Chap 15 Review

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Exercise 1 For the following questions, the quantities involved are:

  • \(a = 25\) ft
  • \(b = 3\) hours
  • \(c = 4\)
  • \(d = 1\) meter
  • \(e = 2.718\)
  1. Is this combination dimensionally valid? \[a / c\] Why or why not?

Division can accommodate any two quantities, regardless of dimension.

\(c\) is a dimensionless quantity.

You can only divide two quantities of the same dimension.

question id: drill-Dimensions-1

  1. Is this combination dimensionally valid? \[\sqrt{a}\] Why or why not?

Invalid. You cannot have a non-integer exponent on a dimension.

Valid. It is simply 5.

Invalid. 25 feet is not a valid quantity.

question id: drill-Dimensions-2

  1. Consider these quantities: Is this combination dimensionally valid? \[b^c\] Why or why not?

Invalid. Exponentiation of a dimensionful quantity isn’t allowed.

Valid. Exponentiation by a dimensionless integer is always allowed.

Invalid. I cannot make any sense out of T\(^4\) as a dimension.

question id: drill-Dimensions-3

  1. Is this combination dimensionally valid? \[c^b\] Why or why not?

Invalid. Exponentiation by a dimensionful quantity isn’t allowed.

Valid. Exponentiation by a dimensionless integer is always allowed.

Valid. You can do what you want with plain (dimensionless) numbers like 4.

question id: drill-Dimensions-4

  1. Is this combination dimensionally valid? \[\sqrt[3]{a^2 d}\] Why or why not?

Invalid. You cannot raise a dimensionful quantity to a non-integer power.

Valid. \(a^2 d\) is a volume: L\(^3\). The cube root of L\(^3\) is L.

Invalid. 25 feet squared is 625 square feet. It makes no sense to multiply square feet by meters.

question id: drill-Dimensions-5

  1. Is this combination dimensionally valid? \[\exp(a d)\] Why or why not?

Invalid. The input to \(\exp()\) must be a dimensionless quantity.

Valid. The \(a\) cancels out the dimension of the \(b\).

Invalid. 25 foot-meters does not mean anything.

question id: drill-Dimensions-6

  1. Is this combination dimensionally valid? \[\exp(c d)\] Why or why not?

Invalid. The input to \(\exp()\) must be a dimensionless quantity.

Valid. The dimension will be L\(^4\)

Invalid. \(c d\) has dimension L.

question id: drill-Dimensions-7

  1. Is this combination dimensionally valid? \[\exp(c/d)\] Why or why not?

Invalid. The input to \(\exp()\) must be a dimensionless quantity.

Valid. The L dimension of \(c\) is cancelled out by the L\(^{-1}\) dimension of \(1/d\)

Invalid. \(c / d\) has dimension L\(^{-1}\).

question id: drill-Dimensions-8

Exercise 2 For the following few problems, keep in mind these physical quantities and their dimension:

  • [Force] = MLT-2
  • [Distance] = L
  • [Area] = L2
  • [Velocity] = L T-1
  • [Acceleration] = L T-2
  • [Momentum] = M L T-1
  1. Given that [Force] = [Pressure][Area], what is the dimension of Pressure?
M L\(^{-1}\) T\(^{-2}\)       M L\(^{1}\) T\(^{-2}\)       M L\(^{-2}\) T\(^{-1}\)      

question id: drill-Dimensions-9

  1. Which one of the following statements is true?

Momentum = Mass * Velocity

Velocity = Mass / Momentum

Momentum = Mass * Acceleration

question id: drill-Dimensions-10

  1. Which one of the following statements is true?

Volume = Distance * Area

Area = Distance / Volume

Force = Momentum / Acceleration

question id: drill-Dimensions-11

  1. Which of the following is true?

Energy = Distance * Force

Force = Energy / Mass

Energy = Momentum * Acceleration

question id: drill-Dimensions-12

  1. Which of the following is true?

Length = Velocity / Acceleration

Length = Force / Momentum

Area = Velocity * Acceleration

question id: drill-Dimensions-13

  1. Which of the following is true?

Time = Force / Momentum

Length = Force / Momentum

Area = Force / Momentum

Mass = Force / Momentum

question id: drill-Dimensions-14

  1. What kind of thing is \[\sqrt[3]{(\text{4in})(\text{2 ft})(\text{1 mile})}\ ?\]
Length       Area       Volume       It is meaningless      

question id: drill-Dimensions-15

  1. What kind of thing is \[\sin(\pi\ \text{seconds})\ ?\]
Length       1 / Length       The number 0       It is meaningless      

question id: drill-Dimensions-16

  1. If \(t\) is measured in seconds and \(A\) is measured in feet, what will be the dimension of \(A \sin(2\pi t/P)\) when \(P\) is two hours?
L       T       L/T      

question id: drill-Dimensions-17

  1. Engineers often prefer to describe sinusoids in terms of their frequency \(\omega\), writing the function as \(\sin(2 \pi \omega t)\), where \(t\) is time.

    What is the dimension of \(\omega\)?
T\(^{-1}\)       T       T\(^2\)      

question id: drill-Dimensions-18

  1. Suppose \(t\) is measured in hours and \(x\) in yards. What will be the dimension of \(P\) in \[\sin(2\pi t x/P)\ ?\]

There is no such \(P\) that will make a valid input to \(\sin()\)

L T

L / T

T / L

question id: drill-Dimensions-19

Exercise 3  

  1. In mathematics/trigonometry, what is the value of \(\sin(180^\circ)\).
0       \(\sqrt{2}\)       1       -1      

question id: drill-04-24-1

  1. In R/mosaic, what is the value of sin(180)?
-0.80       0       0.80       1      

question id: drill-M04-24-2

:::

No answers yet collected