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A small sphere is hung by a string from the ceiling of a van. When the van is stationary, the sphere hangs vertically. However, when the van accelerates, the sphere swings backward so that the string makes an angle of $\theta$ with respect to the vertical. (a) Derive an expression for the magnitude $a$ of the acceleration of the van in terms of the angle $\theta$ and the magnitude $g$ of the acceleration due to gravity. (b) Find the acceleration of the van when $\theta=10.0^{\circ} . \quad(\mathrm{c})$ What is the angle $\theta$ when the van moves with a constant velocity?

$g \tan \theta$

1.73 $\mathrm{m} / \mathrm{s}^{2}$

$0^{\circ}$

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Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

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the expression. The problem is asking this in the first item. It's precisely the explanation that results from applying Newton's second law to the situation. So in order to do that, let us choose our reference frame. I will be using these one a vertical axis that I will call, Why access and a horizontal axis that I call the X Axis. This is my reference frame. Now let me apply Newton's second law in this reference frame. So by doing that on the X axis and I'm applying Newton's second law to the ball that is hanging, we get that the net force on the X direction that well is it goes to the mass off the ball times its acceleration in the X direction first noticed that the acceleration off the ball in the X direction is equals to the acceleration off the van in that direction. So let me call it a then the net force. Next direction is that close to the mask off the ball times A. But the net force in X direction is composed by only one force, which is a component off the tension force. We can they compose the tension force as follows. We can do a horizontal component, which is the axe component on the vertical component, which is the y component. Then we can write this equation as attention. Component X is equal to the mass off the ball times the event acceleration. Now we can relate the X component off the tension with the magnitude off the tension and the angle teeter. To do that, we have to use triangle that is formed by the tension and its components. And the triangle is the following. We have the tension pointing in this direction. We have the Y component and they have the acts component. The form directing will try and go where they have part of news is attention. These side is the Y component and this side is the X component notice by the geometry of the problem that the angle between attention and the direction off the white component s Tita Then this angle off the triangle s Tita, then the relation between the tension, the X component and Tita is given by the sign off teeth. Because T X is the opposite side of triangle to the Uncle Tito. Then the sign off detail is given by the opposite side. T X, divided by they have part the news T then t X Is it close to tee times their sign off? Tita. So T times the sign off Peter is equal to the mass off the ball times acceleration off the van. Still, there is no gravity appearing these expression. How can you make gravity appearances? Expression. We have to use Newton's second law in the other axis to calculate a relation between the magnitude of the tension and the magnitude off the weight force. Before doing that, let me raise the board. Okay? It goes as follows by doing the same in that exists for the y axis we have. That's the net force the Y direction. Is it close to the mask off the ball times its acceleration in the Y direction? Note that the ball is standing in this position, so it's not moving on the wider action and it's not going to move. So its acceleration in that direction is close to zero. And the net force that acts on the ball index direction zero. But this composes net force has been posed by two forces. The y component off the tension on the weight force then the Y component off The tension minus the weak force is close to zero. So the Y component off the tension is the course to the weight force. Now we have to go back to the triangle formed by the tension and its components that remind what is the relation between the y component after tension on the magnitude of the tension? Remember, the triangle was as follows these Waas the Y component is the X disease magnitude and these worst eater and this is a 90 degree angle Then, as we weren't a relation with T, why we have to use the co sign off Peter So the co sign off Tita is given by they are just inside to you. Why? Divided by the have 40 news t then t white, is it Costa Tee times the co sign off Pekka So t times The co sign off Tita is it goes to the weight force which is the mass off the boat times the acceleration of gravity. Then the magnitude of the tension force is given by the mass off the boat times the acceleration of gravity divided by the co sign off Tita. Then we can plaguing this equation into these tension to get the following I am times g times the sign off detail. This is divided by the co sign off detail. Is it close to the mask off the ball times the acceleration off the van. We have the mass off the ball on both sides off this equation so we can simply fight. And now so notice that we have a signed divided by a co sign and this is a tangent. Then we end up with the acceleration of gravity times the tangent off Dita being equals toe the acceleration off the van and this is the final relation. The first item is asking us for the next item. We have to clear the board here. So let me do that Now. For the next item, we have to suppose that Tita is Acosta 10 degrees and calculate the value off acceleration. So now Tita is the course of 10 degrees. What is the acceleration? We can use these expression The acceleration Is it close to G? Remember that he is approximately 9.8 meters per second squared times Detain Geant off 10 degrees. These results in an acceleration off approximately 1.723 meters per second squared. Then we also have to answer the last item which asks, What is the angle, Tita When then then moves. If Acosta velocity So as a van is moving with a constant velocity, its acceleration is it costs zero then g times that tangent off Tita is in course, to zero. We can send G to the other side. So the tangent off Peter is it cost zero. And then when the tension is close to zero, we can conclude that it's because the sign is it goes to zero and the angles that have the sign equals zero R zero degrees on 180 degrees. But notice that in this situation it's impossible to have these 180 show. Our answer is that data is it goes to zero degrees

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