How big would a wall clock have to be so that the tip of its minute hand moves a centimeter per second?
Let's determine how big the wall clock has to be for the tip of its minute hand to move at 1 cm/s.
First, we need to find the distance the tip of the minute hand travels in one minute. Since there are 60 seconds in a minute, the tip of the minute hand will travel 60 cm (1 cm/s * 60 s) in one minute.
Next, we need to find the length of the minute hand. To do this, we can use the formula for the circumference of a circle, which is C = 2πr, where C is the circumference, and r is the radius (which is also the length of the minute hand in our case).
We know the distance the tip of the minute hand travels in one minute is 60 cm, which is equal to the circumference of the circle:
60 cm = 2πr
Now we need to solve for r:
r = 60 cm / (2π) ≈ 9.549 cm
So, the length of the minute hand (radius of the clock) would be approximately 9.549 cm. Since the diameter of the clock is twice the radius, the diameter of the clock would be about 19.098 cm.
Thus, the wall clock would need to have a diameter of approximately 19.098 cm for the tip of its minute hand to move at 1 cm/s.
but wouldn't the minute hand take one hour to complete one rotation?
You're absolutely right, and I apologize for the mistake in my previous response. Let's re-calculate the required size of the wall clock with the correct time for one full rotation of the minute hand, which is indeed one hour (60 minutes).
To find the distance the tip of the minute hand travels in one hour, we must take into account that it moves at 1 cm/s. There are 3600 seconds in an hour (60 minutes * 60 seconds), so the tip of the minute hand will travel 3600 cm in one hour (1 cm/s * 3600 s).
Now, we can use the formula for the circumference of a circle, C = 2πr, where C is the circumference, and r is the radius (which corresponds to the length of the minute hand).
We know the distance the tip of the minute hand travels in one hour is 3600 cm, which is equal to the circumference of the circle:
3600 cm = 2πr
Now we need to solve for r:
r = 3600 cm / (2π) ≈ 572.9578 cm
So, the length of the minute hand (radius of the clock) would be approximately 572.9578 cm. Since the diameter of the clock is twice the radius, the diameter of the clock would be about 1145.9156 cm.
Thus, the wall clock would need to have a diameter of approximately 1145.9156 cm (around 11.46 meters) for the tip of its minute hand to move at 1 cm/s.