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# 1.4E: Exercises

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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.

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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?

$$\displaystyle 150$$ 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)

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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 $$\displaystyle F=100N$$.

$$\displaystyle 200J$$

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.

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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?

$$\displaystyle 1$$ 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

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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

$$\displaystyle \frac{39}{2}$$

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.

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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.

$$\displaystyle ln(243)$$

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.

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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 $$\displaystyle ρ(x)=x^3−2x+5$$

$$\displaystyle \frac{332π}{15}$$

Exercise $$\PageIndex{13}$$

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

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Exercise $$\PageIndex{14}$$

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

$$\displaystyle 100π$$

Exercise $$\PageIndex{15}$$

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

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Exercise $$\PageIndex{16}$$

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

$$\displaystyle 20π\sqrt{15}$$

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?

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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?

$$\displaystyle 6$$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?

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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?

$$\displaystyle 5$$ cm

Exercise $$\PageIndex{21}$$

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

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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?

$$\displaystyle 36$$ 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.

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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?

$$\displaystyle 18,750$$ 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?

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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?

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

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?

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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 $$\displaystyle M=6×10^{24}kg$$.)

$$\displaystyle 8.65×10^5J$$

Exercise $$\PageIndex{29}$$

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

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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.

$$\displaystyle a. 3,000,000$$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.

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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.

$$\displaystyle 23.25π$$ 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.

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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.

$$\displaystyle \frac{AρH^2}{2}$$

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.

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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.