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CircularMotion

CircularMotion - 1 Massimiliano Galeazzi Circular Motion...

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Unformatted text preview: 1 Massimiliano Galeazzi: Circular Motion – Any Reproduction or distribution without the author’s consent is forbidden . Circular Motion By Prof. Massimiliano Galeazzi, University of Miami If you spin a bucket full of water in a vertical circle fast enough, the water will stay in the bucket. How fast do you need to spin it to make sure you don’t get showered? What is the force that makes your car turn when you turn the steering wheel? How big is that force? These are some of the questions we will answer in this chapter. The general topic is that of circular motion, that is, objects moving on a circular trajectory. Along the way we will learn about ferris wheels, cars turning without going off the road, etc. But first, we need to improve our math understanding by reviewing the product between vectors. --------------------------------------------------------------------------------------------------------------------------------------------------- MATH INSERT In chapter 1 we have introduced the concept of vectors and found how vectors can be added and or subtracted. We also mentioned the fact that vectors can be multiplied with each other in two different ways, what are called the scalar (or dot ) product and the vector (or cross ) product . In this chapter we will make use, for the first time, of the dot-product, so it is appropriate to take a look at it before we proceed. 1. The scalar (dot) product between vectors FIG. 1 Let’s consider two vectors, ⃗ = ̂ + ̂ + and ¡⃗ = ̂ + ̂ + . The scalar product between ⃗ and ¡⃗ , denoted ⃗ ∙ ¡⃗ (“A dot B”, hence the term “dot product”), is a scalar quantity equal to the product between the magnitudes of the two vectors, times the cosine of the angle between the two: ⃗ ∙ ¡⃗ = ¢ ⃗ ¢¢ ¡⃗ ¢ cos , [1] where the angle is measured by drawing the two vectors with the tails in the same point (see Fig. 1a-b). We notice that, from Eq. 1, the dot product, like the “regular” product between scalars is commutative, i.e., ⃗ ∙ ¡⃗ = ¡⃗ ∙ ⃗ [2] Also, we can write Eq. [1] as: ⃗ ∙ ¡⃗ = ¢ ⃗ ¢£¢ ¡⃗ ¢ cos ¤ , [3] That is, the scalar product between ⃗ and ¡⃗ is equal to the magnitude of ⃗ times the projection of ¡⃗ in the direction of ⃗ , i.e., ¢ ¡⃗ ¢ cos (see Fig. 1c). Similarly, we can write: 2 Massimiliano Galeazzi: Circular Motion – Any Reproduction or distribution without the author’s consent is forbidden ....
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CircularMotion - 1 Massimiliano Galeazzi Circular Motion...

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