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If \({\bf r}(t)=6t{\bf i}+3t^2{\bf j}+t^3{\bf k}\), what force acts on a particle of mass \(m\) moving along \({\bf r}\) at \(t=0\)?
Let a particle of mass 1 gram (g) follow the path in Exercise 1, with units in seconds and centimeters. What force acts on it at \(t=0\)? (Give the units in your answer.)
A body of mass 2 kilograms moves on a circle of radius 3 meters, making one revolution every 5 seconds. Find the centripetal force acting on the body.
Find the centripetal force acting on a body of mass 4 kilograms (kg), moving on a circle of radius 10 meters (m) with a frequency of 2 revolutions per second (rps).
A satellite is in a circular orbit 500 miles above the surface of the earth. What is the period of the orbit? (You may take the radius of the earth to be 4000 miles, or \({6.436\times10^6}\) meters.)
What is the acceleration of the satellite in Exercise 21? The centripetal force?
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Let \({\bf c}(t)\) be a path, \({\bf v}(t)\) its velocity, and \({\bf a}(t)\) the acceleration. Suppose \({\bf F}\) is a \(C^1\) mapping of \({\mathbb R}^3\) to \({\mathbb R}^3, m >0\), and \({\bf F}({\bf c}(t))=m{\bf a}(t)\) (Newton’s second law). Prove that \[ \frac{d}{{\it dt}}[m{\bf c}(t)\times {\bf v}(t)]={\bf c}(t)\times\, {\bf F}({\bf c}(t)) \] (i.e., “rate of change of angular momentum = torque”). What can you conclude if \({\bf F}({\bf c}(t))\) is parallel to \({\bf c}(t)\)? Is this the case in planetary motion?
Continue the investigations in the previous Exercise to prove Kepler’s law that a planet moving under the influence of gravity about the sun does so in a fixed plane.