Question

The minute hand of a wall clock measures 19 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

The displacement and angle for three intervals are determined: from a quarter after the hour to half past and for the next half hour, both have a magnitude of 38 cm and an angle of 180 degrees, while for the hour after that, the magnitude is zero and the angle is 360 degrees.

Step-by-step explanation:

The challenge involves calculating the magnitude and angle of the displacement vector for the tip of the minute hand on a clock during three different time intervals.

Determination of Magnitude and Angle

1. From a quarter after the hour to half past: The minute hand moves from the 3 to the 6 on the clock, which is a 90-degree angle. Given that the length of the minute hand is 19 cm, the tip traces out an arc of a circle. The magnitude of displacement is equal to twice the radius of the circle, as it moves from one end of the diameter to the other, which is 38 cm. The angle of displacement is 180 degrees as it is a straight line.
2. For the next half hour: Moving from half past to the hour, the minute hand covers another 90-degree arc. The magnitude of the displacement remains 38 cm for the straight line diameter distance, and the angle of displacement is again 180 degrees.
3. For the hour after that: The minute hand completes one full cycle around the clock. The magnitude of the displacement is zero since the hand returns to its original position, and the angle is a full rotation, or 360 degrees.