# [Rotational Inertia, Angular Velocity] problem without given masses.

• Alpha Russ Omega
In summary, the conversation discusses a problem involving a meter stick suspended vertically and its maximum angular velocity when released from rest. The solution involves using rotational inertia and energy conservation to determine the relationship between rotational kinetic energy and potential energy. The conversation ends with the acknowledgement that the original approach using energy conservation was correct.
Alpha Russ Omega
Hello:

I seem to be stuck on this one problem: "A meter stick is suspended vertically at a pivot point 0.26 meters from the top end. It is rotated on the pivot until it is horizontal and then released from rest. What will be its maximum angular velocity (in radians per second)?"

So I figured this problem might involve Rotational Inertia: $$I=\sum_i m_{i} r_{i}^2$$

Would I also need to find torque? (I'm getting confused on how the problem does not give me the mass of the meter stick.) Am I using the right formula?

Any help would be appreciated.

You need to use energy conservation. If the angular velocity is maximal, what does that tell you about rotational kinetic energy? Further on, what does that tell you about potential energy?

## 1. What is rotational inertia and how is it related to angular velocity?

Rotational inertia is a property of an object that determines its resistance to rotational motion. It is similar to inertia in linear motion, but instead refers to the object's resistance to changes in its rotational speed. Angular velocity, on the other hand, is the rate at which an object rotates or changes its angular position. The two are related in that an object with a larger rotational inertia will require more force to change its angular velocity compared to an object with a smaller rotational inertia.

## 2. How can I calculate rotational inertia without given masses?

The rotational inertia of an object can be calculated by using its shape, dimensions, and density. This can be done through the use of mathematical formulas and principles, such as the parallel axis theorem or the moment of inertia equation. These formulas take into account the distribution of mass within the object and its distance from the axis of rotation.

## 3. Can rotational inertia affect an object's stability?

Yes, rotational inertia can affect an object's stability. Objects with a larger rotational inertia will be more stable and less likely to topple over compared to objects with a smaller rotational inertia. This is because objects with a larger rotational inertia require more force to change their angular position, making them more resistant to external forces that could cause them to topple.

## 4. Does the shape of an object affect its rotational inertia?

Yes, the shape of an object can affect its rotational inertia. Objects with a larger mass located further from the axis of rotation will have a larger rotational inertia compared to objects with the same mass but located closer to the axis of rotation. This is why objects with irregular shapes or different mass distributions have different rotational inertias.

## 5. How does rotational inertia affect the motion of objects in space?

In space, there is no external force to slow down or speed up an object's rotation. Therefore, the rotational inertia of an object plays a crucial role in its motion. Objects with a larger rotational inertia will rotate more slowly compared to objects with a smaller rotational inertia, as they require more force to change their rotational speed. This is why spacecrafts and satellites are designed with low rotational inertia to allow for precise control and maneuverability in space.

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