A circular gear turns 60 per hour. Through how many radians does it turn in a 24-hour day?
To determine the number of radians a circular gear turns in a 24-hour day, we first need to find the number of revolutions it makes in that time.
Given:
- The gear turns 60 times per hour
To find the number of revolutions in a 24-hour day:
1 hour = 60 minutes
1 day = 24 hours
Number of revolutions in a day = (60 turns/hour) * (24 hours) = 60 * 24 = 1440 turns
Next, we need to convert the number of revolutions to radians.
1 revolution = 2π radians
Number of radians turned in a day = (1440 turns) * (2π radians/1 turn) = 1440 * 2π
≈ 9026.84 radians
Therefore, a circular gear turns approximately 9026.84 radians in a 24-hour day.
To calculate the number of radians a circular gear turns in 24 hours, we need to know the relationship between the number of revolutions and radians.
One revolution is equal to 2π radians, as the circumference of a circle is 2π times its radius.
In this case, the gear turns 60 revolutions per hour. So, in one hour, the gear turns 60 * 2π radians.
To find the total radians turned in 24 hours, we multiply the radians turned in one hour by the number of hours in a day:
Total radians = 60 revolutions/hour * 2π radians/revolution * 24 hours
Simplifying the equation:
Total radians = (60 * 2π * 24) radians
Now we can calculate the exact value:
Total radians = 1440π radians
So, the circular gear turns through 1440π radians in a 24-hour day.