A 4.0-cm-diameter disk with a 3.0-cm-diameter hole rolls down a 50-cm-long, 20° ramp. What is its speed at the bottom? What percent is this of the speed of a particle sliding down a frictionless ramp?

Verified step by step guidance

1. Determine the moment of inertia of the disk with a hole. The moment of inertia for a solid disk is \( I = \frac{1}{2} m R^2 \), but since there is a hole, subtract the moment of inertia of the missing part. Use \( I_{hole} = \frac{1}{2} m_{hole} r^2 \), where \( m_{hole} \) is the mass of the missing part and \( r \) is its radius.
2. Relate the rotational and translational motion of the rolling disk. Use the rolling condition \( v = \omega R \), where \( v \) is the linear velocity, \( \omega \) is the angular velocity, and \( R \) is the radius of the disk.
3. Apply energy conservation. The total mechanical energy at the top of the ramp (potential energy) is converted into both translational and rotational kinetic energy at the bottom. Use \( m g h = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 \). Substitute \( \omega = \frac{v}{R} \) into the equation.
4. Solve for the linear velocity \( v \) at the bottom of the ramp. Rearrange the energy conservation equation to isolate \( v \), and substitute the expression for the moment of inertia \( I \) derived earlier.
5. To find the percentage of the speed compared to a particle sliding down a frictionless ramp, calculate the speed of the particle using \( v = \sqrt{2 g h} \), where \( h \) is the height of the ramp. Then, compute the ratio \( \frac{v_{disk}}{v_{particle}} \times 100 \% \).

Key Concepts

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 disk with a hole, the moment of inertia can be calculated by subtracting the moment of inertia of the hole from that of the full disk, which affects how the disk rolls down the ramp.

Conservation of Energy

Conservation of energy states that the total energy in a closed system remains constant. In the context of the disk rolling down the ramp, gravitational potential energy is converted into both translational and rotational kinetic energy. Understanding this principle is crucial for calculating the final speed of the disk at the bottom of the ramp.

Rolling Motion

Rolling motion involves both translational and rotational movement, where an object rolls without slipping. The relationship between linear speed and angular speed is given by the equation v = rω, where v is linear speed, r is the radius, and ω is angular speed. This concept is essential for determining the speed of the disk as it rolls down the ramp compared to a particle sliding down.