A 1.00-kg object slides to the right on a surface having a...

A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P8.62a). The object has a speed of vi 5 3.00 m/s when it makes contact with a light spring (Fig. P8.62b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P8.62c). The object is then forced toward the left by the spring (Fig. P8.62d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P8.62e). Find (a) the distance of compression d, (b) the speed v at the unstretched position when the object is moving to the left (Fig. P8.62d), and (c) the distance D where the object comes to rest.

A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P8.62a). The object has a speed of vi 5 3.00 m/s when it makes contact with a light spring (Fig. P8.62b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P8.62c). The object is then forced toward the left by the spring (Fig. P8.62d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P8.62e).
Find (a) the distance of compression d, (b) the speed v at the unstretched position when the object is moving to the left (Fig. P8.62d), and (c) the distance D where the object comes to rest.

Key Concepts

Kinetic Energy

Kinetic energy is the energy of motion, and it plays a central role in energy transformation problems. In this scenario, the object’s initial kinetic energy is redistributed into other forms, such as spring potential energy and heat lost to friction, making it a key element in setting up and solving the energy balance equations.

Conservation of Energy

This concept involves the transformation of energy from one form to another while accounting for losses due to non-conservative forces. In this problem, the object’s kinetic energy is partly converted into spring potential energy and partly dissipated as work done against friction. Recognizing energy conservation allows us to set up energy balance equations to solve for unknown quantities.

Work Done by Non-Conservative Forces

Non-conservative forces, such as friction, do work that results in a dissipation of mechanical energy. The work done by kinetic friction depends on the friction coefficient, the normal force, and the displacement over which the friction acts. Including this work in energy equations is essential to accurately determine the object’s motion and energy changes.

Spring Potential Energy and Hooke's Law

Hooke's law relates the force exerted by a spring to its displacement from equilibrium via the spring constant. The energy stored in a compressed or stretched spring is defined as spring potential energy, which is proportional to the square of the compression or extension distance. This concept is used to determine energy storage in the spring when the object compresses it.

Transcript

00:01 So in this question we have a block of mass that is travelling on a surface with some friction, coefficient of friction and 0 .25.

00:14 Now we are given that it hits a spring.

00:17 So it hits a spring and this spring has a spring constant 50 newtons per meter and it hits the spring with a velocity of 3 meters per second.

00:32 You want to find how far the block would be able to compress this spring by, right? and this is the final state.

00:44 Assuming this is the final state, what is the value of d? so as you know, the initial energy that we have, the initial total energy is just the kinetic energy of the block.

00:57 It's given as half and v square.

01:04 On the other hand, for the final energy, the final energy of the system would be the elastic, potential energy stored in the spring, which is half k times d square, plus the energy that was lost due to friction, right? the work done by friction, which is given as f times d.

01:30 Where f, this is the friction, friction can be written as mu k times n, where n is the normal force and this normal force must be balanced out by the weight.

01:43 So this actually equals m g.

01:49 Putting values in, this is actually about 2 .45 newtons.

01:59 So we equated both of this because the by energy conservation total initial energy must be equal to the total final energy.

02:10 So putting values in, this is 25 d square plus 0 .45d.

02:19 And if we're to rearrange a bit, we bring the half mv square to the right -hand side.

02:23 So we get on the left -hand side is 0...

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