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$\begin{array}{llll}\text { (II) } & \text { Suppose } & \overrightarrow{\mathbf{A}}=1.0 \hat{\mathbf{i}}+1.0 \hat{\mathbf{j}}-2.0 \hat{\mathbf{k}} & \text { and } & \overrightarrow{\mathbf{B}}=\end{array}$$-1.0 \hat{\mathbf{i}}+1.0 \mathbf{j}+2.0 \hat{\mathbf{k}},(a)$ what is the angle between thesetwo vectors? (b) Explain the significance of the sign in part (a).

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A) $3.9 \times 10^{7} \mathrm{J}$B) $370 \mathrm{m}$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

University of Washington

University of Winnipeg

McMaster University

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

02:36

(1I) Suppose $\quad \vec {…

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$15-20$ Find the angle bet…

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Finding the Angle Between …

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Draw cach of the following…

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Find the angle $\theta$ be…

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Two vectors u and v are gi…

01:14

Dot Products and Angles Be…

06:33

Find the angle between vec…

02:16

Find the direction angle f…

02:02

Compute the angle between …

01:11

04:25

so risky starts from rest and sliced honor and 28 degree in client. So starting our dislocation here. Skis studying. And it is this 28 degrees Donald implying and it says that is a friction Gore efficient, which is equals two zero point or nine. This is the coefficient of kinetic friction. The scare is starting from dislocation here and ending up a dislocation. And what we need to do is that we need to find out what is the skier speed at the base, which means at this location need to find out the speed. Now, first of all, in the situation, there is friction present. So we need to use the version of the conservation of energy question, including non conservative friction force, which will come to later on. But first of all, that's dry force diagram for the situation where disposition, we have the normal force, we have mg. And if you resolve mg, you get MG gone stater and we have mg scientist down the plane and force a fiction is acting up the incline. Just force a fiction. So if you're going to find out the force of friction, you know, we know it is equals to mu k f off. And And if we find the net, force is in the wind direction which this is the Y direction here is zero. So we get and Jean course later is equal strength offense. So therefore, our force of friction his equals to mu k n g cost tater Miss, we're going to use later on in the creation. So now let's use the conservation of energy at the two locations at location one and a location do at the top of the bottom. So at this location here at the top, V one is equals to zero and we have the white one, which is the height. The why one is actually equals two l scientist. Oh, because this length off the incline is ill. It says here in the problem that steak here which is 85 meters down the incline So therefore, by applying technimentals, we can get why one over ill is sign off 28 survive on his cell scientist. Okay. And then we have bye to his equals to zero or dislocation. Zero chippy. So we like the conservation of energy at these two locations. So we have have m leven squared. Bless mg. Why one The energy at location one should be equals to the energy at the location to which is V two squad here. Bless mg. Why do now if there was no friction than energy and want to be equals to the energy A tube. But since the restriction, the energy or dislocation to is going to be less as compared to the energy at one. So therefore we can say that the energy at one is equal to the energy to bless the work done by the friction force. So we can just plug in the values here now. So we'll get here half envy one squared, which is zero less mg l scientist. His equals two half Mm. Vito squared less right to a zero. This cancels less work done by friction is f times d Times co scientist. So which is equals to the force of friction times l so solving that we'll get on the next page. All right, the question once again. So we have m g l sine theta is equals to have mm be two squared plus mu k mgl cost ater. So if you equate this, then you'll be getting me too Is equals to do G l scientific data minus mu k cost data. I just love in the values of this equation. Now square root off two times 9.8 times 85 and we have signed 28 minus 0.90 caused 28. If you sold that, you will get 25.49 meters per second, approximately 25 meters per second. Okay for the part B, It says that in the snow is level at the foot of the incline as some coefficient of friction. How far will the ski travel along the level? So for this problem, what is happening here is that he is starting a dislocation and then coming all the way down here and then going somewhere here and coming to rest at this location here. So we have a dislocation. V two is 25.49 and this location we have invited his equals to zero. And then finally, when the object comes to rest, our dislocation we have V trees equals zero and white three is equals to zero. Now, again, we apply the conservation of energy with the fictional approach at this location and a dislocation. So we'll get half and Vito squared plus mg Why do is equals to have and b p squared less mg by three Place the force or friction times some length shovel in the ground. Real treat. So this is you know, so we get half em. We describe plans. Zero is equals to the force of friction along the level. Ground simply calls to mu K and G times l three because the normal force equals two mg. And the force of friction equals Myu care for fen. All right, so we can equate these two equations so they have half. And Vito squared is equal to mu k mg some l street level on the ground. So l three is equals to be to squire divided by two G Um, you okay? Richest 25.49 squared, divided by two times 9.8 times point or nine. If you saw that, you will get 3 70 meters. So which means that as the skier comes down, he travels a distance off 3 70 meters on the level ground before it comes to rest

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