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Mathematics LibreTexts

6.5E: Exercises on Work

  • Page ID
    18207
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    Basic Work Problems

    For exercises 1 - 6, find the work done.

    1) Find the work done when a constant force \( F=12\) lb moves a chair from \( x=0.9\) to \( x=1.1\) ft.

    2) How much work is done when a person lifts a \( 50\) lb box of comics onto a truck that is \( 3\) ft off the ground?

    Answer:
    \(W = 150\) ft-lb

    3) What is the work done lifting a \( 20\) kg child from the floor to a height of \( 2\) m? (Note that \( 1\) kg equates to \( 9.8\) N)

    4) Find the work done when you push a box along the floor \( 2\) m, when you apply a constant force of \( F=100\) N.

    Answer:
    \(W = 200\) J

    5) Compute the work done for a force \( F=\dfrac{12}{x^2}\) N from \( x=1\) to \( x=2\) m.

    6) What is the work done moving a particle from \( x=0\) to \( x=1\) m if the force acting on it is \( F=3x^2\) N?

    Answer:
    \(W = 1\) J

    Spring Work Problems

    7) A \( 12\)-in. spring is stretched to \( 15\) in. by a force of \( 75\) lb. What is the spring constant?

    8) A spring has a natural length of \( 10\) cm. It takes \( 2\) J to stretch the spring to \( 15\) cm. How much work would it take to stretch the spring from \( 15\) cm to \( 20\) cm?

    Answer:
    \(W = 6\) J

    9) A \( 1\)-m spring requires \( 10\) J to stretch the spring to \( 1.1\) m. How much work would it take to stretch the spring from \( 1\) m to \( 1.2\) m?

    10) A spring requires \( 5\) J to stretch the spring from \( 8\) cm to \( 12\) cm, and an additional \( 4\) J to stretch the spring from \( 12\) cm to \( 14\) cm. What is the natural length of the spring?

    Answer:
    The natural length is\( 5\) cm.

    11) A shock absorber is compressed 1 in. by a weight of 1 ton. What is the spring constant?

    12) A force of \( F=\left(20x−x^3\right)\) N stretches a nonlinear spring by \( x\) meters. What work is required to stretch the spring from \( x=0\) to \( x=2\) m?

    Answer:
    \(W = 36\) J

    Cable and Chain Work Problems

    13) Find the work done by winding up a hanging cable of length \( 100\) ft and weight-density \( 5\) lb/ft.

    14) For the cable in the preceding exercise, how much work is done to lift the cable \( 50\) ft?

    Answer:
    \(W = 18,750\) ft-lb

    15) For the cable in the preceding exercise, how much additional work is done by hanging a \( 200\) lb weight at the end of the cable?

    Pyramid & Satellite/Rocket Work Problems

    16) [T] A pyramid of height \( 500\) ft has a square base \( 800\) ft by \( 800\) ft. Find the area \( A\) at height \( h\). If the rock used to build the pyramid weighs approximately \( w=100\,\text{lb/ft}^3\), how much work did it take to lift all the rock?

    Answer:
    \(W= \frac{32}{3}×10^9\) ft-lb

    17) [T] For the pyramid in the preceding exercise, assume there were \( 1000\) workers each working \( 10\) hours a day, \( 5\) days a week, \( 50\) weeks a year. If each of the workers, on average, lifted ten 100-lb rocks \( 2\)ft/hr, how long did it take to build the pyramid?

    18) [T] The force of gravity on a mass \( m\) is \( F=−((GMm)/x^2)\) newtons. For a rocket of mass \( m=1000\) kg, compute the work to lift the rocket from \( x=6400\) to \( x=6500\) km. (Note: \( G=6×10^{−17}\,\text{N m}^2/\text{kg}^2\) and \( M=6×10^{24}\) kg.)

    Answer:
    \(W = 8.65×10^5\) J

    19) [T] For the rocket in the preceding exercise, find the work to lift the rocket from \( x=6400\) to \( x=∞\).

    Pumping Work Problems

    20) [T] Find the work required to pump all the water out of a cylinder that has a circular base of radius \( 5\)ft and height \( 200\) ft. Use the fact that the density of water is \( 62\) lb/ft3.

    21) [T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

    Answer:
    \(W = 23.25π\) million ft-lb

    22) [T] How much work is required to pump out a swimming pool if the area of the base is \( 800 \, \text{ft}^2\), the water is \( 4\) ft deep, and the top is \( 1\) ft above the water level? Assume that the density of water is \( 62\)lb/ft3.

    23) A cylinder of depth \( H\) and cross-sectional area \( A\) stands full of water at density \( ρ\). Compute the work to pump all the water to the top.

    Answer:
    \(W = \dfrac{AρH^2}{2}\)

    24) For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

    25) A cone-shaped tank has a cross-sectional area that increases with its depth: \( A=\dfrac{πr^2h^2}{H^3}\). Show that the work to empty it is half the work for a cylinder with the same height and base.

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.