A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its center with 0.15 J of kinetic energy. What is the speed of a point on the rim?

Verified step by step guidance

1. Convert the given mass of the disk (100 g) into kilograms by dividing by 1000. This is necessary because the SI unit of mass is kilograms.
2. Calculate the radius of the disk by dividing the diameter (8.0 cm) by 2 and converting it into meters by dividing by 100. This ensures the radius is in SI units (meters).
3. Recall the formula for rotational kinetic energy: $K=\frac{1}{2}I{\omega }^2$, where $K$ is the rotational kinetic energy, $I$ is the moment of inertia, and $\omega$ is the angular velocity. Rearrange this formula to solve for $\omega$: $\omega =\sqrt{\frac{2}{K}I}$.
4. Determine the moment of inertia $I$ for a thin disk rotating about its center using the formula: $I=\frac{1}{2}m{r}^2$, where $m$ is the mass and $r$ is the radius of the disk.
5. Once $\omega$ is calculated, find the linear speed of a point on the rim using the relationship: $v=r\omega$, where $v$ is the linear speed, $r$ is the radius, and $\omega$ is the angular velocity.

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation. It is given by the formula KE_rot = (1/2) I ω², where I is the moment of inertia and ω is the angular velocity. For a disk, the moment of inertia can be calculated as I = (1/2) m r², where m is the mass and r is the radius. Understanding this concept is crucial for relating the given kinetic energy to the rotational motion of the disk.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a solid disk, the moment of inertia is calculated using the formula I = (1/2) m r². This concept is essential for determining how the mass and shape of the disk influence its rotational dynamics and kinetic energy.

Linear Speed at the Rim

The linear speed of a point on the rim of a rotating object is related to its angular velocity by the equation v = r ω, where v is the linear speed, r is the radius, and ω is the angular velocity. This relationship allows us to convert between rotational and linear motion, which is necessary for finding the speed of a point on the rim of the disk given its rotational kinetic energy.