HW2 - Physics2210.Homeworkassignment2. B6.4.5. r (t) = i...

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1 Physics 2210. Homework assignment 2. B 6.4.5. r (t) = i cost + j sint + k t . v (t) = ˙ r (t) = i sint + j cost + k . a (t) = ˙ ˙ r (t) = i cost j . Therefore, v (t) = , and | a (t)|=1, and both are constant. The particle spirals forward in k direction (i.e., circular motion in the i and j plane plus a uniform motion in k direction. B 6.4.6. It is straightforward to prove that force F and velocity v are perpendicular to each other. F = q( v × B ) , therefore, F must be perpendicular to velocity v and B. Using the hint from the book, d dt (v 2 ) = d dt ( v v ) = 2 d v dt v = 2 1 m (m d v dt ) v = 2 1 m F v = 0 , where we used the Newton’s second law and that the dot product F and v is zero because these two vectors are perpendicular to each other. This equation implies that v 2 does not vary with time, and hence velocity has a constant magnitude. This then suggests that the magnitude of the force F is a constant, as |F|=qvB. We can also argue that, based on that F is perpendicular to velocity v , the force does not do work to the particle and therefore, the kinetic energy 1/2mv 2 for the particle must be constant with time, and hence v does not vary with time. B 8.1.6. Let m be the mass and r be the radius for the ball at time t, according to the condition given, we have dm/dt=-4 π r 2 k, where k is the proportion constant. Let the ball density be ρ , we have m=4/3 π r 3 ρ , and dr/dt=-a, where a=k/ ρ . Solving this 1 st order differential equation leads to r=-at+C, where C is an integration constant and is 0.5 cm, using the initial condition r(t=0)=r o =0.5 cm.
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This note was uploaded on 11/10/2009 for the course PHYS 2210 at Colorado.

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HW2 - Physics2210.Homeworkassignment2. B6.4.5. r (t) = i...

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