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### 2004-G: Dimensional Analysis 2: Simplifying Dimensions of Quantities

• Topic Cluster: Pure Mathematics
• Topic: Unit Conversion and Dimensional Analysis
• Objective: Simplify dimensions made of different quantities, such as velocity and force, in numerators and denominators.
• Content:
• Level: 3

#### BACK to Ladder Unit Conversion and Dimensional Analysis

• All unit in physics can be written in terms of a selection of unit called the base unit.
• The three most important base unit are:
• Meters
• Kilograms
• Seconds

Each of these unit can be written in terms of base unit:

Symbol
Quantity
Unit
Special Name of Unit
$$t$$
time
seconds (s)
seconds
$$m$$
mass
kilograms (kg)
kilograms
$$x$$ or $$\Delta x$$
position or displacement
meters (m)
meters
$$v$$
velocity
$$\frac{\text{m}}{\text{s}}$$
none
$$a$$
acceleration
$$\frac{\text{m}}{\text{s}^2}$$
none
$$F$$
Force
$$\text{kg} \cdot \frac{\text{m}}{\text{s}^2}$$
Newtons (N)
$$p$$
momentum
$$\text{kg} \cdot \frac{\text{m}}{\text{s}}$$
none
$$K$$ or $$U$$
Kinetic Energy or Potential Energy
$$\text{kg} \cdot \frac{\text{m}}{\text{s}^2}$$
Joules (J)

For each of the following quantities, write the unit in terms of meters, kilograms, and seconds.

COMING SOON!

#### Practice

For each of the following combinations of quantities, determine the unit in terms of meters, kilograms, and seconds. The steps are:

1. Write each quantity in terms of meters, kilograms, and second.
2. If a quantity is in the numerator, add its unit in the numerator. If it is in the denominator, add its unit in the denominator.
3. If a quantity is raised to some power, raise its unit to that power.
4. Simplify the combination of unit, as in a previous pod.
1. $$\frac{\left( \Delta x \right) a }{v^2}$$
2. $$\frac{\left( \Delta t \right) v^3 }{\left( \Delta x \right) a^2}$$
3. $$\frac{m^2 \left( \Delta t \right)^5}{v^3}$$ Note that $$m$$ stands for mass here, not meters!
4. $$\frac{a^3 v^2}{m^2 \left( \Delta x \right)^4}$$

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