Consider a cube. Perhaps a magical or a glowing futuristic cube. You can find the volume of this cube by multiplying the length by width by height. What are the base SI units for volume?
\(\text{m}^3\)
Correct!
\([L]^3\)
Try again. This is the dimensions of volume, but not the base SI units.
\(\text{m}^{-3}\)
Try again. Volume is length multipled by width multiplied by height, this would be in the numerator, not the denominator.
\([L]^{-3}\)
Try again. Volume is length multipled by width multiplied by height, this would be in the numerator, not the denominator. Also, this is notation for dimensions, not base SI units.
Checkpoint3.3.2.
The mass density of an object is defined as the mass divided by the volume. What are the base SI units for mass density?
\(\frac{\text{kg}}{\text{m}^3}\)
Correct!
\(\frac{[M]}{[L]^3}\)
Try again. This is the dimensions of mass density, but not the base SI units.
\(\frac{[L]^3}{[M]}\)
Try again. Mass density is mass divided by volume; mass would be in the numerator and volume in the denominator. Also, this is notation for dimensions, not base SI units.
\(\frac{\text{m}^3}{\text{kg}}\)
Try again. Mass density is mass divided by volume; mass would be in the numerator and volume in the denominator.
Checkpoint3.3.3.
The energy density of a system is defined as the energy divided by the volume. What are the base SI units for energy density?
\(\frac{\text{kg}}{\text{s}^2\text{·m}}\)
Correct!
\(\frac{\text{kg·m}}{\text{s}^2}\)
Try again. Recall energy has base SI units of \(\frac{\text{kg·m}^2}{\text{s}^2}\text{.}\) Energy density is this divided by volume, which has base SI units of \(\text{m}^3\text{.}\) What is \(\frac{\text{m}^2}{\text{m}^3}\text{?}\)
\(\frac{\text{kg·m}^2}{\text{s}^3}\)
Try again. Recall energy has base SI units of \(\frac{\text{kg·m}^2}{\text{s}^2}\text{.}\) Energy density is this divided by volume, which has base SI units of \(\text{m}^3\text{.}\)
\(\frac{\text{s}^2\text{·m}}{\text{kg}}\)
Try again. Oops, you have the energy in the denominator and volume in the numerator.
Checkpoint3.3.4.
The current density in a material is defined as the electric current divided by the area the current is passing through. What are the base SI units for current density?
\(\frac{\text{A}}{\text{m}^2}\)
Correct!
\(\text{A·m}\)
Try again. Recall current has base SI units of \(\text{A}\text{.}\) Current density is this divided by area, which has base SI units of \(\text{m}^2\text{.}\) What is \(\text{A·m}\) compared to \(\frac{\text{A}}{\text{m}^2}\text{?}\)
\(\frac{\text{A}}{\text{m}}\)
Try again. Recall current has base SI units of \(\text{A}\text{.}\) Current density is this divided by area, which has base SI units of \(\text{m}^2\text{.}\)
\(\frac{\text{m}^2}{\text{A}}\)
Try again. Oops, you have the current in the denominator and the area in the numerator.
Checkpoint3.3.5.
Probability is a measure of the likelihood that a particular event will occur. Whether you’re rolling dice, detecting particles, or predicting the path of a rogue comet, probability helps quantify uncertainty. The value of probability ranges from \(0\) (impossible event) to \(1\) (certain event). What are the dimensions of probability?
Dimensionless
Correct! Probability is a pure number with no dimensions. It represents a ratio or likelihood.
\([T]^{-1}\)
Try again. Probability isn’t tied to time. It doesn’t describe a rate or frequency by itself.
\([L]^3\)
Try again. This would describe a volume, not the likelihood of an outcome.
\([L]^{-3}\)
Try again. This would describe some density in three-dimensional space, not the likelihood of an outcome.
Checkpoint3.3.6.
In quantum mechanics, the probability density of an electron tells you how likely you are to find the electron in a particular region of space. Unlike probability itself, probability density spreads that likelihood out over a volume. What are the base dimensions of probability density?
\([L]^{-3}\)
Correct! Probability density represents probability per unit volume.
\([L]^3\)
Try again. This would represent a volume, not a density that spreads probability through space.
Dimensionless
Try again. While total probability is dimensionless, the density must compensate for the volume it’s distributed across.
\([T]^{-1}\)
Try again. Time doesn’t play a direct role in the dimensionality of spatial probability density.
