What Is the Displacement Vector of a Clock's Minute Hand at Different Times?

[-Dragoon-]

Homework Statement

The minute hand on a watch is 2.0 cm in length. What is the displacement vector of the tip of the minute hand:
a) From 8:00 to 8:20 A.M.?
b) From 8:00 to 9:00 A.M.?

Homework Equations

N/A.

The Attempt at a Solution

So, first I drew a diagram of the clock and its initial position. Assuming 3:00 and 9:00 to be the x-axis, and 12:00 and 6:00 to be the Y-axis, and that 6:00 makes a 90° with 3:00, then when the minute hand points to 4:00 (denoting 8:20), it's angle, theta, would then make an angle of 30° with the negative x-axis. So, the displacement of the minute hand would then be:
$\vec{d} = 4 * 2.0 cm = 8 cm$ [30° below the horizontal]
Is this correct?

As for b, the displacement is equal to 0, since the minute hand ends up exactly where it started after a change in distance from 8:00 to 9:00 A.M., correct?

 

[NascentOxygen]

So, the displacement of the minute hand would then be:
$\vec{d} = 4 * 2.0 cm = 8 cm$
Is this correct?

So you have a triangle with two sides of 2 cm, and the third side of 8 cm?

Can you draw that?

 

[-Dragoon-]

So you have a triangle with two sides of 2 cm, and the third side of 8 cm?

Can you draw that?

So, now I would need break the 2 cm side that makes an angle of 30 degrees with the negative x-axis into components, correct?

 

[NascentOxygen]

So, now I would need break the 2 cm side that makes an angle of 30 degrees with the negative x-axis into components, correct?

At 20 mins past, the hand makes an angle with the positive x-axis.

Any method that gets the right answer is okay. I broke the triangle up into two equal triangles.

 

[rwisz]

Dear Student,

Your approach to solving this problem is correct. However, there are a few things that can be improved upon.

Firstly, it is important to include units in your final answer. In this case, the displacement vector would be 8 cm [30° below the horizontal].

Secondly, it would be more accurate to use radians instead of degrees when calculating the displacement vector. This is because radians are a more precise unit of measurement for angles in mathematics and physics. So, instead of using 30°, you could use π/6 radians.

For part b, you are correct in saying that the displacement would be equal to 0, as the minute hand ends up in the same position it started in after one hour. However, it would be helpful to explain why this is the case. This is because the displacement vector is a measure of the change in position, and when the minute hand returns to its original position, there is no change in position, thus resulting in a displacement of 0.

Overall, your solution is correct and demonstrates a good understanding of vectors and displacement. Keep up the good work!