Calculus 3 Labs

Lab 6: Applications of multivariable integrals - part 1


  1. 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.

    1. What is the mass of a small portion of the planet with volume \(\Delta V\)? Explain.

    2. If you stand at the New South Pole, explain why the horizontal components of gravitational force all cancel out.

    3. What is the vertical component of gravitational force between a 1-kg mass at the New South Pole \((0,0,0)\) and a small portion of the planet with volume \(\Delta V\) located at spherical coordinates \((\rho,\phi,\theta)\) ?

    4. Write a Riemann sum using spherical coordinates and \(\Delta V\) approximating the new gravitational force on a 1-kg mass at the New South Pole. Label all expressions in your sum that can be interpreted as a meaningful quantity in this situation, including units.

    5. Convert your sum into an integral. Evaluate the integral, then use the following constants to find a numeric value for the force: \(M=5.972\times10^{24}\) kg, \(R=6.371\times10^{6}\) m, and \(G=6.674\times10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2}\).

      Note that the force on a 1-kg mass at the surface of the Earth used to be 9.8 N, so you should be able to determine if your answer is in the right ballpark.

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  2. Your Chemistry lab has gone horribly wrong, and you have created an unstable vortex of high-energy 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 and increases 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 \(KE=1/2mv^2\).

    1. What is the mass of a portion of the plasma with volume \(\Delta V\)? Explain.

    2. What speed is a small portion of plasma moving if it is located at cylindrical coordinates \((r,\theta,z)\)? Assume \((0,0,0)\) is at the bottom center of the beaker. Explain.

    3. What is the kinetic energy of the portion of plasma described in part b?

    4. Write a Riemann sum using cylindrical coordinates and \(\Delta V\) approximating the Kinetic energy of the plasma in the beaker. Label all expressions in your sum that can be interpreted as a meaningful quantity in this situation, including units.

    5. Convert your sum into an integral. Evaluate the integral to obtain a numerical value for the kinetic energy.