Slide 1
Equations must compare compatible quantities
Dimensional analysis checks whether an equation combines quantities consistently. A length cannot equal a time without an appropriate conversion relationship.
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Dimensions describe kinds of quantity, such as length or mass, while units express their chosen scale. Metres and centimetres have the same dimension even though their numerical values differ for one object.
Slide 2
Follow the units through multiplication
Multiplying speed by time gives length because the time units cancel. Dividing distance by time instead gives speed.
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For 3 metres per second over 4 seconds, the product is 12 metres. Keeping units beside numbers can expose an accidental division or missing factor before a numerical answer looks deceptively plausible.
Slide 3
Addition needs matching dimensions
Terms added in a physical equation must represent compatible kinds of quantity. A conversion may be needed before numerical values can be added.
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Adding 2 metres and 30 centimetres gives 2.3 metres, not 32 metres. The quantities are compatible, but their numerical values cannot be combined until both use the same unit scale.
Slide 4
A correct dimension is only a first test
An equation can have consistent dimensions and still be physically wrong. Dimensionless factors and missing dependencies may escape the check.
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Both mass times speed squared and half that expression have energy dimensions. Dimensional analysis alone cannot determine the one-half factor in classical kinetic energy; additional physics supplies it.
Slide 5
Scaling can guide a model
Dimensions can suggest combinations of variables worth investigating. They help organize experiments without replacing the evidence needed to choose a relationship.
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Dimensionless ratios allow comparisons across different sizes of a system. Matching the relevant ratios can make a scale model informative, but omitted physical effects may introduce additional ratios that also need control.