Dimensions and consistencyAQA A-Level Further Maths: Mind map

Exam questions on Dimensions and consistency

1. 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
2. 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
3. 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