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 + 𝑣 𝑑 + π‘Ž 𝑑 2 + 𝑗 𝑑 6 + 𝑆 𝑑 2 4 + 𝑐 𝑑 1 2 0 ,        οŠͺ  where 𝑠 is a length and 𝑑 is a time.

What is the dimension of 𝑠  ?

  • A L T
  • B L
  • C L T  
  • D T
  • E T L  

What is the dimension of 𝑣  ?

  • A L T
  • B L T  
  • C T
  • D T L  
  • E L T  

What is the dimension of π‘Ž  ?

  • A L T    
  • B T L  
  • C L T 
  • D L T
  • E L T  

What is the dimension of 𝑗  ?

  • A L T 
  • B L T 
  • C L T   
  • D L T  
  • E L T  

What is the dimension of 𝑆  ?

  • A L T   
  • B L T οŠͺ
  • C L T    
  • D L T οŠͺ
  • E L T  οŠͺ

What is the dimension of 𝑐 ?

  • A L T   
  • B L T    
  • C L T οŠͺ 
  • D L T    
  • E 2 L T  

Q2:

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

Is 𝑣 = 2 π‘Ž 𝑠  dimensionally consistent?

  • AYes
  • BNo

Is 𝑠 = 𝑣 𝑑 + 0 . 5 π‘Ž 𝑑   dimensionally consistent?

  • AYes
  • BNo

Is 𝑣 = 𝑠 𝑑 dimensionally consistent?

  • AYes
  • BNo

Is π‘Ž = 𝑣 𝑑 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‘Ÿ πœƒ .

What are the dimensions of 𝑠 ?

  • ALβˆ’1
  • BLβˆ’2
  • CL2
  • DL0
  • EL

What are the dimensions of π‘Ÿ ?

  • AL
  • BL0
  • CLβˆ’2
  • DLβˆ’1
  • EL2

What are the dimensions of πœƒ ?

  • ALβˆ’2
  • BLβˆ’1
  • CL0
  • DL2
  • EL

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 ?

  • A L T  
  • B L
  • C L T  
  • D L T  
  • E L T

What are the dimensions of π‘Ž ?

  • A L
  • B L T
  • C L T  
  • D L T  
  • E L T  

What are the dimensions of ο„Έ 𝑣 𝑑 d ?

  • A L T  
  • B L T  
  • C L T  
  • D L T
  • E L

What are the dimensions of ο„Έ π‘Ž 𝑑 d ?

  • A L T
  • B L
  • C L T  
  • D L T  
  • E L T  

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

  • A L T  
  • B L T  
  • C L T
  • D L T  
  • E L

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?

  • A 𝐿 
  • B 𝐿
  • C 𝐿  
  • D 𝐿  
  • E 𝐿 

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 = 4 πœ‹ 3 π‘Ÿ must be the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

  • A 𝐿  
  • B 𝐿 
  • C 𝐿 οŠͺ
  • D 𝐿  
  • E 𝐿 

Q7:

The diameter of the fireball of a nuclear explosion is approximated to the following formula: 𝐷 = π‘˜ ( 𝑃 𝑑 ) 𝜌 𝑑    . Where 𝑃 is the average power and has the dimension [ 𝑃 ] = 𝑀 𝐿 𝑇    , 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 . 0 6 , 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 .

  • A 1 5 , βˆ’ 1 5 , 2 5
  • B βˆ’ 1 5 , 3 5 , 1 5
  • C βˆ’ 1 , βˆ’ 2 , 2 3
  • D 1 3 , βˆ’ 1 3 , 2 3
  • E 1 2 , 1 3 , 1 4

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

  • A 6 . 3 Γ— 1 0   J
  • B 6 . 3 Γ— 1 0   J
  • C 6 . 3 Γ— 1 0   J
  • D 6 3 0 Γ— 1 0   J
  • E 6 3 Γ— 1 0   J

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

  • A 1.5 kilotons of TNT
  • B 130 kilotons of TNT
  • C 150 kilotons of TNT
  • D 13 kilotons of TNT
  • E 15 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 = 4 πœ‹ must the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

  • A 𝐿  
  • B 𝐿
  • C 𝐿 
  • D 𝐿  
  • E 𝐿 

Q9:

Consider the physical quantities π‘š , 𝑠 , 𝑣 , π‘Ž , and 𝑑 , with dimensions [ π‘š ] = 𝑀 , [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equation 𝑇 = π‘š 𝑠 π‘Ž is dimensionally consistent. find the dimension of the quantity on the left-hand side of the equation.

