1. (a.) A particle moves so that its position vector is given by \( r=\cos 4 \beta t i+\sin 4 \beta t j \), where \( \beta \) is a constant. Show that: (i.) the velocity \( v \) of the particle is orthogonal to \( r \). (ii.) the accelaration \( a \) is directed toward the origin and has magnitude proportional to the distance from the origin. (iii.) \( r \times v \) is a constant vector.
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The velocity \( v \) is the derivative of the position vector \( r \) with respect to time \( t \). \[ v = \frac{d}{dt}(\cos 4\beta t \, i + \sin 4\beta t \, j) \] \[ v = -4\beta \sin 4\beta t \, i + 4\beta \cos 4\beta t \, j \] Show more…
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