Derive dimensions from formula

6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
6

1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
   6
2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).

5 The escape speed of an unpowered object is the minimum speed at which it must be projected to escape the gravitational influence of the Earth if it is projected vertically upwards from the Earth's surface. A formula for the escape speed \(U\) of an unpowered object of mass \(m\) is \(U = \sqrt { \frac { 2 G m } { r } }\) where \(r\) is the radius of the Earth and \(G\) is a constant.

1. Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\). A rocket is a powered object. A rocket is launched with a given launch speed and is then powered by engines which apply a constant force for a period of time after the launch. A student wishes to apply the formula given above to a rocket launch. They wish to model the minimum launch speed required for a rocket to escape the Earth's gravitational influence. They realise that the given formula is for unpowered objects and so they include an extra term in the formula to obtain \(V = \sqrt { \frac { 2 G m } { r } } - \mathrm { kP } ^ { \alpha } \mathrm { W } ^ { \beta } \mathrm { t } ^ { \gamma }\). In their modified formula, \(G\) and \(r\) are the same as before. The other variables are defined as follows.

1 The magnitude of the gravitational force, \(F\), between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(x\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { x ^ { 2 } }$$ where \(G\) is a constant.

1. By using dimensional analysis, find the dimensions of \(G\).
2. The lifetime, \(t\), of a planet is thought to depend on its mass, \(m\), its initial radius, \(R\), the constant \(G\) and a dimensionless constant, \(k\), so that $$t = k m ^ { \alpha } R ^ { \beta } G ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
   Find the values of \(\alpha , \beta\) and \(\gamma\).

1 A tank containing a liquid has a small hole in the bottom through which the liquid escapes. The speed, \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at which the liquid escapes is given by $$u = C V \rho g$$ where \(V \mathrm {~m} ^ { 3 }\) is the volume of the liquid in the tank, \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) is the density of the liquid, \(g\) is the acceleration due to gravity and \(C\) is a constant. By using dimensional analysis, find the dimensions of \(C\).
□
□

1

1. Write down the dimensions of velocity, acceleration and force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(r\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } }$$ where \(G\) is the universal constant of gravitation.
2. Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\).
3. In SI units (based on the kilogram, metre and second) the value of \(G\) is \(6.67 \times 10 ^ { - 11 }\). Find the value of \(G\) in imperial units based on the pound \(( 0.4536 \mathrm {~kg} )\), foot \(( 0.3048 \mathrm {~m} )\) and second.
4. For a planet of mass \(m\) and radius \(r\), the escape velocity \(v\) from the planet's surface is given by $$v = \sqrt { \frac { 2 G m } { r } }$$ Show that this formula is dimensionally consistent.
5. For a planet in circular orbit of radius \(R\) round a star of mass \(M\), the time \(t\) taken to complete one orbit is given by $$t = k G ^ { \alpha } M ^ { \beta } R ^ { \gamma }$$ where \(k\) is a dimensionless constant.
   Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

1 The surface tension of a liquid enables a metal needle to be at rest on the surface of the liquid. The greatest mass \(m\) of a needle of length \(a\) which can be supported in this way by a liquid of surface tension \(S\) is given by $$m = \frac { 2 S a } { g }$$ where \(g\) is the acceleration due to gravity.

1. Show that the dimensions of surface tension are \(\mathrm { MT } ^ { - 2 }\). The surface tension of water is 0.073 when expressed in SI units (based on kilograms, metres and seconds).
2. Find the surface tension of water when expressed in a system of units based on grams, centimetres and minutes. Liquid will rise up a capillary tube to a height \(h\) given by \(h = \frac { 2 S } { \rho g r }\), where \(\rho\) is the density of the liquid and \(r\) is the radius of the capillary tube. \(r\) is the radius of the capillary tube.
3. Show that the equation \(h = \frac { 2 S } { \rho g r }\) is dimensionally consistent.
4. Find the radius of a capillary tube in which water will rise to a height of 25 cm . (The density of water is 1000 in SI units.) When liquid is poured onto a horizontal surface, it forms puddles of depth \(d\). You are given that \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\) where \(k\) is a dimensionless constant.
5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). Water forms puddles of depth 0.44 cm . Mercury has surface tension 0.487 and density 13500 in SI units.
6. Find the depth of puddles formed by mercury on a horizontal surface.

