Worksheet: Dimensional Analysis

In this worksheet, we will practice using dimensional analysis to find the dimensions of unknown quantities and determining whether an equation is dimensionally consistent.

Q1:

Consider the equation 𝑠=𝑠+𝑣𝑑+π‘Žπ‘‘2+𝑗𝑑6+𝑆𝑑24+𝑐𝑑120,οŠͺ where 𝑠 is a length and 𝑑 is a time.

What is the dimension of π‘ οŠ¦?

  • AT
  • BL
  • CLT
  • DLT
  • ETL

What is the dimension of π‘£οŠ¦?

  • AT
  • BTL
  • CLT
  • DLT
  • ELT

What is the dimension of π‘ŽοŠ¦?

  • ALT
  • BLT
  • CTL
  • DLT
  • ELT

What is the dimension of π‘—οŠ¦?

  • ALT
  • BLT
  • CLT
  • DLT
  • ELT

What is the dimension of π‘†οŠ¦?

  • ALT
  • BLTοŠͺ
  • CLT
  • DLTοŠͺ
  • ELTοŠͺ

What is the dimension of 𝑐?

  • ALTοŠͺ
  • BLT
  • CLT
  • DLT
  • ELT

Q2:

Consider the physical quantities 𝑠, 𝑣, π‘Ž, and 𝑑 with dimensions [𝑠]=L, [𝑣]=LT, [π‘Ž]=LT, and [𝑑]=T. Determine whether each of the following equations is dimensionally consistent.

Is 𝑣=2π‘Žπ‘ οŠ¨ dimensionally consistent?

  • AYes
  • BNo

Is 𝑠=𝑣𝑑+0.5π‘Žπ‘‘οŠ¨οŠ¨ dimensionally consistent?

  • AYes
  • BNo

Is 𝑣=𝑠𝑑 dimensionally consistent?

  • AYes
  • BNo

Is π‘Ž=𝑣𝑑 dimensionally consistent?

  • AYes
  • BNo

Q3:

The arc length formula says the length 𝑠 of arc subtended by angle πœƒ in a circle of radius π‘Ÿ is given by the equation 𝑠=π‘Ÿπœƒ.

What are the dimensions of 𝑠?

  • AL
  • BL0
  • CL2
  • DLβˆ’2
  • ELβˆ’1

What are the dimensions of π‘Ÿ?

  • AL0
  • BL2
  • CLβˆ’1
  • DL
  • ELβˆ’2

What are the dimensions of πœƒ?

  • AL2
  • BL
  • CL0
  • DLβˆ’2
  • ELβˆ’1

Q4:

The quantity 𝑠 (displacement) has the dimension L and the quantity 𝑑 (time) has the dimension T. Suppose that the quantity 𝑣 is defined as the derivative of 𝑠 with respect to time and that the quantity π‘Ž is defined as the derivative of 𝑣 with respect to time.

What are the dimensions of 𝑣?

  • ALT
  • BL
  • CLT
  • DLT
  • ELT

What are the dimensions of π‘Ž?

  • ALT
  • BLT
  • CLT
  • DL
  • ELT

What are the dimensions of 𝑣𝑑d?

  • AL
  • BLT
  • CLT
  • DLT
  • ELT

What are the dimensions of ο„Έπ‘Žπ‘‘d?

  • ALT
  • BLT
  • CLT
  • DL
  • ELT

What are the dimensions of the derivative of π‘Ž with respect to time?

  • ALT
  • BLT
  • CL
  • DLT
  • ELT

Q5:

A student is trying to remember a formula from geometry. Assuming that 𝐴 corresponds to area, 𝑉 corresponds to volume, and all other variables are lengths, what missing dimension in the formula 𝑉=𝐴 must be the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

  • A𝐿
  • B𝐿
  • C𝐿
  • D𝐿
  • E𝐿

Q6:

A student is trying to remember a formula from geometry. Assuming that 𝐴 corresponds to area, 𝑉 corresponds to volume, and all other variables are lengths, what missing terms in the formula 𝑉=4𝑝𝑖3π‘Ÿ must the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

  • A𝐿
  • B𝐿
  • C𝐿
  • D𝐿
  • E𝐿οŠͺ

Q7:

The diameter of the fireball of a nuclear explosion is approximated to the following formula: 𝐷=π‘˜(𝑃𝑑)πœŒπ‘‘οŒΊοŒ»οŒΌ. Where 𝑃 is the average power and has the dimension [𝑃]=π‘€πΏπ‘‡οŠ¨οŠ±οŠ§, 𝜌 is the average air density of 1.225 kg/m3, 𝑑 is time from the start of the explosion, and π‘˜,π‘Ž,𝑏, and 𝑐 are constants. Observations show that if π‘˜=2.06, the fireball has a diameter of 260 m after it has been expanding for 25.0 ms.

Using dimensional analysis, find the values of π‘Ž,𝑏, and 𝑐. Assume that π‘˜=1.0.

  • A13,βˆ’13,23
  • B12,13,14
  • Cβˆ’1,βˆ’2,23
  • D15,βˆ’15,25
  • Eβˆ’15,35,15

Find the initial energy release of the explosion in joules (J).

  • A6.3Γ—10 J
  • B63Γ—10 J
  • C630Γ—10 J
  • D6.3Γ—10 J
  • E6.3Γ—10 J

The energy released in large explosions is often cited in units of β€œtons of TNT” abbreviated β€œπ‘‘ TNT”, where 1 𝑑 TNT is about 4.2 GJ. Find the initial energy release of the explosion in kilotons of TNT.

