Further Mechanics 1: Dimensional analysisAQA A-Level Further Maths: Topic test

1. 1
   A fluid of density $\rho$ flows at speed $v$ through a pipe, and the pressure in the fluid is $p$, where pressure is force per unit area.

   (a)

   Which gives the dimensions of $\rho$?

   [1 mark]

   • A$\mathrm{ML^{-3}}$
   • B$\mathrm{M^{-1}L^{3}}$
   • C$\mathrm{ML^{3}}$
   • D$\mathrm{L^{-3}}$

   (b)

   Which of these quantities has the same dimensions as pressure?

   [1 mark]

   • A$\rho v$
   • B$\dfrac{\rho}{v^{2}}$
   • C$\rho v^{2}$
   • D$\rho v^{3}$

   (c)

   Bernoulli's equation states that $p+\frac12\rho v^{2}=\text{constant}$ along a streamline. Show that this equation is dimensionally consistent.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A constant force $F$ acts for a time $t$ on a particle of mass $m$, giving it an impulse $I=Ft$.

   (a)

   Which gives the dimensions of $I$?

   [1 mark]

   • A$\mathrm{MLT^{-2}}$
   • B$\mathrm{MLT^{-1}}$
   • C$\mathrm{ML^{2}T^{-2}}$
   • D$\mathrm{MLT^{-3}}$

   (b)

   Which of these quantities has the same dimensions as $I$?

   [1 mark]

   • Akinetic energy $\frac12mv^{2}$
   • Bforce $ma$
   • Cpower $Fv$
   • Dmomentum $mv$

   (c)

   A student claims that when the particle starts from rest, its final speed $v$ satisfies $Ft=\frac12mv^{2}$. Use dimensions to show that this cannot be correct.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The period $T$ of small oscillations of a simple pendulum is thought to depend only on the mass $m$ of the bob, the length $l$ of the string and the acceleration due to gravity $g$, so that $T=km^{a}l^{b}g^{c}$, where $k$ is a dimensionless constant.

   (a)

   Use dimensional analysis to find $a$, $b$ and $c$.

   [3 marks]

   (b)

   A pendulum of length $0.80$ m has period $1.80$ s at a place where $g=9.8$ m s⁻². Using your formula, find $k$ and predict the period of a pendulum of length $2.0$ m at the same place. Give the period to 2 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The drag force $F$ on a small sphere of radius $r$ moving at speed $v$ through a liquid of viscosity $\eta$ is thought to be given by $F=k\eta^{a}r^{b}v^{c}$, where $k$ is a dimensionless constant. The dimensions of $\eta$ are $\mathrm{ML^{-1}T^{-1}}$.

   (a)

   Use dimensional analysis to find $a$, $b$ and $c$.

   [6 marks]

   (b)

   A student suggests instead that $F=k\eta\dfrac{v^{2}}{r}$. Show, using dimensions, that this formula is not possible. In an experiment a force of $1.70\times10^{-3}$ N acts on a sphere of radius $2.0$ mm moving at $0.050$ m s⁻¹ in a liquid of viscosity $0.90$ kg m⁻¹ s⁻¹. Use your formula from part (a) to find $k$, and then predict the drag force on a sphere of radius $3.0$ mm moving at $0.020$ m s⁻¹ in the same liquid. Give $k$ and the force to 2 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The Young modulus $E$ of a material is defined as stress divided by strain, where stress is force per unit area and strain is the extension divided by the original length.

   (a)

   Which gives the dimensions of stress?

   [1 mark]

   • A$\mathrm{MLT^{-2}}$
   • B$\mathrm{MT^{-2}}$
   • C$\mathrm{ML^{-1}T^{-2}}$
   • D$\mathrm{ML^{-3}T^{-2}}$

   (b)

   Which of these has the same dimensions as $E$, where $\rho$ is density, $v$ is speed, $g$ is the acceleration due to gravity and $h$ is a length?

   [1 mark]

   • A$\rho v^{2}$
   • B$\rho v$
   • C$\rho g$
   • D$\rho gh^{2}$

   (c)

   The speed of a longitudinal wave along a wire is suggested to be $v=\sqrt{\dfrac{E}{\rho}}$, where $\rho$ is the density of the material. Show that this formula is dimensionally consistent.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A stretched string of length $l$, under tension $F$ and with mass per unit length $\mu$, vibrates with frequency $f$.

   (a)

   Which gives the dimensions of $\mu$?

   [1 mark]

   • A$\mathrm{M}$
   • B$\mathrm{ML}$
   • C$\mathrm{ML^{-2}}$
   • D$\mathrm{ML^{-1}}$

   (b)

   Which formula for $f$ is dimensionally consistent?

   [1 mark]

   • A$f=\dfrac{1}{2l}\sqrt{\dfrac{\mu}{F}}$
   • B$f=\dfrac{1}{2l}\sqrt{\dfrac{F}{\mu}}$
   • C$f=\dfrac{1}{2l}\dfrac{F}{\mu}$
   • D$f=\dfrac{l}{2}\sqrt{\dfrac{F}{\mu}}$

   (c)

   Explain why dimensional analysis cannot show that the factor $\frac12$ in the correct formula for $f$ is needed.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A satellite orbits at distance $r$ from the centre of a planet of mass $M$. The orbital period $T$ is thought to depend only on $G$, $M$ and $r$, so that $T=kG^{a}M^{b}r^{c}$, where $k$ is a dimensionless constant. The dimensions of $G$ are $\mathrm{M^{-1}L^{3}T^{-2}}$.

   (a)

   Use dimensional analysis to find $a$, $b$ and $c$.

   [3 marks]

   (b)

   The constant $k$ is found to be $2\pi$. Calculate the period of a satellite orbiting at $r=6.78\times10^{6}$ m about a planet of mass $5.97\times10^{24}$ kg, given that $G=6.67\times10^{-11}$ in SI units. Give your answer in minutes to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The speed $v$ of waves on deep water is thought to depend on the wavelength $\lambda$, the acceleration due to gravity $g$ and the density $\rho$ of the water, so that $v=k\lambda^{a}g^{b}\rho^{c}$, where $k$ is a dimensionless constant.

   (a)

   Use dimensional analysis to find $a$, $b$ and $c$, and comment on the effect of the density of the water on the wave speed.

   [6 marks]

   (b)

   Waves of wavelength $25$ m on deep water are measured to travel at $6.2$ m s⁻¹, where $g=9.8$ m s⁻². Find $k$ to 2 significant figures and predict the speed of waves of wavelength $100$ m. State the effect on the speed of quadrupling the wavelength, and explain why $k$ could not be found by dimensional analysis.

   [6 marks]

   Total for question 8: 12 marks

End of questions