physics 6b hw1 - I. CHAPTER 12 QUESTION 1. Neither case is...

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Unformatted text preview: I. CHAPTER 12 QUESTION 1. Neither case is an example of simple harmonic motion. In the case of the bouncing ball it is easiest to see from the fact that the ball never goes back to its initial position. If the ball starts from rest at a certain height, it will fall, hit the ground and bounce back, but it will never go as high as it started from (i.e. thats why the ball eventually stops bouncing). In the case of the student it is a little more subtle. The motion of the student is not smooth, and his acceleration is not proportional to the displacement, and his velocity is most likely a step function (i.e. he starts from rest, starts moving and keeps moving at the same thing until he stop, his velocity does not follow a sine curve). II. CHAPTER 12 QUESTION 2. The position at time t will be the same as the displacement of the particle, only if the particle was at the equilibrium position at t = 0. In any other case, the displacement can easily be found in the following way: d = x ( t )- x ( t = 0) (which confirms what we said before, if x ( t = 0) = 0 then d=x(t)). III. CHAPTER 12 QUESTION 3. Equation 12 . 6 states the following: x ( t ) = A cos( t + ) All we have to do is rewrite the equation presented in the problem (i.e. x ( t ) =- A cos( t )) in the form of Equation 12 . 6. From trigonometry you should probably remember that cos( a ) =- cos a . Then it is easy to see that the position of the particle should be given by: x ( t ) = A cos( t ) By setting t = 0 we get: x ( t = 0) = A cos( ) =- A 2 FIG. 1: plot of position and velocity. IV. CHAPTER 12 QUESTION 5. Position and velocity are in the same direction half of the time. You can think of this 2 ways. Either you can remember the sign of the sine and the cosine curve in each quadrant: both positive in the first quadrant, sine is positive and cosine is negative in the second, both negative in the third, and finally cosine is positive and sine is negative in the fourth. Then you just have to realize that as long as they have opposite signs (i.e. because the velocity is really the derivative of the position, you get that extra negative sign in front of the sine curve) they are in the same direction. The other way to do this is just drawing arbitrary curves. In Figure 1 I took the position to be a cosine curve with both the amplitude and the frequency equal to one. Similarly for velocity and acceleration we get that that they are the same half the time also (i.e. see Figure 2). Finally by the origin of simple harmonic motion (i.e. Hooks law: a =- k m x ) it should be easy to see that position and acceleration are never in the same direction. 3 FIG. 2: plot of velocity and acceleration....
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physics 6b hw1 - I. CHAPTER 12 QUESTION 1. Neither case is...

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