Find the speed in both radians per second and degrees per second for the hour
hand, the minute hand, and the second hand on a clock.
1 hours = 60 minutes = 3600 seconds
I will do one of them for you, you try the others
angular speed of hour hand:
= 1 rotation/ per 12 hrs
= 2π radians/12(3600) seconds
= (π/21600) radians/second
or
= 360° degrees /12 hrs
= (360/12(3600)° / second
= 1/120 degrees/second
To find the speed of the hour hand in radians per second, we need to understand that the hour hand completes one full revolution (360 degrees) in 12 hours, which is equal to 2π radians.
Since there are 60 minutes in an hour, the hour hand moves 30 degrees (360 degrees divided by 12) for every 60 minutes. This means it moves 0.5 degrees per minute (30 degrees divided by 60 minutes).
To calculate the speed in radians per second, we need to convert the speed from degrees per minute to radians per second. There are 60 seconds in a minute, so we need to divide the speed in degrees per minute by 60 and multiply by π/180 to convert from degrees to radians.
Speed of the hour hand (in radians per second) = (0.5 degrees/minute) * (π/180 radians/degree) * (1 minute/60 seconds) = π/360 radians/second.
To find the speed of the minute hand and the second hand, we can use a similar approach. The minute hand completes one full revolution (360 degrees) in 60 minutes, which is equal to 2π radians. Thus, the speed of the minute hand in radians per second is π/30 radians/second.
Similarly, the second hand completes one full revolution (360 degrees) in 60 seconds, which is also equal to 2π radians. Therefore, the speed of the second hand in radians per second is π/30 radians/second.
To find the speed of the hour, minute, and second hands in degrees per second, we can simply use the conversion factor of 1 minute = 60 seconds, and 1 degree = 60 minutes.
Speed of the hour hand (in degrees per second) = (0.5 degrees/minute) * (1 minute/60 seconds) = 0.00833 degrees/second.
Speed of the minute hand (in degrees per second) = (1 degree/minute) * (1 minute/60 seconds) = 0.01667 degrees/second.
Speed of the second hand (in degrees per second) = (6 degrees/minute) * (1 minute/60 seconds) = 0.1 degrees/second.
So, the speed of the hour hand is π/360 radians/second and 0.00833 degrees/second. The speed of the minute hand and the second hand is π/30 radians/second and 0.01667 degrees/second, and 0.1 degrees/second respectively.