A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline.
(b) How long will it be on the incline before it arrives back at the bottom?

Verified step by step guidance

1. Determine the initial kinetic energy of the hoop when it is moving on the horizontal surface. The kinetic energy is the sum of translational kinetic energy (\(\frac{1}{2} m v^2\)) and rotational kinetic energy (\(\frac{1}{2} I \omega^2\)), where \(I\) is the moment of inertia of the hoop and \(\omega\) is the angular velocity.
2. Calculate the moment of inertia (\(I\)) of the hoop. For a hoop, \(I = m r^2\), where \(m\) is the mass of the hoop and \(r\) is its radius. The angular velocity (\(\omega\)) can be found using the relation \(v = r \omega\).
3. As the hoop moves up the incline, its potential energy increases while its kinetic energy decreases. Use conservation of energy to find the maximum height (\(h\)) reached on the incline by setting the initial kinetic energy equal to the potential energy gained (\(mgh\)).
4. Determine the time it takes for the hoop to reach the maximum height and then return to the bottom. This can be calculated by finding the time to ascend (using the acceleration due to gravity component along the incline, \(g \sin(\theta)\)) and then doubling it, as the time to ascend and descend will be the same.
5. Use the kinematic equation \(h = \frac{1}{2} a t^2\) to solve for the time (\(t\)), where \(a = -g \sin(\theta)\) is the acceleration along the incline. The total time on the incline is twice this value.

Key Concepts

Rolling Motion

Rolling motion occurs when an object rotates about an axis while simultaneously translating along a surface. For a hollow cylinder, both translational and rotational kinetic energy contribute to its overall motion. The relationship between linear speed and angular speed is crucial, as it determines how the cylinder interacts with the incline.

Inclined Plane Dynamics

When an object rolls down an incline, gravitational forces and frictional forces play significant roles in its motion. The angle of the incline affects the component of gravitational force acting along the slope, which influences acceleration. Understanding these forces is essential for calculating the time it takes for the cylinder to return to the bottom.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, kinematic equations can be used to determine the time spent on the incline based on the initial speed and the distance traveled down the slope.