Dimensions of work, energy and power?

Work and energy: ML^{2}T^{-2}. Power: ML^{2}T^{-3}.

Dimensions of pressure and density?

Pressure: ML^{-1}T^{-2}. Density: ML^{-3}.

What does dimensionless mean?

All powers of M, L and T are zero, e.g. angles, ratios, the coefficient of restitution.

When is an equation dimensionally consistent?

Both sides, and every added or subtracted term, have the same dimensions.

Can a dimensionally consistent formula still be wrong?

Yes. It cannot detect wrong numerical factors, e.g. s=ut+at^2.

What dimensions must the argument of \sin or \ln have?

None: it must be dimensionless.

Method to predict x=\lambda a^{p}b^{q}?

Substitute dimensions, equate the powers of M, L and T on both sides, solve for the powers.

Dimensions and consistencyAQA A-Level Further Maths: Flashcards

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What are the three base dimensions in mechanics?

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What are the three base dimensions in mechanics?: Mass $M$ , length $L$ and time $T$ .
Dimensions of velocity?: $LT^{-1}$
Dimensions of acceleration?: $LT^{-2}$
Dimensions of force?: $MLT^{-2}$ , from $F=ma$ .
Dimensions of momentum and impulse?: $MLT^{-1}$
Dimensions of work, energy and power?: Work and energy: $ML^{2}T^{-2}$ . Power: $ML^{2}T^{-3}$ .
Dimensions of pressure and density?: Pressure: $ML^{-1}T^{-2}$ . Density: $ML^{-3}$ .
What does dimensionless mean?: All powers of $M$ , $L$ and $T$ are zero, e.g. angles, ratios, the coefficient of restitution.
When is an equation dimensionally consistent?: Both sides, and every added or subtracted term, have the same dimensions.
Can a dimensionally consistent formula still be wrong?: Yes. It cannot detect wrong numerical factors, e.g. $s=ut+at^2$ .
Dimensions of $G$ in $F=\frac{Gm_1m_2}{r^2}$ ?: $M^{-1}L^{3}T^{-2}$
Dimensions of the spring stiffness $k$ in $F=kx$ ?: $MT^{-2}$
What dimensions must the argument of $\sin$ or $\ln$ have?: None: it must be dimensionless.
Method to predict $x=\lambda a^{p}b^{q}$ ?: Substitute dimensions, equate the powers of $M$ , $L$ and $T$ on both sides, solve for the powers.

Exam questions on Dimensions and consistency

A particle of mass $m$ moves in a circle of radius $r$ with constant speed $v$ , and the quantity $Q=\frac{mv^2}{r}$ is calculated. In this question $M$ , $L$ and $T$ denote the dimensions of mass, length and time.
Show that $Qr$ has the same dimensions as kinetic energy $\frac12mv^2$ .2 marks
A mass $m$ is attached to a spring of stiffness $k$ . When the spring is extended by $x$ the tension in it is $F=kx$ . In this question $M$ , $L$ and $T$ denote the dimensions of mass, length and time.
Show that $\sqrt{\dfrac{m}{k}}$ has the dimensions of time.2 marks
The gravitational force between two particles of masses $m_1$ and $m_2$ , a distance $r$ apart, has magnitude $F=\dfrac{Gm_1m_2}{r^2}$ , where $G$ is the universal gravitational constant. In this question $M$ , $L$ and $T$ denote the dimensions of mass, length and time.
Find the dimensions of $G$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).