University of Wisconsin Green Bay

You have designed a merry-go-round with nearly perfect bearings so that it rotates with virtually no friction. If you wanted to apply a constant force of 180 N to the merry-go-round as you ran around the outside, how fast would you have to run after 5.0 s? The radius of your merry-go-round is 1.8 m and its mass of 32 kg is uniformly distributed.

• In this problem, you are asked to related the kinematic (description of motion) variable velocity to the cause of motion-a dynamic variable. In other words, this is a two part problem. Kinematics and dynamics are linked through acceleration, and so you will work a dynamics problem relating the cause and effect of the motion (force and acceleration) and then you can use the acceleration in a kinematics problem to describe the motion.

Do not worry if you don't recognize both parts of the problem at this point. If you recognize the dynamics problem, you will solve for angular acceleration. At that point, you will see that you are not yet done and need to do another step to find velocity. On the other hand, if you recognize this as a kinematics problem you will quickly see that you need to find angular acceleration before you can begin and so will need to do that pre-step first.

Finally note that in this problem you and the merry-go-round are moving around. In other words, this is an angular motion problem. You can always relate angular and linear variables through the definition of the angular variable.

• The object whose motion you are understanding is the merry-go-round, and the merry-go-round is rotating. And effective picture for solving dynamics problems in the case of rotational motion shows not only the forces on the object of interest but also their location, and also identifies the axis of rotation.

• In one part of this problem, you will relate the cause of motion to its effect. Just as you would use ΣF = ma for a linear problem, you add torques to find angular acceleration in a rotational problem.

Once you know angular acceleration, you can use it to describe the motion. In this problem you want to know velocity after a certain amount of time. In other words, you want the velocity-time relationship for angular motion.

• Now that you have found the angular acceleration, you can proceed to the original question--to find your velocity after 5.0 s. If you had not originally recognized that you would need to do this pre-step, you may have started with the kinematics. You would quickly realize that you needed a to solve the problem and so would have done this as a side step at that point.

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Step 2

Once you know angular acceleration, it is straightforward to find your final angular velocity. Because you are asked for linear velocity, you need to use the definition of angular velocity to relate the two.

• Even though you were told and asked about linear variables, this is a rotational motion problem because the object whose motion you can best understand, the merry-go-round, is rotating. The cause of motion (torque) was used to find the angular acceleration, after which the motion could be described in terms of other variables. As a final step, the rotational velocity can be related to the requested linear velocity.

There is a second consideration to understanding this problem. 56 m/s is too fast to run--125 mph. Can the number be right? 180 N is about 40 pounds of force, so you aren't pushing that hard. But in this case the resulting acceleration is about an additional rotation every second, and in order to keep exerting that constant force, you need to run faster and faster for 5 seconds. A constant acceleration can require you to get faster pretty quickly. The problem suggests that this might not be possible when it asks "how fast would you have to run..."