A smooth circular hoop with a radius of 0.500 m is placed...

A smooth circular hoop with a radius of 0.500 m is placed flat on the floor. A 0.400-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.00 m/s. After one revolution, its speed has dropped to 6.00 m/s because of friction with the floor. (a) Find the energy transformed from mechanical to internal in the particle–hoop–floor system as a result of friction in one revolution. (b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.

A smooth circular hoop with a radius of 0.500 m is placed flat on the floor. A 0.400-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.00 m/s. After one revolution, its speed has dropped to 6.00 m/s because of friction with the floor.
(a) Find the energy transformed from mechanical to internal in the particle–hoop–floor system as a result of friction in one revolution. (b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.

Key Concepts

Uniform Circular Motion and Angular Displacement

This concept involves the motion of an object moving along a circular path with a constant radius. It connects linear quantities, like speed and the work done by friction, with angular quantities, such as the number of revolutions. Understanding how the distance traveled on a circular path relates to the circumference of the circle is essential for calculating quantities like the total number of rotations before an object comes to a stop.

Kinetic Energy and Energy Loss

Kinetic energy, which is calculated as one half the mass times the speed squared, is the measure of an object’s energy due to its motion. In systems where friction acts, kinetic energy is continually reduced as friction does work, and the amount of energy lost per cycle or per unit distance can be determined by comparing energy values at different times.

Mechanical Energy Transformation

This concept involves understanding that in many physical processes, mechanical energy (which includes kinetic and potential energy) can be converted into other forms of energy such as thermal or internal energy. In problems involving friction, a portion of the mechanical energy is dissipated as heat, leading to a decrease in the system’s mechanical energy over time.

Work-Energy Principle

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. This principle is central to analyzing situations where non-conservative forces, like friction, perform work that reduces an object’s kinetic energy, converting it into other forms of energy.

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