- 2.5.1. Determine the mass of a one-dimensional object from its linear density function.
- 2.5.2. Determine the mass of a two-dimensional circular object from its radial density function.
- 2.5.3. Calculate the work done by a variable force acting along a line.
- 2.5.4. Calculate the work done in pumping a liquid from one height to another.
- 2.5.5. Find the hydrostatic force against a submerged vertical plate.
In this section, we examine some physical applications of integration. Let’s begin with a look at calculating mass from a density function. We then turn our attention to work, and close the section with a study of hydrostatic force.
Mass and Density
We can use integration to develop a formula for calculating mass based on a density function. First we consider a thin rod or wire. Orient the rod so it aligns with the with the left end of the rod at and the right end of the rod at (Figure 2.48). Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.
If the rod has constant density given in terms of mass per unit length, then the mass of the rod is just the product of the density and the length of the rod: If the density of the rod is not constant, however, the problem becomes a little more challenging. When the density of the rod varies from point to point, we use a linear density function, to denote the density of the rod at any point, Let be an integrable linear density function. Now, for let be a regular partition of the interval and for choose an arbitrary point Figure 2.49 shows a representative segment of the rod.
The mass of the segment of the rod from to is approximated by
Adding the masses of all the segments gives us an approximation for the mass of the entire rod:
This is a Riemann sum. Taking the limit as we get an expression for the exact mass of the rod:
We state this result in the following theorem.
Mass–Density Formula of a One-Dimensional Object
Given a thin rod oriented along the over the interval let denote a linear density function giving the density of the rod at a point x in the interval. Then the mass of the rod is given by
We apply this theorem in the next example.
Calculating Mass from Linear Density
Consider a thin rod oriented on the x-axis over the interval If the density of the rod is given by what is the mass of the rod?
Applying Equation 2.10 directly, we have
Consider a thin rod oriented on the x-axis over the interval If the density of the rod is given by what is the mass of the rod?
We now extend this concept to find the mass of a two-dimensional disk of radius As with the rod we looked at in the one-dimensional case, here we assume the disk is thin enough that, for mathematical purposes, we can treat it as a two-dimensional object. We assume the density is given in terms of mass per unit area (called area density), and further assume the density varies only along the disk’s radius (called radial density). We orient the disk in the with the center at the origin. Then, the density of the disk can be treated as a function of denoted We assume is integrable. Because density is a function of we partition the interval from along the For let be a regular partition of the interval and for choose an arbitrary point Now, use the partition to break up the disk into thin (two-dimensional) washers. A disk and a representative washer are depicted in the following figure.
We now approximate the density and area of the washer to calculate an approximate mass, Note that the area of the washer is given by
You may recall that we had an expression similar to this when we were computing volumes by shells. As we did there, we use to approximate the average radius of the washer. We obtain
Using to approximate the density of the washer, we approximate the mass of the washer by
Adding up the masses of the washers, we see the mass of the entire disk is approximated by
We again recognize this as a Riemann sum, and take the limit as This gives us
We summarize these findings in the following theorem.
Mass–Density Formula of a Circular Object
Let be an integrable function representing the radial density of a disk of radius Then the mass of the disk is given by
Calculating Mass from Radial Density
Let represent the radial density of a disk. Calculate the mass of a disk of radius 4.
Applying the formula, we find
Let represent the radial density of a disk. Calculate the mass of a disk of radius 2.
Work Done by a Force
We now consider work. In physics, work is related to force, which is often intuitively defined as a push or pull on an object. When a force moves an object, we say the force does work on the object. In other words, work can be thought of as the amount of energy it takes to move an object. According to physics, when we have a constant force, work can be expressed as the product of force and distance.
In the English system, the unit of force is the pound and the unit of distance is the foot, so work is given in foot-pounds. In the metric system, kilograms and meters are used. One newton is the force needed to accelerate kilogram of mass at the rate of m/sec2. Thus, the most common unit of work is the newton-meter. This same unit is also called the joule. Both are defined as kilograms times meters squared over seconds squared
When we have a constant force, things are pretty easy. It is rare, however, for a force to be constant. The work done to compress (or elongate) a spring, for example, varies depending on how far the spring has already been compressed (or stretched). We look at springs in more detail later in this section.
