1.4E: Exercises

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Dec 21, 2020
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- 1.4: Other applications: Work, centroid, and center of mass
- 1.5: Moments and Centers of Mass

This page is a draft and is under active development.

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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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Answers may vary

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