7. The following graph shows the potential energy U(x) of a...

7. The following graph shows the potential energy U(x) of a particle as a function of its position x. [28] U(x) (joules) 8 6 4 2 0 1 2 3 4 5 6 7 x (meters) a) Identify all points of equilibrium for this particle. Suppose the particle has a constant total energy of 4.0 joules, as shown by the dashed line on the graph. b) Determine the kinetic energy of the particle at the following positions i) x = 2.0 m ii) x = 4.0 m c) Can the particle reach the position x = 0.5 m? Explain d) Can the particle reach the position x = 5.0 m? Explain. e) On grid below, carefully draw a graph of the conservative force acting on the particle as a function of x, 0 < x < 7 m. F(x) (newtons) 4 2 0 -2 -4 0 1 2 3 4 5 6 7 x (meters) Extra Credit 1 1 point for answer and 2 more points for justifying why The motion of an object is described by: dv/dt = -kv^2t k is a constant. At time zero the velocity is vo. The expression for velocity is A) v = vo + kt^2/2 B) v = vo + kv^2t^2 C) 1/v = 1/vo + kt^2/2 D) 1/v = kt^2/2 E) 1/v = 1/vo - kt^2/2 F) 1/v = vo - kt^2/2 Extra Credit 2 A particle is acted upon by a force F = Foe^-kx. The particle has a velocity of zero at x = 0, the maximum kinetic energy that the particle can attain is A) Fo/k B) Fo/e^k C) kFo D) (kFo)^2 E) ke^kFo

Transcript

00:01 In this problem, you have a potential energy graph.

00:03 I've tried to draw these reasonably representative of what you have on your paper.

00:08 And we have some questions to have to answer about it.

00:11 First is, where are the equilibrium points at, if any? well, they're at the maximum of the potential energy.

00:19 So, x equals 2 meters and x equals 5 meters.

00:25 If you were to place a product code x equal 2 meters, it would stay there.

00:30 Place it at rest.

00:31 It'll stay there.

00:34 It'll stay there because it feels no force on it remember and we're going to need this later on this is a relationship between force and the potential energy remember graphically what is this slope of a tangent line well what's the slope of tangent line at a maximum endpoint zero so there's no force so this particle sitting there at rest it's got nothing on him that says move one way or another left or right positive x negative x nothing so these are the two points now our in reality, even though it's not asked of here, there are types of equilibrium point, stable, unstable, neutral, and so on.

01:12 This would be an example of a stable one, because if you were to displace it, like an infinitesimal displacement to the right of x equals 2, and then put it there, release it, but what happened to it? this, the tangent line here is positive slope.

01:32 It means the force is negative.

01:33 That means the force would be back.

01:36 Toward x equals 2.

01:38 So you don't get away from x equal 2 in that case, where like x equals 5, this would be an unstable one, because you did the same thing.

01:47 You'd find yourself moving farther away from x equals 5, not going back to it.

01:55 So that was part a to locate the equilibrium points, so 2 and 5.

02:01 Now, once the kinetic energy at 2 meters and 4 meters.

02:07 Now, graphically, what you do, visualize this you draw arrows so this would be u2 and we know you and k must equal e so you have another arrow so this is k of two notes of two arrows if it's pointing up it's positive pointing down is negative kinetic energy arrows can never point down can have negative kinetic energy but potential energy can be negative so these two add to e so e is equal to k2 plus u of two applies k of 2 is equal to 4 joules that's e minus u of 2 which is 1 3 joules so that is the kinetic energy at x equals 2 meters then it wants it 4 of 4 so do the same thing here's your u here's your k notice the k is smaller this is 4 so this k of 4 4 jules minus and i think it was 3 even though my i may be a little off in my spacing.

03:28 This is equal to one jewel.

03:33 So notice anywhere you can draw, you can always draw from the axis to the curve.

03:37 That gives you u and then from the curve to the energy line.

03:42 As long as k is upward pointing, you can do that.

03:46 Anywhere else, you cannot be in.

03:49 Now, the next part c wants to know if you can reach x equals 05.

03:54 Well, here is a point.

03:56 This is what is called a turning point.

04:03 E is equal to you, just visually, the two graphs, two curves cross.

04:08 But effectively, that means what? k is equal to zero.

04:11 What's the length of the k arrow? zero.

04:16 But visually, usually you think of just e equals you.

04:19 So that's what's called a turning point.

04:21 Why? well, first off, let me write this out.

04:26 And it's the only turning point.

04:28 There's no other place where it crosses, where they cross.

04:32 So can reach, well, cannot reach x equals 0 .5 meters cannot.

04:53 Why is what's happening physically? well, i mean, from the graph standpoint, remember, if you're trying to do x equals 0 .5 meters, you have to have a negative k.

05:02 That's not possible.

05:03 But let's not even worry about that.

05:04 Let's talk about physics of this.

05:06 Say you're moving in the negative x direction.

05:11 As you reach x equals 2, your arrow is large.

05:17 But as you move from x equals 2, toward x equals 1, your connect energy arrows are getting smaller and smaller until they reach zero.

05:25 So now you're here at rest.

05:28 Now you're not an equilibrium point, so there's a force on you, and you're going to follow that force.

05:33 That force, if you look at the slope of the tangent line here, it's negative.

05:37 Negative slope, minus, minus, plus.

05:40 So fx is positive.

05:42 That means that object will experience a force in the positive x direction, so you start moving back.

05:48 So you start retracing your steps, getting longer and longer.

05:52 Then once you reach here, you start getting shorter and shorter, but that's a different story, what you're going to do next.

05:57 Now it asks, can you reach x equals five? and the answer is yes, there's no other, there's no other, there's no other turning points.

06:08 There's no crossing points.

06:10 So yeah, you'll come back from the, if you want to from the x equals one turning point, get longer and longer.

06:20 And shorter and shorter and shorter and shorter and shorter but not zero.

06:26 So at x equal five, you're moving.

06:28 You reach it.

06:29 So you can reach it.

06:31 If there was another turning point that'd say x equals four, no, you cannot reach it.

06:42 Can reach x equals five meters.

06:47 So those are those two parts.

06:51 Now, next thing it wants is a graph of the force versus position.

07:00 Technically all these, what we're doing here, this is fx.

07:05 Even though just keep that in mind.

07:10 Now, this is a straight line here.

07:14 This is a straight line here.

07:15 There's little bits of curves here.

07:17 Let's not worry about them.

07:18 They don't gain any real physical insight.

07:23 But i'm worrying about those.

07:24 It's just a mathematical thing.

07:26 But they deal with that.

07:27 So, oh, we're just going to deal with these straight segments and connect them together, linear segments.

07:33 So this, this has a slope, four joules, minus 8 joules, four is one meter, minus zero meter.

07:50 This is minus four joules per meter.

07:53 That is a newtum, but i don't want to use that yet because this is just a slope of a potential energy graph.

07:58 The force is minus of that.

08:00 So on the graph down here, i have to draw close to two, not quite there.

08:08 Close to two, i got to draw round to there.

08:14 The force is four.

08:17 Now, this region here, the slope is equal to three joules minus two joules, four meters minus three meters, and this is one jewel per meter.

08:33 So that means the force, and in case you're not aware of it, when you have a, i should mention it, when you have these lines, the slope of the tangent line at any point, when you have a linear curve, slope of tangent line, any point is the slope of that line also...

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