Suppose we have a variable force that moves an object in a positive direction along the x-axis from point to point To calculate the work done, we partition the interval and estimate the work done over each subinterval. So, for let be a regular partition of the interval and for choose an arbitrary point To calculate the work done to move an object from point to point we assume the force is roughly constant over the interval, and use to approximate the force. The work done over the interval then, is given by
Therefore, the work done over the interval is approximately
Taking the limit of this expression as gives us the exact value for work:
Thus, we can define work as follows.
If a variable force moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is
Note that if F is constant, the integral evaluates to which is the formula we stated at the beginning of this section.
Now let’s look at the specific example of the work done to compress or elongate a spring. Consider a block attached to a horizontal spring. The block moves back and forth as the spring stretches and compresses. Although in the real world we would have to account for the force of friction between the block and the surface on which it is resting, we ignore friction here and assume the block is resting on a frictionless surface. When the spring is at its natural length (at rest), the system is said to be at equilibrium. In this state, the spring is neither elongated nor compressed, and in this equilibrium position the block does not move until some force is introduced. We orient the system such that corresponds to the equilibrium position (see the following figure).
According to Hooke’s law, the force required to compress or stretch a spring from an equilibrium position is given by for some constant The value of depends on the physical characteristics of the spring. The constant is called the spring constant and is always positive. We can use this information to calculate the work done to compress or elongate a spring, as shown in the following example.
The Work Required to Stretch or Compress a Spring
Suppose it takes a force of N (in the negative direction) to compress a spring m from the equilibrium position. How much work is done to stretch the spring m from the equilibrium position?
First find the spring constant, When we know so
and Then, to calculate work, we integrate the force function, obtaining
The work done to stretch the spring is J.
Suppose it takes a force of lb to stretch a spring in. from the equilibrium position. How much work is done to stretch the spring ft from the equilibrium position?
Work Done in Pumping
Consider the work done to pump water (or some other liquid) out of a tank. Pumping problems are a little more complicated than spring problems because many of the calculations depend on the shape and size of the tank. In addition, instead of being concerned about the work done to move a single mass, we are looking at the work done to move a volume of water, and it takes more work to move the water from the bottom of the tank than it does to move the water from the top of the tank.
We examine the process in the context of a cylindrical tank, then look at a couple of examples using tanks of different shapes. Assume a cylindrical tank of radius m and height m is filled to a depth of 8 m. How much work does it take to pump all the water over the top edge of the tank?
The first thing we need to do is define a frame of reference. We let represent the vertical distance below the top of the tank. That is, we orient the vertically, with the origin at the top of the tank and the downward direction being positive (see the following figure).
Using this coordinate system, the water extends from to Therefore, we partition the interval and look at the work required to lift each individual “layer” of water. So, for let be a regular partition of the interval and for choose an arbitrary point Figure 2.53 shows a representative layer.
In pumping problems, the force required to lift the water to the top of the tank is the force required to overcome gravity, so it is equal to the weight of the water. Given that the weight-density of water is N/m3, or lb/ft3, calculating the volume of each layer gives us the weight. In this case, we have
Then, the force needed to lift each layer is
Note that this step becomes a little more difficult if we have a noncylindrical tank. We look at a noncylindrical tank in the next example.
We also need to know the distance the water must be lifted. Based on our choice of coordinate systems, we can use as an approximation of the distance the layer must be lifted. Then the work to lift the layer of water is approximately
Adding the work for each layer, we see the approximate work to empty the tank is given by
This is a Riemann sum, so taking the limit as we get
The work required to empty the tank is approximately 23,650,000 J.
For pumping problems, the calculations vary depending on the shape of the tank or container. The following problem-solving strategy lays out a step-by-step process for solving pumping problems.
- Sketch a picture of the tank and select an appropriate frame of reference.
- Calculate the volume of a representative layer of water.
- Multiply the volume by the weight-density of water to get the force.
- Calculate the distance the layer of water must be lifted.
- Multiply the force and distance to get an estimate of the work needed to lift the layer of water.
- Sum the work required to lift all the layers. This expression is an estimate of the work required to pump out the desired amount of water, and it is in the form of a Riemann sum.
- Take the limit as and evaluate the resulting integral to get the exact work required to pump out the desired amount of water.
We now apply this problem-solving strategy in an example with a noncylindrical tank.
A Pumping Problem with a Noncylindrical Tank
Assume a tank in the shape of an inverted cone, with height ft and base radius ft. The tank is full to start with, and water is pumped over the upper edge of the tank until the height of the water remaining in the tank is ft. How much work is required to pump out that amount of water?
