How big would a wall clock have to be so that the tip of its minute hand moves a centimeter per second?
To determine the size of a wall clock where the tip of the minute hand moves a centimeter per second, we need to know the circumference of the clock face and the length of the minute hand.
Let's assume the clock face has a radius of r. The circumference of the clock face would then be 2 * pi * r. The length of the minute hand would be r * m, where m is the ratio of the length of the minute hand to the radius of the clock face.
The tip of the minute hand moves 360 degrees in 60 minutes, or 6 degrees per minute. This means that in one second, the tip of the minute hand moves 6 degrees / 60 seconds = 0.1 degrees per second.
Let's call the distance traveled by the tip of the minute hand in one second d. This distance can be found using the following formula:
d = 2 * pi * r * m * 0.1 / 360
We know that d = 1 cm, so we can substitute that into the formula and solve for r:
1 = 2 * pi * r * m * 0.1 / 360
360 / (2 * pi * m * 0.1) = r
So, if we know the value of m, we can find the value of r. For example, if m = 0.5, then:
r = 360 / (2 * pi * 0.5 * 0.1) = 719.6 cm
So, for a clock where the tip of the minute hand moves a centimeter per second, with a minute hand length to clock face radius ratio of 0.5, the clock face would need to have a radius of approximately 720 cm.
but wouldn't the minute hand take one hour to complete one rotation?
Yes, you're correct. I apologize for the mistake in my previous answer. The minute hand of a clock takes one hour, or 60 minutes, to complete one full rotation.
This means that in one second, the minute hand moves through an angle of 360 degrees / (60 minutes * 60 seconds/minute) = 0.1 degrees.
Using the same approach as before, we can find the size of the clock face that would be required for the tip of the minute hand to move a centimeter per second:
d = 2 * pi * r * m * 0.1 / 360
1 = 2 * pi * r * m * 0.1 / 360
360 / (2 * pi * m * 0.1) = r
So, if we know the value of m, we can find the value of r. For example, if m = 0.5, then:
r = 360 / (2 * pi * 0.5 * 0.1) = 719.6 cm
So, for a clock where the tip of the minute hand moves a centimeter per second, with a minute hand length to clock face radius ratio of 0.5, the clock face would need to have a radius of approximately 720 cm.