91Ó°ÊÓ

Rotational kinetic energy is the energy an object possesses due to its rotation. Similar to linear kinetic energy, which depends on mass and speed, rotational kinetic energy depends on the moment of inertia and angular velocity.
The formula for rotational kinetic energy (\text{KE}) is: ewline \text{\[ \text{KE} = \frac{1}{2} I \text{\textomega}^2 \] ewline Here, \text{I} is the moment of inertia, and \text{\textomega} is the angular velocity. In the context of the falling meter stick, as the stick falls, its initial potential energy is converted into rotational kinetic energy once it hits the ground. When we equate the potential energy to the final rotational kinetic energy, we get the relationship we need to find the angular velocity of the stick (\text{\textomega}): ewline \text{\[ mg(0.5) = \frac{1}{2} \times \frac{1}{3} mL^2 \text{\textomega}^2 \] ewline By rearranging and solving this, we can determine the angular velocity.

The moment of inertia (\text{I}) is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution of the object relative to the axis of rotation.
For a thin rod rotating about one end, the moment of inertia is given by: ewline \text{\[ I = \frac{1}{3} mL^2 \] ewline Here, \text{m} is the mass of the rod, and \text{L} is its length. The higher the moment of inertia, the harder it is to change the rotational speed of the object. For the falling meter stick problem, understanding the moment of inertia helps us determine how the stick's mass influences its rotational kinetics as it falls. When we substitute this value into our kinetic energy equation, we can find out how the stick's mass and length affect its angular velocity at the moment it hits the floor.

The linear speed (\text{v}) of a point on a rotating object is directly related to its angular velocity (\text{\textomega}) and the distance from the point to the axis of rotation.
The relationship can be given as: ewline \text{\[ v = \text{\textomega} L \] ewline Where \text{\textomega} is the angular velocity and \text{L} is the length of the rod. In our case, the linear speed of the end of the meter stick when it hits the floor is calculated by multiplying the angular velocity by the length of the stick. From our previous calculations, we found \text{\textomega} to be approximately 5.42 rad/s and \text{L} to be 1 meter. Substituting these values in, we get: ewline \text{\[ v = 5.42 \times 1 = 5.42 \text{ m/s} \] ewline This means the end of the meter stick has a linear speed of 5.42 m/s just before it hits the floor.