The earth has a radius of approximately 6400 km. How many radians per second is the earth rotating at the equator?
No offense, but to answer the question of how many radians per second the earth is rotating at the equator, the earth's radius of 6400km and also the reference to "equator" need not be given, since rad/s is an angular speed.
Perhaps you have intention to ask "what is the speed of a particular point on Earth at the equator in km/h or m/s?". I will dwell on this at the last part of this answer.
First of all,
Speed of rotation of earth 1 cycle in 24 hours
= 2 * pi * radian / 24 hrs
= 2 * pi * radian / (24 * 3600 sec)
= 2.3148 * pi * 10^(-5) rad/sec
= 7.2722 * 10^(-5) rad/sec
[This works out to about 0.004167 deg/sec or 15 deg/hour]
Now for the speed of a point on the circumference of the Earth at the equator. Since, by definition, 1 radian will be the angle subtended by an arc which has a length of the radius, the speed of a point on earth on the equator would therefore be given by:
speed (of a point on equator)
= [ 7.2722 * 10^(-8) ] * 6400 km /sec
= 0.4654 km/sec
= 465.4 m/sec .... [ Wow! awesome, isn't it? that's more than 1 round of the stadium in 1 second! And that's faster than the speed of sound which is 333 m/sec]
= 1675.5 km/h .... [ That's more than 10 times the speed of a car at full speed driving on the expressway! ]
To answer this question, you have to first calculate how many seconds it takes for a point at the equator to complete a full rotation. The answer, of course, is 24 hours. So, the earth makes 2pi radians in 24 hours.
24 hours = 24 hours * 60 min / hour * 60 sec / min
24 hours = 86,400 seconds
2 pi radians / 86,400 seconds = 7.27 x 10^-5 rad / sec
Note that the radius of the earth does not enter into the calculation because the question is one of angular velocity, not m/s.
To get the velocity of the point, remember that the distance over an angle on the circumference of a circle is the angle x the radius.
So the velocity of the point is:
7.27 x 10^-5 x 6400 km = 465 m/s = 1040 miles per hour
The earth rotates by 360 degree or 2pi radians in 24 hours.
1hour = 60 minutes.
1minute = 60 seconds.
Therefore 24 hours = 24*60 minutes .
24*60 minutes = 24*60*60 seconds.
Therefore it takes 24*60*60 seconds for earth to rotate 2pi radians. So to obtain the angular speed of the earth in radians, we didivide 2pi radians by 24*60*60 seconds .
Therefore the earth rotates by 2pi/24*60*60 radians per scond , or 0.00007272205217 radians in one second.
Since the radius of the earth is given, any object on the surface of the earth moves by a distance of (2pi/24*60*60)6400km per second, or 465.4211 meter per second.
Radian is a unit of measure of angle. The measure of angle of one complete rotation of earth in terms of radians is 2pi radians. In terms of degrees it is equal to 360 degrees.
Please note that the rotation of earth in terms of radians is same, irrespective of the point on earth where the measurement is made.
The earth makes one complete rotation around its axis every day (24 hours). This means:
Rotation of earth in radians per day = 2pi
We know that:
Number of seconds in a day = 24*60*60 = 86400
pi = 3.14159
Rotation of earth in radians per second = (2*3.14159)/86400
= 0.000072722 radians/s