A 20 kg block on a horizontal surface is attached to a...

A $20 \mathrm{~kg}$ block on a horizontal surface is attached to a horizontal spring of spring constant $k=4.0 \mathrm{kN} / \mathrm{m} .$ The block is pulled to the right so that the spring is stretched $10 \mathrm{~cm}$ beyond its relaxed length, and the block is then released from rest. The frictional force between the sliding block and the surface has a magnitude of $80 \mathrm{~N}$. (a) What is the kinetic energy of the block when it has moved $2.0 \mathrm{~cm}$ from its point of release? (b) What is the kinetic energy of the block when it first slides back through the point at which the spring is relaxed? (c) What is the maximum kinetic energy attained by the block as it slides from its point of release to the point at which the spring is relaxed?

Key Concepts

Conservation of Energy

This concept involves the principle that the total energy in a closed system remains constant, though it can change forms. In problems involving springs and friction, energy is transformed between potential energy stored in the spring, kinetic energy of the moving mass, and energy lost due to non-conservative forces like friction.

Spring Potential Energy

This is the energy stored in a spring when it is displaced from its equilibrium position. It is given by the expression (1/2)kx², where k is the spring constant and x is the displacement. Understanding spring potential energy is crucial in analyzing how energy is transferred and stored in the system before being converted into kinetic energy.

Work Done by Non-Conservative Forces

In the presence of friction, which is a non-conservative force, mechanical energy is not conserved because some energy is dissipated as heat. The work done by friction (force times displacement) reduces the mechanical energy available for conversion between potential and kinetic energy. Accounting for this energy loss is essential in solving problems where friction is involved.

Work-Energy Theorem

This theorem states that the net work done on an object is equal to the change in its kinetic energy. It provides a direct method for relating the forces acting on an object, including both conservative and non-conservative forces, to its kinetic energy. In the context of a mass-spring system with friction, it allows the calculation of kinetic energy at different positions by accounting for work done by both the spring force and friction.

Transcript

00:01 Well, mass of the block, m, is equal to 20 kilograms, 20 kilograms, and sipping constant, k is equal to 4 kilo -newtons per meter.

00:22 And the sipping is stretched to a distance, d, is equal to 10 centimeters beyond its original length and initial speed of the block is zero vi is equal to 0 meter per second so initial speed of the block is zero and the force of friction is equal to 80 newtons it's equal to 80 newtons okay now from the conservation of mechanical energy of the block change in kinetic energy plus change in potential energy plus energy due to friction is equal to zero.

01:27 And changing kinetic energy can be written is kf minus k i, kf minus k i and change in potential energy is half k, k, d, square minus half k d1 square d1 square plus an energy due to friction can be written is force of friction times the displacement x and it's equal to zero now putting the values we have kinetic energy final minus zero initial velocity is zero therefore initial kinetic energy is zero plus one divided by two is k is common and we have d square minus d one one square inside the bracket plus the force of friction times x is equal to zero all right and now putting the values we have final kinetic energy plus 1 divided by 2 multiply by k is 4 kiloons 4 multiply by 10 to the power 3 newtons and d in this case is 0 .08 square minus d1 is 0 .1 is 0 .1 square plus force of fraction is 80 newton's multiply by x is 0 .02 and it's equal to 0.

03:19 All right and therefore kinetic energy final kinetic energy final is equal to 5 .6 joules.

03:34 So we have 5 .6 joules which is the final kinetic energy so this is the final kinetic energy so this was part a.

03:44 Now let's solve part b.

03:47 We'll solve part b.

03:49 Again, we have a change in kinetic energy plus change in potential energy plus energy due to friction is equal to zero.

04:08 And now the block in this case moves 10 centimeters and and the final elastic potential energy of the sip ring is 0 because it returns to its original position...

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