2 The surface tension of a liquid allows a metal needle to be at rest on the surface of the liquid.
The greatest mass \(m\) of a needle of length \(l\) which can be supported in this way by a liquid of surface tension \(S\) is given by the formula \(m = \frac { 2 S l } { g }\) where \(g\) is the acceleration due to gravity.

1. Determine the dimensions of surface tension. Surface tension also allows liquids to rise up capillary tubes. Molly is experimenting with liquids in capillary tubes and she arrives at the formula \(h = \frac { 2 S } { \rho g r }\), where \(h\) is the height to which a liquid of surface tension \(S\) rises, \(\rho\) is the density of the liquid, and \(r\) is the radius of the capillary tube.
2. Show that the equation for \(h\) is dimensionally consistent. In SI units, the surface tension of mercury is \(0.475 \mathrm {~kg} \mathrm {~s} ^ { - 2 }\) and its density is \(13500 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
3. Find the diameter of a capillary tube in which mercury will rise to a height of 10 cm . In another experiment, Molly finds that when liquid of surface tension \(S\) is poured onto a horizontal surface, puddles of depth \(d\) are formed. For this experiment she finds that \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\) where \(k\) is a dimensionless constant.
4. Determine the values of \(\alpha , \beta\) and \(\gamma\).

3 The time period \(T\) of a satellite in circular orbit around a planet satisfies the equation \(G M T ^ { 2 } = 4 \pi ^ { 2 } R ^ { 3 }\),
where

• \(G\) is the universal gravitational constant,
• \(M\) is the mass of the planet,
• \(\quad R\) is the radius of the orbital circle.
  1. Find the dimensions of \(G\).

A student suggests the following formula to model the approach speed between two orbiting bodies. \(v = k G { } ^ { \alpha } { } ^ { \beta } { } _ { r } \gamma _ { m _ { 1 } } m _ { 2 } \left( m _ { 1 } + m _ { 2 } \right)\),
where

3 Solid toy aeroplane nose cones of various sizes are made in the shape shown in Fig. 3.1, where OA is its line of symmetry. \begin{figure}[h] \end{figure} The air resistance against the nose cone as the aeroplane flies through the air is initially modelled by \(R = k r v \eta\), where \(R\) is the air resistance, \(r\) is the radius of the circular flat end of the nose cone, \(v\) is the velocity of the nose cone, \(\eta\) is the viscosity of the air and \(k\) is a dimensionless constant.

1. Use dimensional analysis to show that the dimensions of \(\eta\) are \(\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }\). In an experiment conducted on a particular nose cone, measurements of air resistance are taken for different velocities. The viscosity of the air does not vary during the experiment. The graph in Fig. 3.2 shows the results. Measurements are given using the appropriate S.I. units. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-3_794_1166_1411_427} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. Comment on whether the results of this experiment are consistent with the initial model. It is now suggested that a better model for the air resistance is \(R = K r v \left( \frac { \rho r v } { \eta } \right) ^ { \alpha }\), where \(\rho\) is the density of the air, \(K\) is a dimensionless constant and \(R , r , v\) and \(\eta\) are as before.
3. (A) Find the dimensions of \(\frac { \rho r v } { \eta }\).
   (B) Explain why you cannot use dimensional analysis to find the value of \(\alpha\).

1 Newton's gravitational constant, \(G\), is approximately \(6.67 \times 10 ^ { - 11 } \mathrm {~N} \mathrm {~m} ^ { 2 } \mathrm {~kg} ^ { - 2 }\).

