A sling that is 3 meters long, sweeps out an area of 50 square meters every 5 seconds. a. What is its angular velocity? (in rad/sec) b. What is its speed? (in m/sec) c. How does doubling the length of the sling affect the previous two answers?

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: