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The minute hand of a wall clock measures 11.3 cm from axis to tip. What is the displacement vector of its tip (a) from a quarter after the hour to half past, (b) in the next half hour, and (c) in the next hour?

The minute hand of a wall clock measures 11.3 cm from axis to tip. What is the displacement vector of its tip (a) from a quarter after the hour to half past, (b) in the next half hour, and (c) in the next hour?

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago
Assumptions
Let us assume that the position vector for the tip of the minute hand of the wall clock, at initial position is given by vector \overrightarrow{a}. The magnitude of the position vector is given by a whereas the horizontal and the vertical component of the vector is ax and ay respectively.
When the minute hand displaces from the initial position to come at final, the vector\overrightarrow{b} represents its position vector. The magnitude of the position vector is given by b whereas the horizontal and the vertical component of the vector is bx and by respectively.
Given:
a = 11.3 cm
b = 11.3 cm
(a) The quarter after hour to half past represents the displacement of the minute hand, which is by 15 minutes. This can be seen from the fact that the quarter after hour represents 15 minutes after an hour and the half past represent 30 min. Therefore the elapsed time is 15 minutes.
If the motion of the minute hand is represented on the Cartesian coordinate such that the origin represents the center of the clock, the minute hand can be represented on it.
We assume that, initially, the minute hand subtended angle -90° from positive x axis, measured clockwise (refer figure below).
236-1320_watch1.PNG
The position vector\overrightarrow{a} can be written as:
236-1832_eq1.PNG
The negative sign is due to the fact that the angle made by the minute hand from positive axis is measures clockwise.
Substitute the given value of a , the vector\overrightarrow{a} becomes:
\overrightarrow{a} = -11.3 cm\ \widehat{j}
The quarter after hour to hour past, the minute hand would have displaced by
15 minutes (rotation by ), therefore the position vector of the minute hand can be written as:

236-795_eq2.PNG
The negative sign highlight the fact that the rotation of minute hand is clockwise relative to its initial position, which itself was defined at angle 90° from positive x axis, measured clockwise (refer figure below).

236-584_watch2.PNG
Substitute the value of b to solve for vector\overrightarrow{b} as:
236-1423_eq3.PNG
The displacement vector (say \overrightarrow{s})of the minute hand clock is given as:
\overrightarrow{s} = \overrightarrow{b} - \overrightarrow{a}
Substitute the value of vectors\overrightarrow{b} and \overrightarrow{a} from above to get displacement vector as:
236-240_eq4.PNG
Therefore the displacement vector of the minute hand is -11.3 cm\ \widehat{i}+ 11.3\ cm \widehat{j}.
(b) The minute hand position when it has moved by 30 min relative to the initial position is shown below:

236-1193_watch3.PNG
With the same reference position of the minute hand, the vector\overrightarrow{a} remains the same, that is:
236-939_eq5.PNG
Substitute the given value of to obtain vector \overrightarrow{a}as:
\overrightarrow{a} = -11.3\ cm\ \widehat{j}
However the minute hand has now moved by half hour or 30 minutes which is equal to the rotation by 180° clockwise.
Therefore vector \overrightarrow{b}is given as:
236-777_eq6.PNG
Substitute the given value of b to obtain vector\overrightarrow{b} as:
236-783_eq7.PNG
The displacement vector (say \overrightarrow{s}) of the minute hand clock is given as:
\overrightarrow{s} = \overrightarrow{b}- \overrightarrow{a}

Substitute the value of vectors \overrightarrow{b}and\overrightarrow{a} from above to get displacement vector as:

236-2290_eq8.PNG
Therefore the displacement vector of the minute hand is 22.6 cm \widehat{j}.

(c) In the next hour, the minute hand would have move by 60 min thereby reaching the same position as it was initially. Therefore the net displacement of the minute hand is zero.
The displacement vector in this case would be .0\widehat{i} + 0\widehat{j}

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