the hour after that? Give all angles as positive values measured counterclockwise from the +xdirection (to the right, or 3 o'clock).

Explanation / Answer

circumference of the circle circumscribed by the tip of the minute hand as it moves over the course of exactly one hour is C=2*pi*R = 62.8 cm. everything else below follows from this fact. in one quarter of an hour, the displacement of the tip of the minute hand (arc length) is one quarter of this circumference. 62.8 cm/4 = 15.7 cm at a quarter after the hour, the minute hand is parallel to the x-axis, so the angle is zero degrees by definition. in another half hour, the displacement is another 1/2 of the circumference. 31.4 cm more w.r.t to a quarter past, or 15.7+31.4 = 47.1 cm with respect to the top of the hour. at this point the minute hand is antiparallel to the x-axis. 180 degrees. one hour after this point, the displacement is another one full circumference more than where it was at 3/4 past the top of the original hour. so 62.8 cm more than at the 3/4 point of the original hour, 62.8+31.4 cm w.r.t. to the 1/4 point of the original hour, and 62.8+31.4+15.7 cm w.r.t. the top of the original hour.

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