Objectives
- Reinforce your ability to identify SI units and dimensions in real-world and scientific contexts.
Section 3.2 Explore
Worksheet 3.2.1 Explore: Units and Dimensions
This worksheet is designed to help you explore the concept of units and dimensions in physics.
1.
The quantity position has the following dimensions: \([L]\text{.}\) Position is a measure of how far and in what direction an object is from a reference point. What is the SI unit of position?
Hint.
[L] stands for length.]
Answer.
m, where m is the symbol for meters.
Solution.
Replace [L] with the SI unit symbol for length, m.
So, the SI unit of position is m, where m is the symbol for meters.
2.
The quantity velocity has the following dimensions: \([L][T]^{-1}\text{.}\) Velocity is a measure of how fast an object is moving and in what direction. What is the SI unit of velocity?
Hint.
[L] stands for length, and [T] stands for time. Also, the exponent -1 means that the unit for time is in the denominator (i.e. \([T]^{-1} = \frac{1}{[T]}\))
Answer.
m/s, where m is the symbol for meters and s is the symbol for seconds.
Solution.
Replace [L] with the SI unit for length, which is m, and [T] with the SI unit for time, which is s.
The unit becomes \(\text{m/s}\text{,}\) which is meters per second.
3.
The quantity acceleration has the following dimensions: \([L][T]^{-2}\text{.}\) Acceleration is a measure of how quickly an object changes its velocity and in what direction. What is the SI unit of acceleration?
Hint.
[L] stands for length, and [T] stands for time. Also, the exponent -2 means that the unit for time is in the denominator (i.e. \([T]^{-2} = \frac{1}{[T]^2}\))
Answer.
m/s², where m is the symbol for meters and s is the symbol for seconds.
Solution.
Replace [L] with m and [T] with s.
The unit becomes \(\text{m/s}^2\text{,}\) which is meters per second squared.
4.
The quantity force has the following dimensions: \([M][L][T]^{-2}\text{.}\) Force is a measure of how hard and in what direction an object pushs or pulls on another object. What is the SI unit of force?
Hint.
[M] stands for mass, [L] for length, and [T] for time. Also, the exponent -2 means that the unit for time is in the denominator (i.e. \([T]^{-2} = \frac{1}{[T]^2}\))
Answer.
kg·m/s², where kg is the symbol for kilogram, m is the symbol for meter, and s is the symbol for second.
Solution.
Replace [M] with kg, [L] with m, and [T] with s.
The unit becomes \(\text{kg·m/s}^2\text{,}\) which is kilogram multipled by meter per second.
5.
The quantity energy has the following SI units: \(\frac{kg \, m^2}{s^2}\text{.}\) Energy can’t be summarized by a simple sentence. However, it’s related to many things in our everyday lives, such as the energy stored in food that our body needs to function, the energy stored in batteries that our electronic devices need to function, and even energy stored in the vacuum of space itself! What is the base dimensions of energy?
Hint.
Look at the units: kg (kilogram) is mass, m (meter) is length, and s (second) is time.
Answer.
\([M][L]^2[T]^{-2}\)
Solution.
From the units \(\frac{kg \, m^2}{s^2}\text{,}\) we identify:
- kg → \([M]\)
- \(m^2\) → \([L]^2\)
- \(s^2\) in denominator → \([T]^{-2}\)
So, the base dimensions of energy are \([M][L]^2[T]^{-2}\text{.}\)
6.
The quantity power has the following SI units: \(\frac{kg \, m^2}{s^3}\text{.}\) Power is the rate of energy use or transfer, or in other words power is energy per time. It’s how fast your starship engines burn fuel or how quickly your phone draws energy from an energy source. What is the base dimensions of power?
Hint.
Look at the units: kg (kilogram) is mass, m (meter) is length, and s (second) is time.
Answer.
\([M][L]^2[T]^{-3}\)
Solution.
From the units \(\frac{kg \, m^2}{s^3}\text{,}\) we identify:
- kg → \([M]\)
- \(m^2\) → \([L]^2\)
- \(s^3\) in denominator → \([T]^{-3}\)
So, the base dimensions of power are \([M][L]^2[T]^{-3}\text{.}\)
7.
The molar gas constant \(R\) has SI units of \(\frac{kg \, m^2}{s^2 \, mol \, K}\text{.}\) The molar gas constant connects microscopic energy of molecules and macroscopic quantities like pressure, volume, and temperature - bridging the gap between the tiny molecular world and measurable gas behavior. What is the base dimensions of the molar gas constant?
Hint.
Look at the units: kg (kilogram) is mass, m (meter) is length, s (second) is time, mol (mole) is amount of substance, and K (kelvin) is thermodynamic temperature.
Answer.
\([M][L]^2[T]^{-2}[n]^{-1}[\Theta]^{-1}\)
Solution.
From the units \(\frac{kg \, m^2}{s^2 \, mol \, K}\text{,}\) we identify:
- kg → \([M]\)
- \(m^2\) → \([L]^2\)
- \(s^2\) in denominator → \([T]^{-2}\)
- \(mol\) in denominator → \([n]^{-1}\)
- \(K\) in denominator → \([\Theta]^{-1}\)
So, the base dimensions of the molar gas constant are \([M][L]^2[T]^{-2}[n]^{-1}[\Theta]^{-1}\text{.}\)
8.
The quantity electric charge has SI units of \(A \cdot s\text{.}\) Charge is a fundamental property of matter that governs how particles respond to electromagnetic fields and how they produce those fields themselves. Every lightning bolt, every battery, and every cat rubbing against a sweater has charge involved. What is the base dimensions of electric charge?
Hint.
Look at the units: A (ampere) is electric current, and s (second) is time.
Answer.
\([I][T]\)
Solution.
From the units \(A \cdot s\text{,}\) we identify:
- A → \([I]\)
- S → \([T]\)
So, the base dimensions of charge are \([I][T]\text{.}\)
