Rotational Kinetic Energy Calculation of a Rolling Hoop
What is the formula to calculate the speed of the center of mass of a rolling hoop?
Given data: A hoop is rolling on a flat surface about its center with a moment of inertia of , and a radius of 0.3 m. The hoop has a rotational kinetic energy of 87 J. How can we determine the speed of the center of mass of the rolling hoop?
Calculation of the Speed of the Center of Mass of a Rolling Hoop
The speed of the center of mass of the rolling hoop can be determined using the concept of rotational kinetic energy. The moment of inertia of a hoop is given by I = 2mR^2. The given rotational kinetic energy is 87 J. By rearranging the equation and substituting the known values, the linear speed of the center of mass can be calculated.
When a hoop is rolling on a flat surface about its center, the rotational kinetic energy can be related to its linear speed using the equation: Rotational kinetic energy = 1/2 * moment of inertia * (linear speed)^2. Given the moment of inertia of a hoop as I = 2mR^2, where m is the mass and R is the radius, the equation can be rearranged to solve for the linear speed of the center of mass.
In the provided data, the moment of inertia and the rotational kinetic energy are known. By substituting these values into the equation and solving for the linear speed, we can find that the speed of the center of mass of the rolling hoop is approximately 7.07 m/s.
Understanding the relationship between rotational kinetic energy and linear speed is crucial for calculating the motion of rotating objects. By applying the relevant equations and concepts, we can determine the speed of the center of mass of a rolling hoop and gain insights into its rotational dynamics.