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I do not believe that the omission of the horizontal component of T in that figure was an accidental oversight.
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if he gives the angle and the force of tension then spliting it up between the hor and ver components would be redundant. perhaps you misinterpriated the illustration, the vertical line is proibibly there to show that the angle is off from the vertical.
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For that second illustration Serway states that for the noninertial passenger in the car the sphere is at rest and there is no acceleration. That is why, writes Serway, the observer invokes fictitious force -ma to counter T's horizontal component, and thinks there is zero net force on the ball (italics his). Serway states that the interpretation of the situation is different in each of the two reference frames.
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from the vantage point of the passenger, the ball is remaining still, that is why he states that there is a net zero force on the ball. and that is why there is needed an extra force, to account for the angle on the string.
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I suspect that Serway has not had daily experience commuting to work by rail. If he did, he certainly would know that anyone standing vertically with feet together and not holding on to anything will fall down in an accelerating railroad car. I do not believe the depicted position of that observer to be an accidental oversight or mistake.
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you are right that this is not an accident, the point of the illustration is to depect the actions on the ball in different perspectives. How the person is standing is unimportant. Also, previously he stated that the passenger was non-inertial, which is just another way of saying that this object has noting to do with the example it is just here for reference.
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As with the train, the force causing acceleration of a reference frame is simultaneously applied to all objects in the reference frame. Because the source of that force is external to the bodies in the reference frame, that force is a part of the environment of those bodies. Therefore, observed forces on bodies in an accelerating reference frame do result from interaction between such bodies and their environment and, by the definition of a real force given above, those observed forces are real. The force that accelerates a reference frame is often a forgotten force.
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I do not think that you have the concept of reference frame right. whatever is outside the reference frame is ignored because is is not
in the reference frame, this includes the force accelerating the reference frame. the reference frame is assumed to be the environment.
the force accelerating the train is not forgotten it is ignored because it is outside the reference frame, and therefore not in the environment
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If the accelerometer consisted, instead, of a vertically immobile, horizontal, helical compression spring, fastened at one end to the rear wall of the boxcar, and with a weight on the free end, both observers would see that the spring was compressed from its rest position.
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replacing the string with a spring causes no change in the example, because they are both ways of showing the forces present on the ball.
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Regarding the observer in the accelerating boxcar Serway wrote, in a disparaging sense, that the observer thought there was zero net force on the ball. According to Newton?s third law, for every force there is always an equal and opposite force. That law followed to its logical conclusion means that no matter how many forces act on a body, for each one there is always an equal and opposite force; and, therefore, the net force acting on any body is always zero. Surely Serway did not intend to slight Newton?s third law; however, that is what can inadvertently happen if one forgets the inertial force exerted by a body against an external force causing it to accelerate.
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that is an incorrect conclusion to Newton's third law, and is accually in direct conflict with Newton's second law. in Newton's second law he states that acceleration is caused be a
non-zero net force. if the net force acting on any body was always zero, then there would be no acceleration of any body.
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At last centrifugal force becomes required and real, has a formula, and is equal and in opposition to centripetal force.
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just because it has a formula, doesn't make it real. it just means that it can be measured.