A large fake cookie sliding on a horizontal surface is...

A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant $k=400 \mathrm{~N} / \mathrm{m} ;$ the other end of the spring is fixed in place. The cookie has a kinetic energy of $20.0 \mathrm{~J}$ as it passes through the spring's equilibrium position. As the cookie slides, a frictional force of magnitude $10.0 \mathrm{~N}$ acts on it. (a) How far will the cookie slide from the equilibrium position before coming momentarily to rest? (b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?

Key Concepts

Energy Conservation with Nonconservative Forces

The principle of energy conservation states that the total energy in a closed system remains constant, but when nonconservative forces like friction are present, mechanical energy is not fully conserved. Instead, energy is transformed into other forms such as heat. This requires us to account for the work done by nonconservative forces when applying the work-energy theorem to analyze energy transformations in a system.

Work Done by Friction

The work done by friction is the energy lost to non-conservative forces as an object moves across a surface. In problems involving friction, the work done by this force is calculated as W_friction = friction force × distance traveled, and it represents energy that is not recoverable within the system. This dissipation of energy affects the overall energy balance in dynamic systems.

Elastic Potential Energy

Elastic potential energy is the energy stored in a stretched or compressed spring, which is governed by Hooke’s law. The energy stored in the spring is given by the formula U = 1/2 k x², where k is the spring constant and x is the displacement from the equilibrium position. This energy can be converted back into kinetic energy as the spring returns to its equilibrium state.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is quantified by the expression KE = 1/2 m v², where m is the mass of the object and v is its velocity. This energy is responsible for the ability of an object to do work as it moves, and it can be transferred or transformed into other forms of energy during physical processes.

Transcript

00:01 So here we can say that we're going to, if you wanted to see a fig, if you wanted to see like the sketch of the system, it's nice to reference figure 8 .3 and add friction.

00:15 So if you add friction to figure 8 .3, of course, friction is always in the direction opposite of motion.

00:21 This would be a good figure in order to take a look at the system.

00:25 At this point, we can then say that we're going to use equation 831.

00:30 And see that the change in thermal energy would be equal to the force of kinetic friction multiplied by d, the distance traveled.

00:39 We can then relate the initial kinetic energy, k -sub -i, to the resting potential energy.

00:47 So we can say k -s -i plus the initial potential energy would be equal to f, rather we can say force of friction k times d, plus the kinetic energy resting plus the resting potential energy.

01:07 We know that initially we have no initial potential.

01:10 And of course, the resting kinetic energy, the resting kinetic energy, this is an oxymoron and wouldn't actually exist.

01:19 So this is of course equaling zero.

01:21 So we have that the initial kinetic energy equals the resting potential energy plus the change in thermal energy.

01:28 And we can then substitute.

01:31 We have 20 .0 joules.

01:35 This would be equal to the force of friction kinetic multiplied by d plus 1⁄2kd squared.

01:47 So we can say that here we know that the force of friction kinetic is equaling 10 .0 neutins.

01:55 And we know that here the spring constant is equaling 400 newtons per meter squared.

02:04 So essentially we're trying to find d and we can then say that one half of 400, this would be we can say 200 d squared plus we can say force of friction is 10...

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