The Earth travels in a circular orbit around the Sun at 29.79 km/sec. If the radius of
the orbit is 1.496 � 108 km, what is the angular velocity in radians/sec? (Round to
three significant digits.)
Is that radius 1496108 km ?
so the circumference of the orbit would be
9400323.8 km
at a rate of 29.79 km/sec
it would take 9400323.8/29.79 seconds
or appr. 315553 seconds
angular velocity = 2π/315553 rad/sec
or .00002 radians/sec
To find the angular velocity in radians per second, we need to relate the linear velocity to the angular velocity.
The linear velocity v can be calculated using the formula:
v = ω * r
Where:
v is the linear velocity
ω (omega) is the angular velocity
r is the radius of the orbit
In this case, the linear velocity v is given as 29.79 km/s, and the radius of the orbit r is given as 1.496 * 10^8 km.
Plugging in the values into the formula, we have:
29.79 km/s = ω * 1.496 * 10^8 km
To solve for ω, we rearrange the equation:
ω = v / r
Substituting the given values, we get:
ω = 29.79 km/s / 1.496 * 10^8 km
Now, we need to convert the units so that they match. We want the result in radians per second, so we need to convert kilometers to meters:
1 km = 1000 meters
Thus,
ω = (29.79 km/s * 1000 m/km) / (1.496 * 10^8 km)
Simplifying the expression gives us:
ω = (29.79 * 1000) / (1.496 * 10^8) rad/s
Now, we can evaluate the expression to find the angular velocity ω in radians per second.