Directions: Questions 1, 2, and 3 are short free-response...

Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part. In the diagram above, a spring-block system is oscillating on a flat horizontal surface. Part of the surface is frictionless, and part of the surface is not frictionless. The block starts at rest at position $x=0 \mathrm{cm}$. The block is then pushed to the left, compressing the spring, until it reaches the position $x=- 50 \mathrm{cm}$. After being pushed in, the block is released and allowed to move naturally. (a) Draw a graph of the system’s position as a function of time starting from the moment it is released and explain your reasoning behind the graph you draw. Be sure to label important values on the graph. (b) Given that the mass $m=2 \mathrm{kg}$ and the spring constant $k=100$ $\mathrm{N} / \mathrm{m},$ what is the magnitude of work done by the non-frictionless surface?

Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.
In the diagram above, a spring-block system is oscillating on a flat horizontal surface. Part of the surface is frictionless, and part of the surface is not frictionless. The block starts at rest at position $x=0 \mathrm{cm}$. The block is then pushed to the left, compressing the spring, until it reaches the position $x=- 50 \mathrm{cm}$. After being pushed in, the block is released and allowed to move naturally.
(a) Draw a graph of the system’s position as a function of time starting from the moment it is released and explain your reasoning behind the graph you draw. Be sure to label important values on the graph.
(b) Given that the mass $m=2 \mathrm{kg}$ and the spring constant $k=100$ $\mathrm{N} / \mathrm{m},$ what is the magnitude of work done by the non-frictionless surface?

Key Concepts

Work-Energy Principle

The work-energy principle relates the work done by all forces acting on an object to the change in its kinetic energy. This principle is vital when quantifying the energy losses due to friction in oscillatory systems, as it connects the work done by friction with the reduction in the energy available for motion.

Friction and Non-Conservative Forces

Friction is a non-conservative force that dissipates energy as heat. In systems where friction acts over part of the path, it reduces the total mechanical energy, making it crucial to account for energy losses when applying energy conservation principles.

Simple Harmonic Motion

Simple harmonic motion describes oscillatory motion where the restoring force is directly proportional to the negative of the displacement. This produces sinusoidal motion in time, which is essential for understanding periodic behavior in systems like a block attached to a spring.

Hooke's Law and Elastic Potential Energy

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium (F = -kx). The work done during compression or extension is stored as elastic potential energy, given by U = 1/2 kx², a key element in analyzing the energy transformations in oscillating systems.

Transcript

00:01 For this example, we're going to look at the motion of a mass on a spring when there's a little bit of friction involved.

00:10 And as our example, we're going to start off with a mass on a frictionless horizontal surface.

00:21 And we'll start our mass oscillating somewhere to the left of the equilibrium position.

00:33 We'll start it at minus 50 centimeters.

00:38 And we're going to put the friction over to the right of the equilibrium, starting at 30 centimeters and going all along to the right.

01:05 And what we know will happen is as the mass slides into that frictional region, it's going to slow down, lose energy, and therefore each time it goes into that region, its amplitude is going to decrease.

01:23 And it'll decrease both in the graph that we see to the right of the 30 centimeters and also in its final amplitude off on the left.

01:38 Both of those will decrease.

01:40 However, if the amplitude falls beneath 30 centimeter, the mass will continue to oscillate fine with, continued amplitude of slightly less than 30 centimeters, because it will no longer fall in that region of friction.

01:59 So we're going to make a graph of position versus time for that mass and just kind of see what that motion looks like.

02:19 And we should probably scale the time.

02:24 We'll go ahead and use the period of the motion, the period is related to the angular frequency through the square root of k over m.

02:38 And let's say for the point of calculation that k is 100 newtons per meter and m is 2 kilograms, then omega is close to 7 rads per second, and that's related to the period p through 2 pi radians and the period p.

03:13 And solving for that, we get a period close to a second.

03:19 So this is not a rapid oscillation.

03:25 But on this graph, the two important xs, if you will, are the 50 centimeters, where it starts out negative.

03:40 And it's never going to get quite to positive 50 centimeters, but we'll put that up there anyway.

03:54 And the other important quantity is the minus 30, which delineates the region of the friction.

04:20 And at the time, we'll try to show the periods marked off in quarters.

04:34 So there's one full period.

04:36 I'll show that a little bit more clearly.

04:54 So about three full periods, i think, will be enough for us to see what this motion looks like.

05:02 And there's one period, two period, three period.

05:07 Okay, so the mass starts at minus 50 centimeters...

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