Question

If the kinetic energy of an object is K = 1/2 * m * v², then what is the angular kinetic energy of the object?

1) 1/2 * I * Ω²
2) 1/2 * m * v²
3) 1/2 * m * r² * Ω²
4) 1/2 * I * v²

Answer 1

The angular kinetic energy of an object is represented by the formula 1/2 * I * Ω², making option 1 the correct answer.

Step-by-step explanation:

The angular kinetic energy of an object is given by the formula KErot = 1/2 * I * Ω², where 'I' is the moment of inertia and 'Ω' (Omega) is the angular velocity.

Thus, the correct answer to the question is option 1) 1/2 * I * Ω². This formula is similar to the linear kinetic energy formula K = 1/2 * m * v², but for rotational motion, where the moment of inertia replaces mass, and angular velocity replaces the linear velocity.

Answer 2

1) K = 1/2 · I · ω²

Step-by-step explanation:

Knowing that kinetic energy of an object is given as:

`\boxed{\left\begin{array}{ccc}\text{\underline{Kinetic Energy:}}\\\\ K = (1)/(2)mv^2 \\\\\text{Where:}\\\bullet \ K \ \text{is the kinetic energy}\\\bullet \ m \ \text{is the mass of the object}\\\bullet \ v \ \text{is the velocity of the object}\end{array}\right}`

Lets derive a formula for rotational kinetic energy from the linear version of kinetic energy. We know the relationship between linear velocity and angular velocity is:

`\boxed{ \left \begin{array}{ccc} \text{\underline{Linear Velocity:}} \\\\ v = r\omega \\\\ \text{Where:} \\ \bullet \ v \ \text{is the linear velocity} \\ \bullet \ r \ \text{is the radius or distance from the rotation axis} \\ \bullet \ \omega \ \text{is the angular velocity} \end{array} \right.}`

Substitute this into the linear kinetic energy formula:

`\Longrightarrow K = (1)/(2)mv^2\\\\\\\\\Longrightarrow K = (1)/(2)m(r\omega)^2\\\\\\\\\Longrightarrow K = (1)/(2)mr^2\omega^2`

Notice that the formula for moment of inertia shows up. Moment of inertia (for "point-like" objects) is given as:

`\boxed{ \left \begin{array}{ccc} \text{\underline{Moment of Inertia:}} \\\\ I = mr^2 \\\\ \text{Where:} \\ \bullet \ I \ \text{is the moment of inertia} \\ \bullet \ m \ \text{is the mass of the object} \\ \bullet \ r \ \text{is the distance from the axis of rotation} \end{array} \right.}`

We can now plug this in:

`\therefore K = (1)/(2)I\omega^2`

Thus, we have derived a formula for rotational kinetic energy. The only option that matches this equation is 1st option, therefore it is the correct option.

`\boxed{ \left \begin{array}{ccc} \text{\underline{Rotational Kinetic Energy:}} \\\\ K = (1)/(2)I\omega^2 \\\\ \text{Where:} \\ \bullet \ K \ \text{is the rotational kinetic energy} \\ \bullet \ I \ \text{is the moment of inertia} \\ \bullet \ \omega \ \text{is the angular velocity} \end{array} \right.}`