# 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.

**Q2: **

Consider the physical quantities , , , and with dimensions , , , and . Determine whether each of the following equations is dimensionally consistent.

Is dimensionally consistent?

- AYes
- BNo

Is dimensionally consistent?

- AYes
- BNo

Is dimensionally consistent?

- AYes
- BNo

Is dimensionally consistent?

- AYes
- BNo

**Q3: **

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

What are the dimensions of ?

- AL
- BL
^{0} - CL
^{2} - DL
^{−2} - EL
^{−1}

What are the dimensions of ?

- AL
^{0} - BL
^{2} - CL
^{−1} - DL
- EL
^{−2}

What are the dimensions of ?

- AL
^{2} - BL
- CL
^{0} - DL
^{−2} - EL
^{−1}

**Q4: **

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

What are the dimensions of ?

- A
- B
- C
- D
- E

What are the dimensions of ?

- A
- B
- C
- D
- E

What are the dimensions of ?

- A
- B
- C
- D
- E

What are the dimensions of ?

- A
- B
- C
- D
- E

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

- A
- B
- C
- D
- E

**Q5: **

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

- A
- B
- C
- D
- E

**Q6: **

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

- A
- B
- C
- D
- E

**Q7: **

The diameter of the fireball of a nuclear explosion is approximated to the following formula:
.
Where is the average power and has the dimension ,
is the average air density of
1.225 kg/m^{3},
is time from the start of the explosion,
and , and are constants.
Observations show that if ,
the fireball has a diameter of 260 m
after it has been expanding for 25.0 ms.

Using dimensional analysis, find the values of , and . Assume that .

- A
- B
- C
- D
- E

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

- A J
- B J
- C J
- D J
- E J

The energy released in large explosions is often cited in units of “tons of TNT” abbreviated “ TNT”, where 1 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 corresponds to area, corresponds to volume, and all other variables are lengths, what missing terms in the formula must the right-hand side of the formula be multiplied by to make the formula dimensionally consistent?

- A
- B
- C
- D
- E

**Q15: **

Consider the equation , where the dimension of is length and the dimension of is length per time, and and are constants.

What is the dimension of ?

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

What is the SI unit of ?

- Am
^{2} - Bm/s
- Cs
- Dm
- Em/s
^{2}

What is the dimension of ?

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

What is the SI unit of ?

- Am
- Bm/s
^{2} - Cm
^{2} - Dm/s
- Es