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1.5E: Exercises

  • Page ID
    128817
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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) [Technology Required] 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
    \(W= \frac{32}{3} \times 10^9\) ft-lb

    17) [Technology Required] 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) [Technology Required] 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: \( G=6 \times 10^{−17}\,\text{N m}^2/\text{kg}^2\) and \( M=6 \times 10^{24}\) kg.)

    Answer
    \(W = 8.65 \times 10^5\) J

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

    Pumping Work Problems

    20) [Technology Required] 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 weight density of water is \( 62\) lb/ft3.

    21) [Technology Required] 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
    \(W = 23.25 \pi \) million ft-lb

    22) [Technology Required] 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 weight density of water is \( 62\) lb/ft3.

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

    Answer
    \(W = \dfrac{A \rho H^2}{2}\)

    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: \( A=\dfrac{ \pi r^2h^2}{H^3}\). Show that the work to empty it is half the work for a cylinder with the same height and base.

    Answer
    Answers may vary.

    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.


    This page titled 1.5E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Roy Simpson.

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