Lab 7: Applications of multivariable integrals  part 2

Your physics lab has gone horribly wrong, and you accidentally detached the entire southern hemisphere of the planet which is now spinning off to some distant corner of the solar system! Feeling a little guilty, you embark on a trek to survey the damage and geography of your newly hemispherical planet. First you travel to the New South Pole, the location that used to be the center of the earth.
Use the following variables to represent physical quantities: the mass of the earth was previously a uniformly distributed \(M\) kg, and the radius was previously \(R\) meters. Newton's law of gravitation says that the force between two point masses \(m_1\) and \(m_2\) a distance \(r\) apart is \(F=Gm_1m_2/r^2\) where \(G\) is a universal gravitational constant.
Write a paragraph with full sentences explaining how to construct an integral to compute the gravitational force on a 1kg mass at the New South Pole. Your sentences can (and should) include mathematical expressions and equations, but they should still make sense as sentences. Use the letters \(M\), \(R\), and \(G\) in your narrative rather than their numerical values. Conclude with the computation of this integral. Then use the following constants to find a numeric value: \(M=5.972\times10^{24}\) kg, \(R=6.371\times10^{6}\) m, and \(G=6.674\times10^{11} \text{ m}^3\text{kg}^{1}\text{s}^{2}\).

Your Chemistry lab has gone horribly wrong, and you have created an unstable vortex of highenergy plasma in a beaker. The beaker is 0.2 m tall with a radius of .05 m and completely filled with 1 kg of plasma that is rotating at a frightening 100 revolutions per second at the top, increasing linearly to an even more frightening 200 revolutions per second at the bottom. As your TA flees the room, you calmly assess the danger by determining the kinetic energy of the plasma.
In the following, use the fact that an object of mass \(m\) traveling at speed \(v\) has kinetic energy \(K=1/2mv^2\).
Write a paragraph with full sentences explaining how to construct an integral to compute the kinetic energy of the plasma in the beaker. Your sentences can (and should) include mathematical expressions and equations, but they should still make sense as sentences. Conclude with the computation of this integral.

Fluid traveling at a velocity \(v\) across a surface area \(A\) produces a flow rate of \(F=vA\). (What are the units?) Poiseuilleâ€™s law says that in a pipe of radius \(R\), the viscosity of a fluid causes the velocity to decrease from a maximum at the center (\(r=0\)) to zero at the sides (\(r=R\)) according to the function \[v(r)=v_{max}\left(1\frac{r^2}{R^2}\right).\] Find the rate that water flows in a 4inch diameter pipe if \(v_{max} = 4.4 \)ft/s. Explain your work.