Lab 7: Modeling with definite integrals

Instructions: Teach the other two members of your new group how to answer Parts a-c of the question. Then work together to answer Part d for all three of your contexts. Submit your individual write-up for all parts of the three questions.

The kinetic energy of an object with mass \(m\) and constant speed \(v\) is \(K = \frac{1}{2} mv^2\), at least in the case where the entire object is moving at the same speed. Suppose a 0.1 m long rod has a mass of 0.03 kg with uniform density. It is rotating around one of its ends at a rate of one revolution per minute, much like the second hand of a clock.
1. Write a definite integral that gives the kinetic energy of the rod in Joules (kg\(\cdot\)m\(^2\)/s\(^2\)).
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. If the rod is only half as long but moves twice as fast (and still has mass 30 g), does the kinetic energy increase or decrease?
The density of oil in a circular oil slick on the surface of the ocean at a distance \(r\) meters from the center of the slick is given by \(\delta(r) = \frac{50}{1 + r}\) kg/m\(^2\).
1. If the slick extends from \(r = 0\) to \(r = 10,000\) m, write a definite integral that gives the total mass of oil in the slick.
  
  Hint: Check that \(\frac{d}{dr}\left(r - \ln(1 + r)\right) = \frac{r}{1 + r}\).
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. Is more than half, or less than half, of the oil contained within a 5,000 m radius?
The force of gravity that the earth exerts on an object diminishes as the object gets further away from the earth. The energy required to lift an object 1 foot at sea level is greater than the energy required to lift the same object 1 foot at the top of Mt. Everest. However, the difference in altitudes is so small in comparison to the radius of the earth that the difference in energy is negligible. On the other hand, when an object is rocketed into space, the fact that the force of gravity diminishes with distance from the center of the earth is critical. According to Newton, the force of gravity on a given mass is proportional to the reciprocal of the square of the distance of that mass from the center of the earth. That is, there is a constant \(k\) such that the gravitational force at distance \(r\) from the center of the earth is given by \[ F(r) = \frac{k}{r^2}.\] The energy required to move an object a distance \(d\) is \(E = Fd\), if the force is constant over the distance \(d\).
1. Write a definite integral that gives the energy in Joules (1 J = 1 Nm) required to lift a 1-kg payload from the surface of the earth to the moon, which is about 362,570 km away at its closest point.
  
  (Hint: The earth's surface is at a distance of 6,371 km from its center. At this value of \(r\), the force of gravity on the 1 kg object is 9.8 N. Use this to determine the constant \(k\).)
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. How far can a payload be lifted using half the energy required to take it to the moon?
The energy required to move an object a distance \(x\) while exerting a constant force \(F\) is \(E = Fx\). Suppose that you have two magnets and a wire. One magnet is attached to the end of the wire and the other can slide along the wire. If the magnets are arranged so that they repel each other, then it will require force to push the movable magnet toward the fixed magnet. The amount of force needed to move the magnet increases as the two get closer together. In fact, the force at a distance \(d\) is proportional to \(1/d^2\).
1. Using a constant of proportionality \(k\) between the force and distance, write a definite integral that gives the energy required to move the magnet from 5 cm away to 3 cm away in Joules (kg\(\cdot\)m\(^2\)/s\(^2\)), and evaluate the integral (your answer will depend on \(k\)).
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. Which will require more energy, to move the magnet from 5 cm away to 3 cm away, or from 3 cm away to 2 cm away?
An exponential model for the density of the earth's atmosphere says that if the temperature of the atmosphere were constant, then the density of the air as a function of height, \(h\) (in meters), above the surface of the earth would be given by \[ \delta(h) = 1.28 e^{-0.00124h} \text{ kg/m}^3.\]
1. Write a definite integral that gives the mass of the portion of the atmosphere from \(h=0\) to \(h=10,000\) m (i.e., the first 10 kilometers above sea level). Assume the radius of the earth is 6400 km. Use a computer algebra system to find an antiderivative.
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. Use this model to compute the total mass of the earth's atmosphere.
The gravitational attraction between two particles of mass \(m_1\) and \(m_2\) at a distance \(r\) apart is \[ F(r) = \frac{Gm_1 m_2}{r^2}.\]
1. Write a definite integral that that gives the gravitational attraction between a thin uniform rod of mass \(M\) and length \(l\) and a particle of mass \(m\) lying in the same line as the rod at a distance \(a\) from one end.
2. Evaluate the integral and explain the meaning of your result.
3. Describe the meaning of each factor of your integral and give the units it is measured in.
4. Two long, thin, uniform rods of length \(l_1\) and \(l_2\) lie on a straight line with a gap between them of length \(A\). Suppose their masses are \(M_1\) and \(M_2\), respectively. What is the force of attraction between the rods? (Use the result of Part b.)