\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 3.5 Calculus 2

Work
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3.5 Work

3.5.1 Work by a Constant Force

  • In physics, the work \(W\) done by a force of constant magnitude \(F\) over a displacement \(d\) by the formula \(W=Fd\).
  • Example Calculate the work done by a crane in lifting a \(3000\) pound wrecking ball \(25\) feet.
  • Example Estimate the work done in lifting a leaky bucket of water \(1\) meter off the ground if it weighs approximately \(4\) newtons on the ground, \(3.8\) newtons at \(25\) cm, \(3.5\) newtons at \(50\) cm, and \(2.3\) newtons at \(75\) cm.

3.5.2 Work by a Variable Force

  • If the force \(F(x)\) acting on an object varies with respect to the position \(x\) of the object, then work done in moving the object from \(a\) to \(b\) is defined by \(W=\int_a^b F(x)\,dx\).
  • Example Find the work done in lifting a leaky bucket of water \(1\) meter off the ground if it weighs \(4-3x^2\) newtons when it is \(x\) meters above the ground.
  • Example How much work is done in pulling up \(20\) feet of hanging chain if it weighs \(1\) pound per \(4\) feet?
  • Example Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, \(F(x)=kx\) where \(x\) is the difference between the spring’s natural length and its current length. If a spring has natural length \(10\) inches, and it requires \(15\) pounds of force to hold the spring at \(13\) inches, how much work is required to stretch the spring an additional \(2\) inches?

3.5.3 Work and Pumping Liquid

  • To compute the work in pumping liquid, we proceed by computing a work differential \(dW\) for each infintesimal cross-section of liquid at height \(y\), and then evaluating \(W=\int_{y=a}^{y=b} dW\) where \(y=a\) is the lowest point of liquid and \(y=b\) is the highest.
    • \(dV = (\text{area})dy\)
    • \(dF = (\text{density})dV\)
    • \(dW = (\text{distance})dF\)
  • Example Assume salt water weighs \(10,000\) newtons per cubic meter. How much work is required to pump out a conical tank pointed downward of height \(6\) meters and radius \(3\) meters, if it is initially filled with \(4\) feet of salt water?

Review Exercises

  1. Estimate the work done in pushing a plow \(6\) meters through increasingly packed dirt; this movement requires \(1\) newton of force at the beginning, \(5\) newtons of force after \(2\) meters, and \(9\) netwons of force after \(4\) meters.
  2. Compute the exact amount of work done in pushing a plow \(6\) meters through increasingly packed dirt; this movement requires \(F(x)=1+2x\) newtons of force after \(x\) meters.
  3. Find the work done in lifting a leaky bucket from the ground to a height of four feet, assuming it weighs \(25-x\) pounds at \(x\) feet above the ground.
  4. A cable weighing \(4\) pounds per foot holds a \(500\) pound bucket of coal at the bottom of a \(300\) foot mine shaft. Show that the total work done in lifting the cable and bucket is \(330,000\) foot-pounds.
  5. Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, \(F(x)=kx\) where \(x\) is the difference between the spring’s natural length and its current length. Show that if a spring has natural length \(20\) cm, and it requires \(25\) newtons of force to hold the spring at \(15\) cm, then the work required to stretch the spring from its natural length to \(26\) cm is \(90\) N-cm.
  6. A uniformly weighted \(100\) foot rope weighs \(50\) pounds. Suppose it is fully extended into a well, tied to a leaky bucket of water. This bucket weighs \(10\) pounds and initially holds \(30\) pounds of water, but loses \(1\) pound of water every \(2\) feet. Show that the work done in lifting the rope and bucket is \(4400\) ft-lbs. (Hint: When does the bucket run out of water?)
  7. Assume salt water weighs \(10\) kilonewtons (kN) per cubic meter. A cylindrical tank with a radius of \(3\) meters and a height of \(10\) meters holds \(8\) meters of salt water. Show that the work required to pump out the salt water to the top of the tank is \(4320\pi\) kN-m (kJ).
  8. Assume salt water weighs \(10000\) newtons per cubic meter. A pyramid-shaped tank of height \(4\) meters is pointed upward, with a square base of side length \(4\) meters, and is completely filled with salt water. Show that the work done in completely pumping all the water in the tank up to the point of the pyramid is \(10000\int_0^4(4-y)^3\,dy\) J.
  9. Assume that a cubic inch of Juicy Juice™ weighs \(D\) oz. Suppose a perfectly spherical coconut-shaped cup with radius \(R\) inches is completely filled with Juicy Juice™. Show that drinking the entire beverage using a straw which extends \(S\) inches above the top of the container requires \(\frac{4}{3}D\pi R^3(R+S)\) inch-ounces of work.
  10. What is the work required to push a heavy box \(3\) meters over an irregular surface, assuming it requires \(F(x)=3+2x-x^2\) newtons of force to move at \(x\) meters?
  11. What integral gives the work in ft-lbs required to pull up a hanging \(30\)-pound \(15\)-foot chain?
  12. What integral gives the work in kN-m required to pump out all salt-water to the top of a cubical tank with side length \(4\) meters, if it is initially half-full? Assume the density of salt water is \(10\) kilonewtons per cubic meter.

Solutions 1-6

Solutions 7-12

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 6.5