A 750 g disk and a 760 g ring, both 15 cm in diameter, are rolling along a horizontal surface at 1.5 m/s when they encounter a 15° slope. How far up the slope does each travel before rolling back down?

Verified step by step guidance

1. Convert the masses of the disk and ring from grams to kilograms: \( m_{\text{disk}} = 0.750 \; \text{kg} \) and \( m_{\text{ring}} = 0.760 \; \text{kg} \). Also, convert the diameter to radius: \( r = \frac{15}{2} \; \text{cm} = 0.075 \; \text{m} \).
2. Determine the total energy of each object at the base of the slope. The total energy is the sum of translational kinetic energy and rotational kinetic energy: \( E_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega = \frac{v}{r} \) is the angular velocity.
3. Substitute the moments of inertia for the disk \( I_{\text{disk}} = \frac{1}{2} m r^2 \) and the ring \( I_{\text{ring}} = m r^2 \) into the energy equation. Simplify to find the total energy for each object in terms of \( m \), \( v \), and \( r \).
4. At the highest point on the slope, all the kinetic energy is converted into gravitational potential energy: \( E_{\text{total}} = m g h \), where \( h \) is the height reached. Use the energy conservation principle to solve for \( h \): \( h = \frac{E_{\text{total}}}{m g} \).
5. Relate the height \( h \) to the distance traveled up the slope \( d \) using the slope angle \( \theta \): \( h = d \sin(\theta) \). Solve for \( d \): \( d = \frac{h}{\sin(\theta)} \). Substitute the values for \( \theta = 15^\circ \), \( g = 9.8 \; \text{m/s}^2 \), and the calculated \( h \) for each object to find the distance traveled up the slope.

Key Concepts

Moment of Inertia

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 and a ring, the moment of inertia is calculated differently: the disk has a moment of inertia of (1/2)mr², while the ring has mr², where m is mass and r is the radius. Understanding these differences is crucial for analyzing their rolling motion.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the kinetic energy of the rolling objects will convert into gravitational potential energy as they ascend the slope. The total mechanical energy at the bottom of the slope must equal the total mechanical energy at the highest point reached before rolling back down.

Rolling Motion

Rolling motion involves both translational and rotational motion, where an object rolls without slipping. The velocity of the center of mass and the angular velocity are related through the radius of the object. For the disk and ring, the rolling motion affects how they convert kinetic energy into potential energy as they climb the slope, influencing the distance each travels before rolling back down.