  • A 𝐿 𝑇   
  • B 𝑀 𝐿 𝑇
  • C 𝑀 𝑇 
  • D 𝑀 𝐿 𝑇  
  • E 𝑀 𝑇  

Q10:

Suppose [ 𝐴 ] = 𝐿 , [ 𝜌 ] = 𝑀 𝐿  –  , and [ 𝑑 ] = 𝑇 .

What is the dimension of ο„Έ 𝜌 𝐴 d ?

  • A 𝑀 𝐿  
  • B 𝑀 𝐿 𝑇
  • C 𝑀 𝐿
  • D 𝑀 𝐿 
  • E 𝑀 𝐿  

What is the dimension of d d 𝐴 𝑑 ?

  • A 𝐿 𝑇   
  • B 𝐿 𝑇   
  • C 𝐿 𝑇
  • D 𝐿 𝑇   
  • E 𝐿 𝑇  

What is the dimension of ο„Έ ο€½ 𝐴 𝑑  d d ?

  • A 𝑀 𝐿 𝑇 
  • B 𝑀 𝐿 𝑇    
  • C 𝑀 𝐿 𝑇  
  • D 𝑀 𝐿 𝑇    
  • E 𝑀 𝐿 𝑇

Q11:

Consider the physical quantities π‘š , 𝑠 , 𝑣 , π‘Ž , and 𝑑 , with dimensions [ π‘š ] = 𝑀 , [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equation 𝑃 = 𝑣 π‘Ž is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • A 𝐿 𝑇 
  • B 𝐿 𝑇
  • C 𝐿 𝑇   
  • D 𝐿 𝑇  
  • E 𝐿 𝑇 

Q12:

Consider the physical quantities π‘š , 𝑠 , 𝑣 , π‘Ž , and 𝑑 , with dimensions [ π‘š ] = 𝑀 , [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equation 𝐾 = 3 π‘š 𝑠  is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • A 3 𝑀 𝐿 
  • B 𝑀 𝐿 
  • C 3 𝑀 𝐿
  • D 𝑀 𝐿 
  • E 𝑀 𝐿

Q13:

Consider the physical quantities π‘š , 𝑠 , 𝑣 , π‘Ž , and 𝑑 , with dimensions [ π‘š ] = 𝑀 , [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equation π‘Š = π‘š π‘Ž 2 𝑠 is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • A 𝑀 𝑇  
  • B 𝑀 𝑇
  • C 𝑀 𝐿 𝑇  
  • D 1 2 𝑀 𝑇  
  • E 1 2 𝑀 𝐿 𝑇  

Q14:

Consider the physical quantities π‘š , 𝑠 , 𝑣 , π‘Ž , and 𝑑 , with dimensions [ π‘š ] = 𝑀 , [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equation 𝐿 = 𝑣 𝑠  is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

  • A 𝐿 𝑇  
  • B 𝐿 𝑇   οŠͺ
  • C 𝐿 𝑇
  • D 𝐿 𝑇 οŠͺ  
  • E 𝐿 𝑇   

Q15:

Consider the equation π‘₯ = π‘š 𝑣 + 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
  • BLength per time
  • CTime
  • DMass per Time
  • ELength

What is the SI unit of π‘š ?

  • A m/s
  • B s
  • C m
  • D m2
  • E m/s2

What is the dimension of 𝑐 ?

  • ALength per time
  • BMass
  • CLength per time squared
  • DLength
  • ETime

What is the SI unit of 𝑐 ?

  • A m/s2
  • B s
  • C m2
  • D m
  • E m/s

Q16:

The physical quantities 𝑠 , 𝑣 , π‘Ž , and 𝑑 have the dimensions [ 𝑠 ] = 𝐿 , [ 𝑣 ] = 𝐿 𝑇   , [ π‘Ž ] = 𝐿 𝑇   , and [ 𝑑 ] = 𝑇 . The equations equation 1, equation 2, and equation 3 are 𝑠 = 𝑣 𝑑 + 0 . 5 π‘Ž 𝑑  , 𝑠 = 𝑣 𝑑 + 0 . 5 π‘Ž 𝑑  , and 𝑣 = ο€Ύ π‘Ž 𝑑 𝑠  s i n  respectively.

Is equation 1 dimensionally consistent?

  • Ayes
  • Bno

Is equation 2 dimensionally consistent?

  • Ano
  • Byes

Is equation 3 dimensionally consistent?

  • Ayes
  • Bno

Q17:

What is the dimension of the quantity surface tension?

  • AM Lβˆ’1Tβˆ’2
  • BM Lβˆ’1
  • CMβˆ’1Tβˆ’2
  • DM2Lβˆ’2
  • EM2Lβˆ’1T

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