A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline. How far up the incline will it go?

Verified step by step guidance

1. Step 1: Identify the type of energy involved. The hollow cylinder has both translational kinetic energy (due to its linear motion) and rotational kinetic energy (due to its rolling motion). The total mechanical energy is conserved as it moves up the incline, assuming no energy loss due to friction or air resistance.
2. Step 2: Write the expression for the total initial energy. The total energy is the sum of translational kinetic energy and rotational kinetic energy: \( E_{initial} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 \), where \( m \) is the mass of the cylinder, \( v \) is its linear velocity, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity. For a hollow cylinder, \( I = m r^2 \), and \( \omega = \frac{v}{r} \).
3. Step 3: Simplify the total initial energy. Substitute \( I = m r^2 \) and \( \omega = \frac{v}{r} \) into the equation: \( E_{initial} = \frac{1}{2} m v^2 + \frac{1}{2} m r^2 \left( \frac{v}{r} \right)^2 \). Simplify to get \( E_{initial} = m v^2 \).
4. Step 4: Write the expression for the potential energy at the highest point on the incline. At the maximum height, all the kinetic energy is converted into gravitational potential energy: \( E_{final} = m g h \), where \( h \) is the height, \( g \) is the acceleration due to gravity, and \( m \) is the mass of the cylinder.
5. Step 5: Relate the height \( h \) to the distance \( d \) along the incline. Use trigonometry: \( h = d \sin(\theta) \), where \( \theta = 18° \). Set \( E_{initial} = E_{final} \), so \( m v^2 = m g h \). Cancel \( m \) and substitute \( h = d \sin(\theta) \) to solve for \( d \): \( d = \frac{v^2}{g \sin(\theta)} \).

Key Concepts

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the kinetic energy of the rolling hoop will convert into gravitational potential energy as it ascends the incline. This relationship allows us to calculate how high the hoop will rise based on its initial speed.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. For a rolling object like a hollow cylinder, we must also consider its rotational kinetic energy, which is given by KE_rot = 1/2 Iω², where I is the moment of inertia and ω is the angular velocity. This combined energy will be relevant in determining how far up the incline the hoop travels.

Inclined Plane Dynamics

When an object moves up an incline, the angle of the incline affects the gravitational force acting on it. The component of gravitational force acting parallel to the incline will decelerate the object. Understanding the relationship between the incline angle and the forces involved is crucial for calculating the distance the hoop travels before coming to a stop.