Radians (& Circular motion) for Physicists
Main formulae:
Angular velocity () = angle turned through or
Time taken
Speed = arc = =
Time
Period = time for 1 revolution
Frequency (f) =
A particle moving round a circular path of radius r with constant angular speed and tangential speed v has acceleration of magnitude , or , directed towards the centre of the circle.
Use either your calculator for or take it as 3.142; take g = 9.8 ms-2
1) Convert each of the following angles in degrees into radians:
a)180ob) 60oc) 25od) 2o
2) Convert from radians to degrees:
a) b) c) 0.30d) 8e) 5
3) A wheel rotates at a frequency of 20 Hz. What is its angular velocity?
4) The earth orbits the sun along an approximately circular path having a radius of 1.50 x 108 km and each orbit takes about 365 days to complete. Calculate:
a) the angular velocity
b) the speed of this movement
5) The moon is approximately 3.8 x 105 km from the Earth and the diameter of the Moon is approximately
1.7 x 103 km. Calculate an approximate value for the angle in radians subtended by the Moon from the Earth.
6) Astronauts are trained to withstand the effects of high acceleration in a “centrifugal” machine. They sit or lie in cabins at the end of long metal arms, which rotate them about a vertical axis in a horizontal circle. If the radius of the circle is 12 metres, and the acceleration to be experienced horizontally is 10g, how long should it take for the arms to make one revolution?
7) The moon Io orbits the planet Jupiter every 1.528 x 105 seconds. The orbit is approximately circular and of mean radius 4.22 x108 metres. (Model the motion of Io relative to Jupiter as circular with constant speed.) Calculate:
a) the speed with which Io orbits Jupiter
b) the magnitude of the acceleration of Io as it orbits Jupiter.
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