Question

The minute hand of a wall clock measures 10 cm from 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

Answer 1

Answer:

  (a)  10√2 cm

  (b)  -135°

  (c)  20 cm

  (d)  90°

  (e)  0

  (f)  0 (any)

Step-by-step explanation:

(a, b) At 1/4 past the hour, the minute hand points to "3" on the clock face. Its angle relative to the +x axis is 0°.

At 1/2 past the hour, the minute hand points to "6" on the clock face. Its angle relative to the +x axis is -90°. The positions of the two hands and the vector between 3 and 6 form an isosceles right triangle whose hypotenuse is 10√2 cm in length.

The internal acute angles of that triangle are 45°. The external angle of that triangle at "3" is the angle the vector from "3" to "6" makes with the +x axis. Measured from the +x axis, it is -135°.

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(c, d) The displacement from straight down to straight up is twice the length of the minute hand, 20 cm. The angle is straight up, 90°.

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(e, f) The net displacement after 1 hour is zero. The minute hand is in the same (straight up) position it was an hour ago. (The angle can be anything you like, as it is irrelevant when the magnitude is zero.)

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A diagram may help you visualize these clock hand positions.