a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume

this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, $F = k x,$ where $k$ is a constant

the pressure exerted by water on a submerged object

the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance

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Calculus Volume 1 · 6.5 Physical Applications

Physical Applications: practice problems

19 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 219
For the following exercises, find the work done.
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?
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Exercises 221
For the following exercises, find the work done.
Find the work done when you push a box along the floor $2$ m, when you apply a constant force of $F = 100 N .$
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Exercises 223
For the following exercises, find the work done.
What is the work done moving a particle from $x = 0$ to $x = 1$ m if the force acting on it is $F = 3 x^{2}$ N?
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Exercises 225
For the following exercises, find the mass of the one-dimensional object.
A car antenna that is $3$ ft long (starting at $x = 0)$ and has a density function of $ρ (x) = 3 x + 2$ lb/ft
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Exercises 227
For the following exercises, find the mass of the one-dimensional object.
A pencil that is $4$ in. long (starting at $x = 2)$ and has a density function of $ρ (x) = 5 / x$ oz/in.
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Exercises 229
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
An oversized hockey puck of radius $2$ in. with density function $ρ (x) = x^{3} - 2 x + 5$
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Exercises 231
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
A plate of radius $10$ in. with density function $ρ (x) = 1 + cos (π x)$
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Exercises 233
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
A disk of radius $5$ cm with density function $ρ (x) = 3 x$
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Exercises 235
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
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?
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Exercises 237
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
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?
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Exercises 239
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
A force of $F = 20 x - x^{3}$ N stretches a nonlinear spring by $x$ meters. What work is required to stretch the spring from $x = 0$ to $x = 2$ m?
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Exercises 241
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
For the cable in the preceding exercise, how much work is done to lift the cable $50$ ft?
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Exercises 243
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
[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 $w = 100 lb/ft^{3},$ how much work did it take to lift all the rock?
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Exercises 245
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
[T] The force of gravity on a mass $m$ is $F = - ((GM m) / x^{2})$ newtons. For a rocket of mass $m = 1000 kg,$ compute the work to lift the rocket from $x = 6400$ to $x = 6500$ km. State your answers with three significant figures. (: and
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Exercises 247
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
[T] A rectangular dam is $40$ ft high and $60$ ft wide. Assume the weight density of water is 62.5 lbs/ft³. Compute the total force $F$ on the dam when
the surface of the water is at the top of the dam and
the surface of the water is halfway down the dam.
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Exercises 249
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
[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.
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Exercises 251
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
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.
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Exercises 253
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
A cone-shaped tank has a cross-sectional area that increases with its depth: $A = (π r^{2} h^{2}) / H^{3} .$ Show that the work to empty it is half the work for a cylinder with the same height and base.
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Review Exercises 435
True or False? Justify your answer with a proof or a counterexample.
The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.
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Key terms · 4

density function: a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume
Hooke’s law: this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, $F = k x,$ where $k$ is a constant
hydrostatic pressure: the pressure exerted by water on a submerged object
work: the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance

Learn 6.5 on the whiteboard

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Physical Applications: practice problems

19 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 219

For the following exercises, find the work done.

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?

Exercises 221

For the following exercises, find the work done.

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

Exercises 223

For the following exercises, find the work done.

What is the work done moving a particle from $x = 0$ to $x = 1$ m if the force acting on it is $F = 3 x^{2}$ N?

Exercises 225

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

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

Exercises 227

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

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

Exercises 229