The tank is depicted in Figure 2.54. As we did in the example with the cylindrical tank, we orient the vertically, with the origin at the top of the tank and the downward direction being positive (step 1).
The tank starts out full and ends with ft of water left, so, based on our chosen frame of reference, we need to partition the interval Then, for let be a regular partition of the interval and for choose an arbitrary point We can approximate the volume of a layer by using a disk, then use similar triangles to find the radius of the disk (see the following figure).
From properties of similar triangles, we have
Then the volume of the disk is
The weight-density of water is lb/ft3, so the force needed to lift each layer is approximately
Based on the diagram, the distance the water must be lifted is approximately feet (step 4), so the approximate work needed to lift the layer is
Summing the work required to lift all the layers, we get an approximate value of the total work:
Taking the limit as we obtain
It takes approximately ft-lb of work to empty the tank to the desired level.
A tank is in the shape of an inverted cone, with height ft and base radius 6 ft. The tank is filled to a depth of 8 ft to start with, and water is pumped over the upper edge of the tank until 3 ft of water remain in the tank. How much work is required to pump out that amount of water?
Hydrostatic Force and Pressure
In this last section, we look at the force and pressure exerted on an object submerged in a liquid. In the English system, force is measured in pounds. In the metric system, it is measured in newtons. Pressure is force per unit area, so in the English system we have pounds per square foot (or, perhaps more commonly, pounds per square inch, denoted psi). In the metric system we have newtons per square meter, also called pascals.
Let’s begin with the simple case of a plate of area submerged horizontally in water at a depth s (Figure 2.56). Then, the force exerted on the plate is simply the weight of the water above it, which is given by where is the weight density of water (weight per unit volume). To find the hydrostatic pressure—that is, the pressure exerted by water on a submerged object—we divide the force by the area. So the pressure is
By Pascal’s principle, the pressure at a given depth is the same in all directions, so it does not matter if the plate is submerged horizontally or vertically. So, as long as we know the depth, we know the pressure. We can apply Pascal’s principle to find the force exerted on surfaces, such as dams, that are oriented vertically. We cannot apply the formula directly, because the depth varies from point to point on a vertically oriented surface. So, as we have done many times before, we form a partition, a Riemann sum, and, ultimately, a definite integral to calculate the force.
Suppose a thin plate is submerged in water. We choose our frame of reference such that the x-axis is oriented vertically, with the downward direction being positive, and point corresponding to a logical reference point. Let denote the depth at point x. Note we often let correspond to the surface of the water. In this case, depth at any point is simply given by However, in some cases we may want to select a different reference point for so we proceed with the development in the more general case. Last, let denote the width of the plate at the point
Assume the top edge of the plate is at point and the bottom edge of the plate is at point Then, for let be a regular partition of the interval and for choose an arbitrary point The partition divides the plate into several thin, rectangular strips (see the following figure).
Let’s now estimate the force on a representative strip. If the strip is thin enough, we can treat it as if it is at a constant depth, We then have
Adding the forces, we get an estimate for the force on the plate:
This is a Riemann sum, so taking the limit gives us the exact force. We obtain
Evaluating this integral gives us the force on the plate. We summarize this in the following problem-solving strategy.
- Sketch a picture and select an appropriate frame of reference. (Note that if we select a frame of reference other than the one used earlier, we may have to adjust Equation 2.13 accordingly.)
- Determine the depth and width functions, and
- Determine the weight-density of whatever liquid with which you are working. The weight-density of water is lb/ft3, or 9800 N/m3.
- Use the equation to calculate the total force.
Finding Hydrostatic Force
A water trough 15 ft long has ends shaped like inverted isosceles triangles, with base 8 ft and height 3 ft. Find the force on one end of the trough if the trough is full of water.
Figure 2.58 shows the trough and a more detailed view of one end.
Select a frame of reference with the oriented vertically and the downward direction being positive. Select the top of the trough as the point corresponding to (step 1). The depth function, then, is Using similar triangles, we see that (step 2). Now, the weight density of water is lb/ft3 (step 3), so applying Equation 2.13, we obtain
The water exerts a force of 748.8 lb on the end of the trough (step 4).
A water trough 12 m long has ends shaped like inverted isosceles triangles, with base 6 m and height 4 m. Find the force on one end of the trough if the trough is full of water.
