Choose three of the small objects made of the same kind of wood shown in the figure below. Write an equation for the mass of each object, using unique symbols for any quantities that are different from each other.
The wood the objects are made from has a density of approximately \(\rho = 700 \mathrm{~kg/m^3}\text{.}\) Estimate a numerical value for the mass of each object. Make sense of your answers.
Consider the system consisting of all three objects together. Write an equation for the total mass of the system. Simplify your equation as much as possible before finding a numerical value.
Why is the equation you wrote in part A incorrect for the mass of this system? Are there any other systems for which you think the equation in part A would be incorrect?
Why do you think \(dm\) is an appropriate symbol for the mass of this object? Why is it important that there is a differential symbol \(d\) on both sides of the equation?
Write an equation for \(dV\) using the coordinate system below. Which other quantities in this equation should also be written with the differential symbol \(d\text{?}\)
Suppose the curved line above can be written \(y = \frac{x^2}{L}\text{,}\) where \(L\) is the horizontal length of the object. Label \(x\) and \(L\) on the figure above. What is the difference between these two quantities? Where is the origin? Why would the simpler equation \(y = x^2\) be incorrect for this line?
The task of finding \(dV\) in an equation like \(dm = \rho dV\) (both as its own equation and by sketching the parts of \(dV\) on a figure) is known as chopping up space, or, more simply, chopping. Why do you think this is an appropriate name for this task? Why do you think it is important?
Now that you know the mass of each small piece, \(dm\text{,}\) describe a conceptual procedure you could use to determine the total mass \(m\) of the object.
Carry out the procedure you described above using the information from the previous parts. You should end up with both an equation for m in terms of known quantities and a numerical answer.
The task of integrating both sides of an equation like \(dm = \rho dV\) (including appropriate limits) is a form of adding, or, alternatively, accumulating.
You now have a three-step strategy that can be used to construct symbolic equations for quantities that are made from changing quantities: Chop-Multiply-Add. Now you will have a chance to apply this strategy in a new context.
The object shown below has a mass density that increases from left to right. The density is uniform in the other directions. The horizontal length of the object is \(L\) and the height and depth are both \(a\text{.}\)
List a few different ways you could chop this object into small pieces. Write a symbolic equation for the volume of an example small piece for each way you think of. Make sure to use the differential \(d\) symbol appropriately in your equations.
Choose the way of chopping up the object that allows your small pieces to be as large as possible while allowing you to treat the density of each piece as uniform. Sketch an arbitrary example on the figure above and label both its size and position (make sure to specify your coordinate system).
Assume the mass density \(\rho\) varies linearly with respect to \(x\) (the horizontal position), from \(\rho_o\) at the left edge to \(3\rho_o\) at the right edge.
Start by writing a symbolic equation for \(\rho\) in terms of \(x\text{,}\) using what you know about the equation for a straight line. Make sure your equation agrees with your chosen reference frame.
Write a symbolic equation for dm, the mass of the small piece you labeled on the figure in the Chop part. It can often help to add a label for dm to your figure. Your equation should only be written in terms of known variables (such as \(L\text{,}\)\(a\text{,}\) or \(\rho_o\)) or coordinate variables (such as \(x\text{,}\)\(y\text{,}\)\(z\text{,}\) or \(dx\text{,}\)\(dy\text{,}\)\(dz\)).
Use the diagram you drew on the previous page to determine the appropriate limits of integration. Why would it be incorrect to do an indefinite integral here?
Evaluate your integral expression. Do you need to use an integral calculator or is this an integral that is manageable to do by hand? (Hint: pay careful attention to which quantities are constant and which are not.)
How did this example differ from the original example you explored in the first three sections? What was similar? What do you want to remember about this process for future activities that ask you to use a chop-multiply-add strategy?
Chop: Start by chopping something up into small pieces: usually regions of space or intervals of time. Label each relevant piece with a differential, such as \(dx\) or \(dt\text{.}\)
Multiply: Multiply appropriate quantities, including the infinitesimal quantity identified in the Chop step, to find an infinitesimal version fo the desired quantity. For example, mass might be written in terms of linear mass density \(\lambda\) as \(dm = \lambda dx\text{.}\)
