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Dimensional Analysis (Grade 10)

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Dimensional Analysis

A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons.

Hint: Convert pints to quarts then convert quarts to gallons.

The dwarf seahorse, Hippocampus zosterae, swims at a rate of 52.68 feet per hour. Use dimensional analysis to convert this speed to inches per minute.

Hint: Use the conversion factor to 12 in/ 1 ft convert feet to inches, and use the conversion factor 1 hr/ 60 min to convert hours to minutes.

Use dimensional analysis to convert the speed of a train, traveling 95 km/h, to cm/s.

The current price of gold is $1,215 per ounce, in US dollars (USD). If 1 USD = 0.89 EUR (euros), what is the price of gold in euros per gram? Hint: 1 ounce equals 28.3 grams.

A water tank is leaking water at a rate of 0.5 L/min. Using dimensional analysis, find the rate in cubic meters per hour.
Hint: [math]1 \ "ml" = 1 \ "cm"^3[/math]

Use dimensional analysis to find how many minutes are in a year. Round to the nearest minute if necessary.
  1. 525,600 minutes
  2. 21,900 minutes
  3. 8,760 minutes
  4. 913 minutes
Micah and two friends are taking a trip from Houston, TX to St Louis, IL, which is about 778 miles. The car they are driving gets about 26.4 miles per gallon and the cost of gas is about $2.50 per gallon. They will share the costs equally, what is the cost per person of the trip (assuming gas is the only cost). Use dimensional analysis to choose the expression below which correctly gives the cost per person of this trip.
  1. [math]3 xx 778 xx 26.4 xx 2.5[/math]
  2. [math]1/3 xx 778 xx 26.4 xx 2.5[/math]
  3. [math]1/3 xx 778 xx 1/26.4 xx 2.5[/math]
  4. [math]1/3 xx 778 xx 1/26.4 xx 1/2.5[/math]
Rob is writing a physics test. For one of the problems, he needs the gravitational force formula for the force of gravity between two objects. He's not sure he remembers it correctly. He believes it is
[math]F_g = (G \ m_1 \ m_2)/r[/math],
where [math]F_g[/math] is the force, measured in Newtons; [math]G[/math] is the gravitational constant, measured in [math]"N"*"m"^2//"kg"^2[/math]; [math]m_1, m_2[/math] are the masses of the two objects measured in kilograms; and [math]r[/math] is the distance between the objects, measured in meters. Using dimensional analysis, is his formula correct? Why or why not?
  1. Yes, since [math]("N" * "m")/("kg"^2) xx "kg"/1 xx "kg"/1 xx 1/("m") = "N"[/math].
  2. No, since [math]("N" * "m"^2)/("kg"^2) xx "kg"/1 xx "kg"/1 xx 1/("m") = "N"*"m"[/math].
  3. No, since [math]("N" * "m"^2)/("kg"^2) xx 1/"kg" xx 1/"kg" xx "m"/1 = ("N" * "m"^3)/ ("kg"^4)[/math].
  4. No, since [math]("N" * "m"^2)/("kg"^2) xx "kg"/1 xx "kg"/1 xx "m"/1 = "N" * "m"^3[/math].
Near the end of class, Jillian's physics teacher writes a formula up on the board. Jillian quickly writes it down before leaving. Later that night while doing homework, she is unsure if she correctly copied the formula. What she wrote is:
[math]d = d_0 + v_0t + 1/2 at^2[/math]
where [math]d, d_0[/math] are distances measured in meters; [math]v_0[/math] is velocity, measured in meters per second; [math]t[/math] is time, measured in seconds; and [math]a[/math] is acceleration, measured in meters per second squared.

She decides she will use dimensional analysis to determine if it is correct or not. She reasons that each term has to have the same units, and since the term on the left side of the equation and the first term on the right side of the equation are in meters, the other two terms need to be as well. The second term on the right side of the equation is

[math]"m"/"s" *"s"/1 = "m"[/math]

and the last term is

[math]"m"/"s"^2 * "s"^2/1 = "m"[/math].

Since all terms are in meters, she decides that the equation she wrote down is right. Is she correct, and if not, what mistake did she make?
  1. Yes, she is correct.
  2. No. She assumed that all terms need to have the same units, when all terms need to be without units.
  3. No. Although the variable [math]t[/math] is squared, the units are not. Therefore, the units of the last term are m/s, which are different than the rest of the terms.
  4. No. Jillian did the dimensional analysis incorrectly. The units of the second term on the right side come out to [math]"m"//"s"^2[/math] and the units of the last term are [math]"m"//"s"^4[/math].
Using dimensional analysis, the following calculation can be performed to convert between 3 pounds and its equivalent in grams (using the rate of 1 oz equals 28.3 g).

[math](3 \ "lb")/1 xx (16 \ "oz") / (1 \ "lb") xx (28.3 \ "g")/(1 \ "oz") = 1,358.4 \ "g"[/math]

Mathematically, why can 3 be multiplied by the factors [math] (16 \ "oz") / (1 \ "lb")[/math] and [math](28.3 \ "g")/(1 \ "oz") ?[/math]
  1. Because this is dimensional analysis, regular rules of math do not apply.
  2. Since the necessary units cancel out, there is no problem.
  3. Because each of these factors is equal to one (the numerator and denominator are equal, but in different units).
  4. By the Multiplicative Property of Equality.

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