A 0.400-kg particle slides around a horizontal track. The...

A 0.400-kg particle slides around a horizontal track. The track has a smooth vertical outer wall forming a circle with a radius of 1.50 m. The particle is given an initial speed of8.00 m/s. After one revolution, its speed has dropped to 6.00 m/s because of friction with the rough floor of the track. (a) Find the energy converted from mechanical to internal in the system due to friction in one revolution. (b) Calculate the coefficient of kinetic friction. (c) What is the total number of revolutions the particle makes before stopping?

Key Concepts

Work-Energy Principle

The work-energy principle states that the work done by all forces acting on an object results in a change in its kinetic energy. In this context, the negative work done by friction (a non-conservative force) decreases the mechanical energy of the particle, which is then converted into internal energy. This principle is used to calculate the energy lost per revolution as the speed decreases.

Energy Dissipation

Energy dissipation refers to the process by which mechanical energy is converted into other forms of energy, such as thermal energy, due to non-conservative forces like friction. In the scenario of the sliding particle, energy dissipation is observed as a loss of kinetic energy, which not only slows down the particle but also helps in determining the number of revolutions it will make before coming to a stop.

Uniform Circular Motion

Uniform circular motion describes the movement of an object traveling at a constant speed along a circular path. Although the speed is constant in magnitude, the direction of the velocity changes continuously, meaning there is always a centripetal acceleration directed towards the center of the circle. Understanding uniform circular motion is important for analyzing the forces acting on the particle as it moves along the curved track.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity of the object. This concept is crucial for understanding how the particle’s energy changes as it slows down due to friction, indicating how much of its mechanical energy is converted to other forms, such as thermal energy.

Friction

Friction is the resistive force that opposes the motion of objects in contact. It causes energy dissipation in the form of heat and is characterized by its dependency on the nature of the surfaces and the normal force between them. Friction is the key factor that converts some of the particle’s kinetic energy into internal energy as it moves along the rough track.

Coefficient of Kinetic Friction

The coefficient of kinetic friction is a dimensionless quantity that represents the ratio of the kinetic friction force to the normal force acting on an object. It is used to determine the magnitude of the frictional force that opposes the motion of the particle. This coefficient is critical in calculating how quickly the particle loses energy and slows down when in contact with the rough surface.

Transcript

00:04 So in this problem, we are concerned with energy loss and energy transfer due to friction.

00:12 So the first part of the problem asks us to find the amount of kinetic energy transferred into internal energy, given an initial velocity and a final velocity after going around in a circle, for one revolution.

00:31 So in order to find that, we just simply calculate our different.

00:36 In initial and final kinetic energy.

00:40 So we know that our expression for kinetic energy, which we'll call k -e, is equal to 1 half mass times velocity square.

01:01 So to find the amount of energy loss and transferred into internal energy, we'll take delta k -e, which is the change in kinetic energy equal to one half mass parentheses final velocity squared minus the initial velocity squared.

01:41 And in the problem, we're given our values for the mass and these final and initial velocities.

01:48 And we are getting that this value is negative 5 .6 joules, which means that the total amount of energy, lost from mechanical to internal, is simply 5 .6 joules.

02:15 All right.

02:18 So the second part of the problem asks us to consider this change in kinetic energy and then use that to solve for the coefficient of kinetic friction.

02:37 So the most straightforward way to approach this is using our work energy theorem, which states that the change in kinetic energy is equal to the work done on that system.

02:53 And the system in question here is simply a point particle.

03:01 So we know that the only work done on the particle here is work due to friction.

03:07 And we know this change kinetic energy from the last part of the problem.

03:11 So our expression for work can be thought of as the frictional force times the distance the particle travels.

03:29 During this time.

03:33 So we know that this frictional force is going to be coefficient of kinetic friction times mass, times gravity, which is your normal force.

03:46 And we know the distance that it travels in this time is simply one revolution around the circle, which is 2 pi r.

04:02 So if we rearrange this expression and just solve for the coefficient, we're left with the coefficient is equal to the change in kinetic energy, over the mass times gravity times two pi...

Please give Ace some feedback