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In this lesson, we will learn how to use dimensional analysis to find the dimensions of unknown quantities and determine whether an equation is dimensionally consistent.

Q1:

Consider the equation where π is a length and π‘ is a time.

What is the dimension of π 0 ?

What is the dimension of π£ 0 ?

What is the dimension of π 0 ?

What is the dimension of π 0 ?

What is the dimension of π 0 ?

What is the dimension of π ?

Q2:

Consider the physical quantities π , π£ , π , and π‘ with dimensions [ π ] = L , [ π£ ] = L T β 1 , [ π ] = L T β 2 , and [ π‘ ] = T . Determine whether each of the following equations is dimensionally consistent.

Is π£ = 2 π π 2 dimensionally consistent?

Is π = π£ π‘ + 0 . 5 π π‘ 2 2 dimensionally consistent?

Is π£ = π π‘ dimensionally consistent?

Is π = π£ π‘ dimensionally consistent?

Q3:

The arc length formula says the length π of arc subtended by angle π in a circle of radius π is given by the equation π = π π .

What are the dimensions of π ?

What are the dimensions of π ?

What are the dimensions of π ?

Q4:

Consider the physical quantities π , π , π£ , π , and π‘ , with dimensions [ π ] = π , [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equation πΎ = 3 π π 2 is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

Q5:

Suppose [ π΄ ] = πΏ , [ π ] = π πΏ 2 β 3 , and [ π‘ ] = π .

What is the dimension of οΈ π π΄ d ?

What is the dimension of d d π΄ π‘ ?

What is the dimension of οΈ ο½ π΄ π‘ ο d d ?

Q6:

What is the dimension of the quantity surface tension?

Q7:

A student is trying to remember a formula from geometry. Assuming that π΄ corresponds to area, π corresponds to volume, and all other variables are lengths, what missing terms in the formula π = 4 π 3 π must be the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

Q8:

Consider the physical quantities π , π , π£ , π , and π‘ , with dimensions [ π ] = π , [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equation π = π π 2 π is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

Q9:

The physical quantities π , π£ , π , and π‘ have the dimensions [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equations equation 1, equation 2, and equation 3 are π = π£ π‘ + 0 . 5 π π‘ 2 , π = π£ π‘ + 0 . 5 π π‘ 2 , and π£ = οΎ π π‘ π ο s i n 2 respectively.

Is equation 1 dimensionally consistent?

Is equation 2 dimensionally consistent?

Is equation 3 dimensionally consistent?

Q10:

Consider the physical quantities π , π , π£ , π , and π‘ , with dimensions [ π ] = π , [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equation πΏ = π£ π 3 is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

Q11:

The diameter of the fireball of a nuclear explosion is approximated to the following formula: π· = π ( π π‘ ) π π‘ π π π . Where π is the average power and has the dimension [ π ] = π πΏ π 2 β 1 , π is the average air density of 1.225 kg/m^{3}, π‘ is time from the start of the explosion, and π , π , π , and π 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 , π , and π . Assume that π = 1 . 0 .

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

The energy released in large explosions is often cited in units of βtons of TNTβ abbreviated β π‘ TNTβ, where 1 π‘ TNT is about 4.2 GJ. Find the initial energy release of the explosion in kilotons of TNT.

Q12:

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

What are the dimensions of π£ ?

What are the dimensions of π ?

What are the dimensions of οΈ π£ π‘ d ?

What are the dimensions of οΈ π π‘ d ?

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

Q13:

Consider the physical quantities π , π , π£ , π , and π‘ , with dimensions [ π ] = π , [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equation π = π π π is dimensionally consistent. find the dimension of the quantity on the left-hand side of the equation.

Q14:

Consider the physical quantities π , π , π£ , π , and π‘ , with dimensions [ π ] = π , [ π ] = πΏ , [ π£ ] = πΏ π β 1 , [ π ] = πΏ π β 2 , and [ π‘ ] = π . The equation π = π£ π is dimensionally consistent. Find the dimension of the quantity on the left-hand side of the equation.

Q15:

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

Q16:

Consider the equation π₯ = π π£ + π , where the dimension of π₯ is length and the dimension of π£ is length per time, and π and π are constants.

What is the dimension of π ?

What is the SI unit of π ?

What is the SI unit of π ?

Q17:

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

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