Dimensions and consistencyAQA A-Level Further Maths: Subtopic test

1. 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.

   (a)

   What are the dimensions of $\frac{v^2}{r}$?

   [1 mark]

   • A$L^{2}T^{-2}$
   • B$LT^{-2}$
   • C$LT^{-1}$
   • D$L^{3}T^{-2}$

   (b)

   What are the dimensions of $Q$?

   [1 mark]

   • A$ML^{2}T^{-2}$
   • B$MLT^{-1}$
   • C$MLT^{-2}$
   • D$MT^{-2}$

   (c)

   Show that $Qr$ has the same dimensions as kinetic energy $\frac12mv^2$.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   What are the dimensions of $k$?

   [1 mark]

   • A$MT^{-2}$
   • B$MLT^{-2}$
   • C$ML^{-1}T^{-2}$
   • D$ML^{2}T^{-2}$

   (b)

   What are the dimensions of $\frac{k}{m}$?

   [1 mark]

   • A$T^{-1}$
   • B$T^{2}$
   • C$MT^{-2}$
   • D$T^{-2}$

   (c)

   Show that $\sqrt{\dfrac{m}{k}}$ has the dimensions of time.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Find the dimensions of $G$.

   [3 marks]

   (b)

   A satellite orbits a planet of mass $M_P$ in a circle of radius $r$. Its orbital period $\tau$ is thought to depend only on $G$, $M_P$ and $r$, so that $\tau=\lambda G^{a}M_P^{b}r^{c}$, where $\lambda$ is a dimensionless constant. Use dimensional analysis to find $a$, $b$ and $c$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle of mass $m$ is released from rest and falls vertically through air, where $g$ is the acceleration due to gravity. When its speed is $v$ the air resistance has magnitude $kv^2$, where $k$ is a constant. The particle approaches a terminal speed $V$. In this question $M$, $L$ and $T$ denote the dimensions of mass, length and time.

   (a)

   (i) Find the dimensions of $k$.
   (ii) Determine, showing your working, which of $V=\sqrt{\dfrac{mg}{k}}$ and $V=\dfrac{mg}{k}$ is dimensionally consistent.

   [6 marks]

   (b)

   The time $\tau$ taken for the particle to reach half of its terminal speed is thought to depend only on $m$, $g$ and $k$, so that $\tau=\mu\,m^{a}g^{b}k^{d}$, where $\mu$ is a dimensionless constant. Find $a$, $b$ and $d$, and write down the formula for $\tau$.

   [6 marks]

   Total for question 4: 12 marks

End of questions