Rotational Kinetics

We are able to solve this problem using our basic kinematic equations. Firstly, the final angular velocity is calculated through the equation:

Since σ _o = 0 , α = 5 and t = 15 ,

The second quantity we are asked for is the total angular displacement:

Using the right hand rule, we curl our fingers to follow the counterclockwise path of the hurricane and, if we are viewing from above, we find that our thumb points towards us. Thus the angular velocity vector points out into space, perpendicular to the earth's surface.

Again, we use our kinematic equations. In this case, we are given σ _o , α and Δμ and are asked to find σ _f . Thus we use the following equation:

Because the object is moving in a circle, it experiences a radial acceleration: a _R σ ² r = 25(2) = 50 m/s ² . In addition, the object experiences angular acceleration, resulting in an acceleration in a tangential direction: a _T = αr = 8 m/s ² . We know that these two values will always be perpendicular. Thus to find the magnitude of the total acceleration on the object we treat a _T and a _R as perpendicular components of a , just like x and y components:

As is clear from the magnitude of the acceleration, almost all of the acceleration is in the radial direction, as the tangential acceleration is insignificant compared to the rate at which the direction of the object is changing as it moves in a circle.

To solve this equation we must use both kinematic equations and relations between angular and linear variables. We know that the ball leaves the stick with a velocity of 10 m/s, in a direction tangential to the rotation of the stick. Thus we can infer that an instant before it was released, the ball was accelerated to this speed. We can then use the relation v = σr To calculate our final angular velocity:

We also know that the initial angular velocity is zero, and the angular displacement over this period of time is 90 ^o , or rad. Thus we can manipulate a kinematic equation to solve for our angular acceleration:

Again, we can assume that the acceleration is constant, and use the following equation:

We know that . Since we want to solve for the total angular displacement, or , we rearrange this equation: However, we are asked for the number of revolutions, not the number of radians. Since there are radians in every revolution, we divide our number by : Thus the merry-go-round revolves about 4 times in that period.