By Kleppner D., Kolenkow R.
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Extra resources for An Introduction to Mechanics
Since its components define a vector, we can specify a vector entirely by its components. Thus A = (A x , Ay ) or, more generally, in three dimensions, A = (A x , Ay , Az ). Prove for yourself that A = A x 2 + Ay 2 + Az 2 . If two vectors are equal A = B, then in the same coordinate system their corresponding components are equal. A x = Bx Ay = By The single vector equation A = B symbolically represents three scalar equations. The vector A has a meaning independent of any coordinate system. However, the components of A depend on the coordinate system being used.
For example, if we are dealing with the familiar Cartesian coordinate system of three dimensions, the base vectors lie along the x, y, and z axes. We shall designate the x unit vector ˆ (Sometimes the by ˆi, the y unit vector by ˆj, and the z unit vector by k. ) The base vectors have the following properties, as you can readily verify: z k jˆ y iˆ ˆi · ˆi = ˆj · ˆj = kˆ · kˆ = 1 ˆi · ˆj = ˆj · kˆ = kˆ · ˆi = 0 x ˆi × ˆj = kˆ ˆj × kˆ = ˆi kˆ × ˆi = ˆj. As shown in the drawing, we can write any three-dimensional vector in terms of its components and the base vectors: z A = A xˆi + Ay ˆj + Az kˆ Az kˆ To find the component of a vector in any direction, take the dot product with a unit vector in that direction.
T, the time required to execute one complete cycle, is called the period. y v dr dt = rω(− sin ωt ˆi + cos ωt ˆj). v= r ωt x We can show that v is tangent to the trajectory by calculating v · r: v · r = r2 ω(− sin ωt cos ωt + cos ωt sin ωt) = 0. Because v is perpendicular to r, the motion is tangent to the circle, as we expect. It is easy to show that the speed |v| = r ω is constant. y dv dt = r ω2 (− cos ωt ˆi − sin ωt ˆj) a= r a ωt = −ω2 r. x The acceleration is directed radially inward and is known as the centripetal acceleration.
An Introduction to Mechanics by Kleppner D., Kolenkow R.