a) 2.22 rad/s
b) 6.66 m/s
c) 0.56 rad/s, 3.36 m/s
Explanation:
a)
The angular velocity of an object in rotation is equal to the rate of change of the angular displacement:
where
is the angular displacement
is the time elapsed
First of all, we calculate the full area of the circle. The radius is the length of the sling:
r = 3 m
So the total area is
This area corresponds to an angle of radians.
Here the area swept out by the sling is
So the angle corresponding to this area is
While the time elapsed is
Therefore, the angular velocity is:
b)
For an object in rotational motion, the relationship between angular velocity and linear speed is:
where
v is the linear speed
is the angular velocity
r is the radius
In this problem:
is the angular velocity
r = 3 m is the radius (length of the sling)
So, the speed is:
c)
In this case, the length of the sling is doubled, so
r = 6 m
First of all, we can re-calculate the area of the full circle:
Therefore, the angular displacement of the sling this time is:
And so, the angular velocity this time is
The linear speed instead will be given by
And by substituting the new values, we get: