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Mathematics LibreTexts

1.4E: Exercises

  • Page ID
    18542
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    Physical Applications

    For the following exercises, find the work done.

    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{2}\)

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

    Answer

    \(\displaystyle 150\) ft-lb

    Exercise \(\PageIndex{3}\)

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

    Answer

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    Exercise \(\PageIndex{4}\)

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

    Answer

    \(\displaystyle 200J\)

    Exercise \(\PageIndex{5}\)

    Compute the work done for a force \(\displaystyle F=12/x^2\)N from \(\displaystyle x=1\) to \(\displaystyle x=2\) m.

    Answer

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    Exercise \(\PageIndex{6}\)

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

    Answer

    \(\displaystyle 1\) J

    For the following exercises, find the mass of the one-dimensional object.

    Exercise \(\PageIndex{7}\)

    A wire that is \(\displaystyle 2\)ft long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=x^2+2x\) lb/ft

    Answer

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    Exercise \(\PageIndex{8}\)

    A car antenna that is \(\displaystyle 3\) ft long (starting at \(\displaystyle x=0)\) and has a density function of \(\displaystyle ρ(x)=3x+2\) lb/ft

    Answer

    \(\displaystyle \frac{39}{2}\)

    Exercise \(\PageIndex{9}\)

    A metal rod that is \(\displaystyle 8\)in. long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=e^{1/2x}\) lb/in.

    Answer

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    Exercise \(\PageIndex{10}\)

    A pencil that is \(\displaystyle 4\)in. long (starting at \(\displaystyle x=2\)) and has a density function of \(\displaystyle ρ(x)=5/x\) oz/in.

    Answer

    \(\displaystyle ln(243)\)

    Exercise \(\PageIndex{11}\)

    A ruler that is \(\displaystyle 12\)in. long (starting at \(\displaystyle x=5\)) and has a density function of \(\displaystyle ρ(x)=ln(x)+(1/2)x^2\) oz/in.

    Answer

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    For exercises 12 - 16, find the mass of the two-dimensional object that is centered at the origin.

    Exercise \(\PageIndex{12}\)

    An oversized hockey puck of radius \(\displaystyle 2\)in. with density function \(\displaystyle ρ(x)=x^3−2x+5\)

    Answer

    \(\displaystyle \frac{332π}{15}\)

    Exercise \(\PageIndex{13}\)

    A frisbee of radius \(\displaystyle 6\)in. with density function \(\displaystyle ρ(x)=e^{−x}\)

    Answer

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    Exercise \(\PageIndex{14}\)

    A plate of radius \(\displaystyle 10\)in. with density function \(\displaystyle ρ(x)=1+cos(πx)\)

    Answer

    \(\displaystyle 100π\)

    Exercise \(\PageIndex{15}\)

    A jar lid of radius \(\displaystyle 3\)in. with density function \(\displaystyle ρ(x)=ln(x+1)\)

    Answer

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    Exercise \(\PageIndex{16}\)

    A disk of radius \(\displaystyle 5\)cm with density function \(\displaystyle ρ(x)=\sqrt{3x}\)

    Answer

    \(\displaystyle 20π\sqrt{15}\)

    Exercise \(\PageIndex{17}\)

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

    Answer

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    Exercise \(\PageIndex{18}\)

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

    Answer

    \(\displaystyle 6\)J

    Exercise \(\PageIndex{19}\)

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

    Answer

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    Exercise \(\PageIndex{20}\)

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

    Answer

    \(\displaystyle 5\) cm

    Exercise \(\PageIndex{21}\)

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

    Answer

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    Exercise \(\PageIndex{22}\)

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

    Answer

    \(\displaystyle 36\) J

    Exercise \(\PageIndex{23}\)

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

    Answer

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    Exercise \(\PageIndex{24}\)

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

    Answer

    \(\displaystyle 18,750\) ft-lb

    Exercise \(\PageIndex{25}\)

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

    Answer

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    Exercise \(\PageIndex{26}\)

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

    Answer

    \(\displaystyle \frac{32}{3}×10^9ft-lb\)

    Exercise \(\PageIndex{27}\)

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

    Answer

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    Exercise \(\PageIndex{28}\)

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

    Answer

    \(\displaystyle 8.65×10^5J\)

    Exercise \(\PageIndex{29}\)

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

    Answer

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    Exercise \(\PageIndex{30}\)

    A rectangular dam is \(\displaystyle 40\) ft high and \(\displaystyle 60\) ft wide. Compute the total force \(\displaystyle F\) on the dam when

    a. the surface of the water is at the top of the dam and

    b. the surface of the water is halfway down the dam.

    Answer

    \(\displaystyle a. 3,000,000\)lb, \(\displaystyle b. 749,000\)lb

    Exercise \(\PageIndex{31}\)

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

    Answer

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    Exercise \(\PageIndex{32}\)

    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

    \(\displaystyle 23.25π\) million ft-lb

    Exercise \(\PageIndex{33}\)

    How much work is required to pump out a swimming pool if the area of the base is \(\displaystyle 800\)ft2, the water is \(\displaystyle 4\) ft deep, and the top is \(\displaystyle 1\) ft above the water level? Assume that the density of water is \(\displaystyle 62\)lb/ft3.

    Answer

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    Exercise \(\PageIndex{34}\)

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

    Answer

    \(\displaystyle \frac{AρH^2}{2}\)

    Exercise \(\PageIndex{35}\)

    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.

    Answer

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    Exercise \(\PageIndex{36}\)

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

    Answer

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