The 25-lb block has an initial speed of v0=10 ft / s when it is midway between springs A and B.

The 25-lb block has an initial speed of v0=10 ft / s when it...

Key Concepts

Work?Energy Principle

This principle states that the work done by the forces acting on an object equals the change in its kinetic energy. In this problem, it is used to equate the block’s initial kinetic energy with the work done against friction (and through interactions with springs) to determine when and where its energy is fully dissipated, thereby helping to calculate the total distance traveled.

Frictional Energy Dissipation

Friction converts the kinetic energy of the moving block into thermal energy, causing it to gradually slow down. The friction force is determined by the coefficient of kinetic friction and the normal force (which, on a horizontal surface, equals the block’s weight), and the work done by friction is the product of this frictional force and the distance over which it acts.

Spring Potential Energy

When the block compresses a spring, its kinetic energy is temporarily stored as potential energy in the spring according to the equation (1/2)kx², where k is the spring constant. This energy is later released, converting back into kinetic energy as the spring recoils, and plays a role in the block’s oscillatory motion.

Energy Conservation in the Presence of Non?Conservative Forces

In systems with only conservative forces, total mechanical energy remains constant. However, when non?conservative forces like friction are involved, some mechanical energy is lost from the system. In this scenario, the energy dissipated by friction over multiple oscillations eventually stops the motion of the block, and accounting for this loss is essential in determining the total distance traveled before coming to rest.

Transcript

00:01 We have a block that's traveling to the right at an initial velocity of 5 meters per second, and it's going to crash into that spring, and you want to find the distance or the compression of that spring before it launches a block in the opposite direction.

00:22 So that distance will say it's called sa.

00:26 We're given the stiffness of that spring as k -a is equal to 200 -newton.

00:31 Meters.

00:32 The mass of the block is 8 kilograms and the block is experiencing kinetic friction.

00:40 So the coefficient of kinetic friction is 0 .25.

00:46 So because we're dealing with movement here, there's got to be some sort of kinetic energy going on.

00:52 And we want to find the distance.

00:55 And we know that embedded in work is the distance because work is force times distance.

01:01 So we can use the work energy principle, and that's given as the initial kinetic energy, plus the work being done by the forces is equal to the final kinetic energy.

01:16 Well, when the block strikes the spring, the spring is going to compress, so the block momentarily stops moving before launching back.

01:25 So the initial kinetic, or the final kinetic energy is zero because the velocity final is zero.

01:34 So now we're just left with the initial kinetic energy, which is one half m squared plus the work being done by the forces and then all of that equal to zero.

01:51 We're going to call this equation one.

01:55 So now let's look at the work being done.

01:59 So work is force times the distance.

02:05 So we know that there's a force due to the spring, and there's a force due to the kinetic friction, and both of them are in the opposite direction of the movement of the block.

02:17 The block is moving to the right, friction in the opposite direction, and the spring's going to compress and launch it with the force in the opposite direction.

02:25 So we have minus, and then the force of the spring, times the distance, so we have one -half, k, the stiffness of the spring, times s .a.

02:38 Squared minus the force due to friction, which is coefficient kinetic friction times the normal force.

02:48 The normal force is equal to the weight of the object because they're in the wide direction, and those are the only forces in the wide direction...

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