Dimensional Analysis
1) Which of the following equations are dimensionally correct?
(a) vf = vi + ax
(b) y = (2 m)cos(kx), where k = 2 m-1.
2) A fellow student has proposed that an object accelerates under the influence of the gravity of the Sun according to the formula:
a = GM/r
where a is the magnitude of the acceleration measured in feet per second squared, M isthe mass of the Sun, the physical constant G is 6.67 x 10-11 m3 / s2 kg and 'r' is the distance to the Sun's center. Can this formula be correct?
3) In quantum mechanics, the fundamental constant called Planck's constant, h, has dimensions of [ML2T-1].
What combination of the Planck's constant 'h', a mass 'm' and the speed of light 'c' will result in a physical quantity that has the dimensions of length?
Unit Conversion
4) An auditorium measures 40.0 m x 20.0 m x 12.0 m. The density of air is 1.20 kg/m3. What are
(a) the volume of the room in cubic feet and
(b) the weight of air in the room in pounds?
Assume that it takes 7.00 minutes to fill a 30.0-gal gasoline tank.
(a) Calculate the rate at which the tank is filled in gallons per second.
(b) Calculate the rate at which the tank is filled in cubic meters per second.
(c) Determine the time interval, in hours, required to fill a 1-m3 volume at the same rate. (1 U.S. gal = 231 in.3)
5) One gallon of paint (volume = 3.78 10-3 m3) covers an area of 25.0 m2. What is the thickness of the paint on the wall? Estimation
6) Soft drinks are commonly sold in aluminum containers. To an order of magnitude, how many such containers are thrown away or recycled each year by U.S. consumers? How many tons of aluminum does this represent? In your solution state the quantities you measure or estimate and the values you take for them.
7) To an order of magnitude, how many piano tuners are in New York City? The physicist Enrico Fermi was famous for asking questions like this on oral Ph.D. qualifying examinations. Make this calculation for Seattle (remember this is an estimate!). Justify your assumptions!
Scaling
8) A sculptor builds a model of a horse. She discovers that to cast her small scale model she needs 2kg of bronze. When she is done, she finds that she can give it two coats of finishing polyurethane varnish using exactly one small can of varnish. The final statue is supposed to be 5 times as large as the model in each dimension. The final statue is supposed to be 5 times as large as the model in each dimension.
How much bronze will she need? How much varnish should she buy?
9) We know from our dimensional analysis that if an object maintains its shape but changes its size, its area changes as the square of its length and its volume changes as the cube of its length. Suppose you are a parent and your child is sick and has to take some medicine. You have taken this medicine previously and you know its dose for you. You are 5'10" tall and weigh 180 lb and your child is 2'11" tall and weighs 30lbs. Estimate an appropriate dosage for your child's medicine in the following cases. Be sure to discuss your reasoning.
a) The medicine is one that will enter the child's bloodstream and reach every cell in the body. Your dose is 250mg.
b) The medicine is one that is meant to coat the child's throat. Your dose is 15ml.