## Activity Based Physics Thinking Problems in Mechanics: Dimensional Analysis

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### Activity Based Physics Thinking Problems in Mechanics: Dimensional Analysis

 1) A student makes the following argument: "I can prove a dollar equals a penny. Since a dime (10 cents) is one-tenth of a dollar, I can write: 10¢ = 0.1 \$ Square both sides of this equation. Since the squares of equals are equal, 100 ¢ = 0.01 \$. Since 100 ¢ = 1 \$ and 0.01 \$ = 1 ¢ it follows that 1\$ = 1 ¢." What's wrong with the argument?

2) Each of the following equations has been written down by a student during an examination. Do a dimensional analysis of each equation and explain why the equation cannot be correct.

 The University of Maryland's mascot is a terrapin (a snapping turtle). It is represented on campus by a bronze statue in front of the library named Testudo. Unfortunately, students rub his nose for luck on tests, and he is beginning to wear out. A sculptor decides to propose a replacement. She builds a small scale model and discovers she needs 2 kg of bronze. When she is finished, she finds that she can give it two coats of finishing polyurethane varnish using exactly one small can of varnish. Her final statue will be 5 times as big as her model in all dimensions. How much bronze and how much varnish will she need for the final product?

3) In the US we specify how efficiently a car uses gas by citing how many miles per gallon it gets (MPG). In Europe they cite liters per 100 kilometers (LPK). Note the conversion factors:

• 1 mi = 1.6 km
• 1 gal = 3.8 liters

(a) Does a more efficient car have a larger or smaller MPG number? Explain how you know.
(b) Does a more efficient car have a larger or smaller LPK number? Explain how you would know.
(c) If M is the number of miles per gallon a car gets, and L is the number of liters it takes to go 100 km, find a mathematical equation that lets you convert from M to L.

4) Two Terrapins (U. of Md. students) decide to go to Jerry's for a pizza. When they get there they find Jerry's is having a special:

SPECIAL:
one 20" pizza \$15 (toppings extra)
REGULAR PRICE:
one 10" pizza \$5
one 20" pizza \$18
 Raphael goes: "Great! Let's get a large one." Donatello goes: "Don't be dumb. Let's get three of the small ones for the same price. That'll give us more pizza and be cheaper." Raphael goes: "Why would it be a special if it's less pizza than we could get for regular price? Let's get the large." Who's right? Which would you buy? What would the difference be if you were buying them at Ledo's (square pizzas)?
 5) A fan is driven by an electric motor which delivers a constant torque t. Take the axis of rotation and the torque to be fixed in the z-direction. The air gives a retarding torque which has the form tair=-w2 where B is a constant and w is the magnitude of the angular velocity of the fan. Let I be the moment of inertia of the moving parts of the fan about its axis of rotation. Write the equation for the angular motion of the fan which gives the angular acceleration a =dw/dt. Give an argument that the fan will reach a terminal angular velocity, wT, and give an expression for wT in terms of the parameters B, I, and t What are the dimensions of the parameter B? Suppose we consider the geometrical shape of the fan to be fixed, but change its size, i.e. the radius, R, and the mass, M. These can be changed independently, for example by changing the material of which the fan is made. Assume that B depends on the density of air r, and the radius R, but not on M. Using dimensional analysis, how does B depend on r and R? Why is it reasonable to assume that B does not depend on M? Suppose we build a second fan, exactly like the first except the radius is 2 R instead of R, and the motor gives a torque tbigger instead of t. If these two fans are to have the same terminal angular velocity, what must the ratio tbigger/t be? Contributed by Richard Prange, University of Maryland

6) Each of the mathematical expressions below was given by a student on his or her way to the solution of an exam problem. Assume each of the symbols stands for what we use them for in this class (as indicated in each line). If you believe the equation given could possibly be a correct physical equation, write Y. If you think that it could not possibly be a correct physical equation, write N and give a brief explanation of why you think so.

(a)  (mass, gravitational field strength, coefficient of friction, normal force)

(b)  (tension, mass, gravitational field strength, moment of inertia, radius)

(c)  (force, pressure, area)

(d)  (velocity, universal gravitational constant, mass, radius)

These problems written and collected by E. F. Redish. Terrapin photo courtesy of University of Maryland InforM. These problems may be freely used in classrooms. They may be copied and cited in published work if the Activity Based Phyics Thinking Problems in Physics site is mentioned and the URL given. Web page edited by K. A. Vick.

To contribute problems to this site, send them to redish@physics.umd.edu.

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Maintained by University of Maryland PERG
Comments and questions may be directed to E. F. Redish