Chapter Opener: Finding Hydrostatic Force
We now return our attention to the Hoover Dam, mentioned at the beginning of this chapter. The actual dam is arched, rather than flat, but we are going to make some simplifying assumptions to help us with the calculations. Assume the face of the Hoover Dam is shaped like an isosceles trapezoid with lower base ft, upper base ft, and height ft (see the following figure).
When the reservoir is full, Lake Mead’s maximum depth is about 530 ft, and the surface of the lake is about 10 ft below the top of the dam (see the following figure).
- Find the force on the face of the dam when the reservoir is full.
- The southwest United States has been experiencing a drought, and the surface of Lake Mead is about 125 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?
- We begin by establishing a frame of reference. As usual, we choose to orient the vertically, with the downward direction being positive. This time, however, we are going to let represent the top of the dam, rather than the surface of the water. When the reservoir is full, the surface of the water is ft below the top of the dam, so (see the following figure).
To find the width function, we again turn to similar triangles as shown in the figure below.
From the figure, we see that Using properties of similar triangles, we get Thus,
Using a weight-density of lb/ft3 (step 3) and applying Equation 2.13, we get
Note the change from pounds to tons lb = ton) (step 4). This changes our depth function, and our limits of integration. We have The lower limit of integration is The upper limit remains Evaluating the integral, we get
When the reservoir is at its average level, the surface of the water is about 50 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?
To learn more about Hoover Dam, see this article published by the History Channel.
Section 2.5 Exercises
For the following exercises, find the work done.
Find the work done when a constant force lb moves a chair from to ft.
How much work is done when a person lifts a lb box of comics onto a truck that is ft off the ground?
What is the work done lifting a kg child from the floor to a height of m? (Note that kg equates to N)
Find the work done when you push a box along the floor m, when you apply a constant force of
Compute the work done for a force N from to m.
What is the work done moving a particle from to m if the force acting on it is N?
For the following exercises, find the mass of the one-dimensional object.
A wire that is ft long (starting at and has a density function of lb/ft
A car antenna that is ft long (starting at and has a density function of lb/ft
A metal rod that is in. long (starting at and has a density function of lb/in.
A pencil that is in. long (starting at and has a density function of oz/in.
A ruler that is in. long (starting at and has a density function of oz/in.
For the following exercises, find the mass of the two-dimensional object that is centered at the origin.
An oversized hockey puck of radius in. with density function
A frisbee of radius in. with density function
A plate of radius in. with density function
A jar lid of radius in. with density function
A disk of radius cm with density function
A -in. spring is stretched to in. by a force of lb. What is the spring constant?
A spring has a natural length of cm. It takes J to stretch the spring to cm. How much work would it take to stretch the spring from cm to cm?
A -m spring requires J to stretch the spring to m. How much work would it take to stretch the spring from m to m?
A spring requires J to stretch the spring from cm to cm, and an additional J to stretch the spring from cm to cm. What is the natural length of the spring?
A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?
A force of N stretches a nonlinear spring by meters. What work is required to stretch the spring from to m?
Find the work done by winding up a hanging cable of length ft and weight-density lb/ft.
For the cable in the preceding exercise, how much work is done to lift the cable ft?
For the cable in the preceding exercise, how much additional work is done by hanging a lb weight at the end of the cable?
[T] A pyramid of height ft has a square base ft by ft. Find the area at height If the rock used to build the pyramid weighs approximately how much work did it take to lift all the rock?
[T] For the pyramid in the preceding exercise, assume there were workers each working hours a day, days a week, weeks a year. If the workers, on average, lifted 10 100 lb rocks ft/hr, how long did it take to build the pyramid?
[T] The force of gravity on a mass is newtons. For a rocket of mass compute the work to lift the rocket from to km. (Note: and
[T] For the rocket in the preceding exercise, find the work to lift the rocket from to
[T] A rectangular dam is ft high and ft wide. Compute the total force 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.
[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius ft and height ft. Use the fact that the density of water is lb/ft3.
[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.
[T] How much work is required to pump out a swimming pool if the area of the base is ft2, the water is ft deep, and the top is ft above the water level? Assume that the density of water is lb/ft3.
A cylinder of depth and cross-sectional area stands full of water at density Compute the work to pump all the water to the top.
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.
A cone-shaped tank has a cross-sectional area that increases with its depth: Show that the work to empty it is half the work for a cylinder with the same height and base.