The minute hand of a wall clock measures 11 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? Give all angles as positive values measured counterclockwise from the +x direction (to the right, or 3 o'clock).

a. 12

To determine the displacement vectors for each time interval, we can use the formula:

displacement = radius * angle

where the radius is the length of the minute hand (11 cm) and the angle is determined by the time elapsed.

(a) Magnitude and (b) angle from a quarter after the hour to half past:

- To find the time interval from a quarter after the hour to half past, it is a duration of 15 minutes or 1/4 of an hour.
- The angle covered in this interval is 90 degrees, as the minute hand moves 1/4 of the way around the clock face (1/4 * 360 degrees = 90 degrees).
- Therefore, the magnitude of the displacement vector is calculated as:
magnitude = radius * angle
magnitude = 11 cm * 90 degrees = 990 cm

- The angle from the +x direction (counterclockwise) can be found by subtracting the angle covered by the minute hand in the initial position:
angle = 90 degrees - 0 degrees = 90 degrees

Therefore, the magnitude and angle for the time interval from a quarter after the hour to half past are:
(a) magnitude: 990 cm
(b) angle: 90 degrees

(c) Magnitude and (d) angle for the next half hour:

- The next half-hour interval is 30 minutes or 1/2 of an hour.
- The angle covered in this interval is 180 degrees since the minute hand moves halfway around the clock face (1/2 * 360 degrees = 180 degrees).
- Therefore, the magnitude of the displacement vector is calculated as:
magnitude = radius * angle
magnitude = 11 cm * 180 degrees = 1980 cm

- The angle from the +x direction (counterclockwise) can be found by subtracting the angle covered by the minute hand in the initial position:
angle = 180 degrees - 0 degrees = 180 degrees

Therefore, the magnitude and angle for the next half-hour are:
(c) magnitude: 1980 cm
(d) angle: 180 degrees

(e) Magnitude and (f) angle for the hour after that:

- The time interval for the hour after that is 60 minutes or 1 hour.
- The angle covered in this interval is 360 degrees since the minute hand completes a full revolution.
- Therefore, the magnitude of the displacement vector is calculated as:
magnitude = radius * angle
magnitude = 11 cm * 360 degrees = 3960 cm

- The angle from the +x direction (counterclockwise) can be found by subtracting the angle covered by the minute hand in the initial position:
angle = 360 degrees - 0 degrees = 360 degrees

Therefore, the magnitude and angle for the hour after that are:
(e) magnitude: 3960 cm
(f) angle: 360 degrees

To solve this problem, we can use the concept of circular motion and trigonometry. The minute hand of the wall clock revolves around the clock's center in a circular path, and we need to calculate the magnitude and angle of the displacement vector for each time interval.

Let's tackle each time interval one by one:

(a) Magnitude of displacement vector from a quarter after the hour to half past:
To calculate the magnitude, we need to find the distance covered by the minute hand in this time interval. From a quarter after the hour to half past, the minute hand moves a distance of half the circumference of a circle with a radius of 11 cm. Hence, the magnitude is given by:
Magnitude = (1/2) * 2 * π * 11 cm = π * 11 cm

(b) Angle from a quarter after the hour to half past:
The angle is the measure of rotation of the minute hand in this time interval. The minute hand moves in the counterclockwise direction, so the angle is positive and measured counterclockwise from the +x direction. From a quarter after the hour to half past, the minute hand moves a quarter of a full revolution, which is 90 degrees. Hence, the angle is 90 degrees.

(c) Magnitude of displacement vector for the next half hour:
To calculate the magnitude, we need to find the distance covered by the minute hand in this time interval. In half an hour, the minute hand moves a distance equal to half the circumference of a circle with a radius of 11 cm. Hence, the magnitude is the same as before: π * 11 cm.

(d) Angle for the next half hour:
The angle is the measure of rotation of the minute hand in this time interval. The minute hand moves in the counterclockwise direction, so the angle is positive and measured counterclockwise from the +x direction. In the next half hour, the minute hand moves half a revolution, which is 180 degrees. Hence, the angle is 180 degrees.

(e) Magnitude of displacement vector for the hour after that:
To calculate the magnitude, we need to find the distance covered by the minute hand in this time interval. In an hour, the minute hand moves a distance equal to the full circumference of a circle with a radius of 11 cm. Hence, the magnitude is 2 * π * 11 cm.

(f) Angle for the hour after that:
The angle is the measure of rotation of the minute hand in this time interval. The minute hand moves in the counterclockwise direction, so the angle is positive and measured counterclockwise from the +x direction. In an hour, the minute hand completes a full revolution, which is 360 degrees. Hence, the angle is 360 degrees.

So, the answers to the given questions are:
(a) Magnitude = π * 11 cm
(b) Angle = 90 degrees
(c) Magnitude = π * 11 cm
(d) Angle = 180 degrees
(e) Magnitude = 2 * π * 11 cm
(f) Angle = 360 degrees

recall that every 5 minutes covers 30°

That will give you the angle traversed.