In Fig. 8-52, a 3.5 kg block is accelerated from rest by a...

In Fig. 8-52, a 3.5 kg block is accelerated from rest by a compressed spring of spring constant 640 $\mathrm{N} / \mathrm{m}$ . The block leaves the spring at the spring's relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction $\mu_{k}=0.25 .$ The frictional force stops the block in distance $D=7.8 \mathrm{m}$ . What are (a) the increase in the thermal energy of the block-floor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring?

In Fig. 8-52, a 3.5 kg block is accelerated from rest by a compressed spring of spring constant 640 $\mathrm{N} / \mathrm{m}$ . The block leaves the spring at the spring's relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction $\mu_{k}=0.25 .$ The frictional force stops the block in distance $D=7.8 \mathrm{m}$ . What are (a) the increase in the thermal energy
of the block-floor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring?

Key Concepts

Work Done by Friction

Work done by friction refers to the process by which frictional forces convert the mechanical energy of a moving object into thermal energy. This concept is vital in problems where energy is dissipated as heat, such as when an object slides along a rough surface, reducing its kinetic energy.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is determined by the object's mass and the square of its velocity. In scenarios where objects are accelerated, maximum kinetic energy represents the peak energy state before other forces, such as friction, begin to work on decelerating the object.

Spring Potential Energy

Spring potential energy is the energy stored within a spring when it is either compressed or stretched. Based on Hooke's law, this energy is a function of the spring constant and the displacement from its equilibrium position. It describes the work that can be done by the spring when it returns to its natural length.

Conservation of Energy

This principle states that energy in a closed system is conserved and can change forms, such as transforming stored potential energy into kinetic energy or thermal energy. It is crucial for analyzing systems where energy is transferred or dissipated, ensuring that all energy inputs and outputs balance when accounting for losses like friction.

Transcript

00:02 In this problem, we're given a 3 .5 kilogram block that travels over a smooth surface that i have highlighted in blue, and then it transitions to a rough surface that i've highlighted in orange with a kinetic coefficient of friction of 0 .25.

00:24 It's connected via a spring that has a spring constant of 640 newtons per meter, and the block has a mass of 3 .5 kilograms.

00:36 Now, for the first part, we need to find out the total thermal energy dissipated as the block travels a distance d, and d is given as 7 .8 meters, so d is equal to 7 .8 meters, and then it comes to rest.

00:54 So for that, we need to calculate the frictional force exerted when the block reaches the orange surface.

01:03 The kinetic friction is given by f equals neo sub k times the contact force r.

01:11 Now let's consider the forces acting on the block vertically.

01:17 So it's being pulled down by the weight that is equal to the mass times gravity, and it's being pushed up by the contact force are.

01:31 Now, there is no net vertical acceleration, so the contact force must be equal to the weight.

01:40 That means that r equals mg.

01:44 And we can go ahead and make the substitution in our first formula.

01:48 So that gives f equals muz of k times m g.

01:54 Now note that all of these are constants.

01:58 The coefficient of kinetic friction is a constant.

02:00 The mass of the block is a constant.

02:03 And the gravitational acceleration is also constant.

02:08 Now when we have a constant force, it's work done.

02:13 Is the force times the distance traveled under the influence of the force.

02:21 So the work done is actually equal to mu sub k times m times g times d.

02:30 And we have all of these values so mu equals 0 .25.

02:35 The mass of the block is 3 .5 kilograms.

02:39 Gravitational acceleration is 9 .8 meters per second squared and d is 7 .8 meters now it's a simple matter of performing the multiplication, which we can do using a calculator...

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