  • A150 kilotons of TNT
  • B1.5 kilotons of TNT
  • C13 kilotons of TNT
  • D15 kilotons of TNT
  • E130 kilotons of TNT

Q8:

A student is trying to remember a formula from geometry. Assuming that 𝐴 corresponds to area, 𝑉 corresponds to volume, and all other variables are lengths, what missing terms in the formula 𝐴=4πœ‹ must the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

  • A𝐿
  • B𝐿
  • C𝐿
  • D𝐿
  • E𝐿

Q9:

Consider the physical quantities π‘š, 𝑠, 𝑣, π‘Ž, and 𝑑, with dimensions [π‘š]=𝑀, [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equation 𝑇=π‘šπ‘ π‘Ž is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • AπΏπ‘‡οŠ¨οŠ±οŠ¨
  • Bπ‘€πΏπ‘‡οŠ±οŠ§
  • Cπ‘€π‘‡οŠ±οŠ¨
  • Dπ‘€π‘‡οŠ¨
  • E𝑀𝐿𝑇

Q10:

Suppose [𝐴]=𝐿,[𝜌]=π‘€πΏοŠ¨β€“οŠ©, and [𝑑]=𝑇.

What is the dimension of ο„ΈπœŒπ΄d?

  • A𝑀𝐿𝑇
  • Bπ‘€πΏοŠ¨
  • Cπ‘€πΏοŠ±οŠ¨
  • Dπ‘€πΏοŠ±οŠ§
  • E𝑀𝐿

What is the dimension of dd𝐴𝑑?

  • A𝐿𝑇
  • BπΏπ‘‡οŠ±οŠ¨οŠ§
  • CπΏπ‘‡οŠ¨οŠ¨
  • DπΏπ‘‡οŠ¨οŠ±οŠ§
  • EπΏπ‘‡οŠ±οŠ§οŠ¨

What is the dimension of 𝐴𝑑dd?

  • A𝑀𝐿𝑇
  • Bπ‘€πΏπ‘‡οŠ±οŠ§οŠ±οŠ§
  • Cπ‘€πΏπ‘‡οŠ±οŠ¨
  • Dπ‘€πΏπ‘‡οŠ±οŠ§οŠ±οŠ§
  • Eπ‘€πΏπ‘‡οŠ¨

Q11:

Consider the physical quantities π‘š,𝑠,𝑣,π‘Ž, and 𝑑, with dimensions [π‘š]=𝑀, [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equation 𝑃=π‘£π‘Ž is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • AπΏπ‘‡οŠ¨οŠ±οŠ©
  • BπΏπ‘‡οŠ©
  • CπΏπ‘‡οŠ±οŠ©
  • DπΏπ‘‡οŠ¨
  • E𝐿𝑇

Q12:

Consider the physical quantities π‘š,𝑠,𝑣,π‘Ž, and 𝑑, with dimensions [π‘š]=𝑀, [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equation 𝐾=3π‘šπ‘ οŠ¨ is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • Aπ‘€πΏοŠ¨
  • B𝑀𝐿
  • C3𝑀𝐿
  • Dπ‘€πΏοŠ¨
  • E3π‘€πΏοŠ¨

Q13:

Consider the physical quantities π‘š,𝑠,𝑣,π‘Ž, and 𝑑, with dimensions [π‘š]=𝑀, [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equation π‘Š=π‘šπ‘Ž2𝑠 is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • A12π‘€π‘‡οŠ±οŠ¨
  • Bπ‘€π‘‡οŠ±οŠ¨
  • C12π‘€πΏπ‘‡οŠ±οŠ§
  • D𝑀𝑇
  • Eπ‘€πΏπ‘‡οŠ±οŠ§

Q14:

Consider the physical quantities π‘š,𝑠,𝑣,π‘Ž, and 𝑑, with dimensions [π‘š]=𝑀, [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equation 𝐿=π‘£π‘ οŠ© is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • AπΏπ‘‡οŠ©οŠ±οŠ§
  • B𝐿𝑇οŠͺ
  • CπΏπ‘‡οŠ±οŠ§οŠͺ
  • DπΏπ‘‡οŠ±οŠ©
  • E𝐿𝑇

Q15:

Consider the equation π‘₯=π‘šπ‘£+𝑐, where the dimension of π‘₯ is length and the dimension of 𝑣 is length per time, and π‘š and 𝑐 are constants.

What is the dimension of π‘š?

  • AMass per Time
  • BLength
  • CTime
  • DLength per time
  • EMass

What is the SI unit of π‘š?

  • Am2
  • Bm/s
  • Cs
  • Dm
  • Em/s2

What is the dimension of 𝑐?

  • ALength
  • BLength per time squared
  • CLength per time
  • DTime
  • EMass

What is the SI unit of 𝑐?

  • Am
  • Bm/s2
  • Cm2
  • Dm/s
  • Es

Q16:

The physical quantities 𝑠, 𝑣, π‘Ž, and 𝑑 have the dimensions [𝑠]=𝐿, [𝑣]=πΏπ‘‡οŠ±οŠ§, [π‘Ž]=πΏπ‘‡οŠ±οŠ¨, and [𝑑]=𝑇. The equations equation 1, equation 2, and equation 3 are 𝑠=𝑣𝑑+0.5π‘Žπ‘‘οŠ¨, 𝑠=𝑣𝑑+0.5π‘Žπ‘‘οŠ¨, and 𝑣=ο€Ύπ‘Žπ‘‘π‘ οŠsin respectively.

Is equation 1 dimensionally consistent?

  • Ayes
  • Bno

Is equation 2 dimensionally consistent?

  • Ayes
  • Bno

Is equation 3 dimensionally consistent?

  • Ano
  • Byes

Q17:

What is the dimension of the quantity surface tension?

  • AM2Lβˆ’2
  • BMβˆ’1Tβˆ’2
  • CM2Lβˆ’1T
  • DM Lβˆ’1Tβˆ’2
  • EM Lβˆ’1

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