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. Calculate the moment of inertia (I) for the disk. Since the disk is rotating about an axis through its center, use the formula for the moment of inertia of a disk, which is \(I = \frac{1}{2} m r^2\), where \(m\) is the mass of the disk and \(r\) is the radius.
2. Convert the mass of the disk from grams to kilograms and the diameter to radius in meters. The mass \(m = 100\) g = 0.1 kg and the radius \(r = \frac{8.0}{2}\) cm = 0.04 m.
3. Substitute the values of \(m\) and \(r\) into the moment of inertia formula to find \(I\).
4. Use the rotational kinetic energy formula \(K = \frac{1}{2} I \omega^2\) to solve for the angular velocity \(\omega\). Here, \(K\) is the kinetic energy given as 0.15 J.
5. Calculate the linear speed \(v\) of a point on the rim using the relationship between linear speed and angular velocity, which is \(v = r \omega\). Substitute the value of \(r\) and the calculated \(\omega\) to find \(v\).

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 geometry of the disk affect its rotational dynamics and kinetic energy.

Linear Speed from Angular Velocity

The linear speed of a point on the rim of a rotating object can be determined from its angular velocity using the relationship v = r ω, where v is the linear speed, r is the radius, and ω is the angular velocity. This concept connects rotational motion to linear motion, allowing us to find the speed of a point on the disk's edge once we have calculated the angular velocity from the kinetic energy.