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

- 1 mi = 1.6 km
- 1 gal = 3.8 liters
- 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*, w_{T}, and give an expression for w_{T}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 t_{bigger}instead of t. If these two fans are to have the same terminal angular velocity, what must the ratio t_{bigger}/t be?
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.
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: (a) Does a more efficient car have a larger or smaller MPG number? Explain
how you know.
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 (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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Last modified June 21, 2002