1. Find the dimensions of \(G\). The escape velocity, \(v\), of a body from a planet's surface, is given by the formula \(\mathrm { v } = \mathrm { kG } ^ { \alpha } \mathrm { M } ^ { \beta } \mathrm { r } ^ { \gamma }\),
   where \(M\) is the planet's mass, \(r\) is the planet's radius and \(k\) is a dimensionless constant.
2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

6

1. Write down the dimensions of force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(d\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is the universal gravitational constant.
   In SI units the value of \(G\) is \(6.67 \times 10 ^ { - 11 } \mathrm {~kg} ^ { - 1 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 2 }\).
2. Write down the dimensions of \(G\).
3. Determine the value of \(G\) in imperial units based on pounds, feet, and seconds. Use the facts that 1 pound \(= 0.454 \mathrm {~kg}\) and 1 foot \(= 0.305 \mathrm {~m}\). For a planet of mass \(M\) and radius \(r\), it is suggested that the velocity \(v\) needed for an object to escape the gravitational pull of the planet, the 'escape velocity', is given by the following formula. \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\),
   where \(k\) is a dimensionless constant.
4. Show that this formula is dimensionally consistent. Information regarding the planets Earth and Mars can be found in the table below.

   	Earth	Mars
   Radius (m)	6371000	3389500
   Mass (kg)	\(5.97 \times 10 ^ { 24 }\)	\(6.39 \times 10 ^ { 23 }\)
   Escape velocity ( \(\mathrm { m } \mathrm { s } ^ { - 1 }\) )	11186

5. Using the formula \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\), determine the escape velocity for planet Mars.

2 The universal law of gravitation states that \(F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } }\) where \(F\) is the magnitude of the force between two objects of masses \(m _ { 1 }\) and \(m _ { 2 }\) which are a distance \(r\) apart and \(G\) is a constant. Find the dimensions of \(G\).

6 This question is about modelling the relation between the pressure, \(P\), volume, \(V\), and temperature, \(\theta\), of a fixed amount of gas in a container whose volume can be varied. The amount of gas is measured in moles; 1 mole is a dimensionless constant representing a fixed number of molecules of gas. Gas temperatures are measured on the Kelvin scale; the unit for temperature is denoted by K . You may assume that temperature is a dimensionless quantity. A gas in a container will always exert an outwards force on the walls of the container. The pressure of the gas is defined to be the magnitude of this force per unit area of the walls, with \(P\) always positive. An initial model of the relation is given by \(P ^ { \alpha } V ^ { \beta } = n R \theta\), where \(n\) is the number of moles of gas present and \(R\) is a quantity called the Universal Gas Constant. The value of \(R\), correct to 3 significant figures, is \(8.31 \mathrm { JK } ^ { - 1 }\).

1. Show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\) and \([ R ] = \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 }\).
2. Hence show that \(\alpha = 1\) and \(\beta = 1\). 5 moles of gas are present in the container which initially has volume \(0.03 \mathrm {~m} ^ { 3 }\) and which is maintained at a temperature of 300 K .
3. Find the pressure of the gas, as predicted by the model. An improved model of the relation is given by \(\left( P + \frac { a n ^ { 2 } } { V ^ { 2 } } \right) ( V - n b ) = n R \theta\), where \(a\) and \(b\) are constants.
4. Determine the dimensions of \(b\) and \(a\). The values of \(a\) and \(b\) (in appropriate units) are measured as being 0.14 and \(3.2 \times 10 ^ { - 5 }\) respectively.
5. Find the pressure of the gas as predicted by the improved model. Suppose that the volume of the container is now reduced to \(1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 3 }\) while keeping the temperature at 300 K .
6. By considering the value of the pressure of the gas as predicted by the improved model, comment on the validity of this model in this situation.

6 A ball is thrown with speed \(u\) at an angle of \(45 ^ { \circ }\) to the horizontal from a point \(O\) When the horizontal displacement of the ball is \(x\), the vertical displacement of the ball above \(O\) is \(y\) where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6

1. Use dimensional analysis to find the dimensions of \(k\) 6
2. State what can be deduced about \(k\) from the dimensions that you found in part (a).

3 This question is about modelling the relation between the pressure, \(P\), volume, \(V\), and temperature, \(\theta\), of a fixed amount of gas in a container whose volume can be varied. The amount of gas is measured in moles; 1 mole is a dimensionless constant representing a fixed number of molecules of gas. Gas temperatures are measured on the Kelvin scale; the unit for temperature is denoted by K . You may assume that temperature is a dimensionless quantity. A gas in a container will always exert an outwards force on the walls of the container. The pressure of the gas is defined to be the magnitude of this force per unit area of the walls, with \(P\) always positive. An initial model of the relation is given by \(P ^ { \alpha } V ^ { \beta } = n R \theta\), where \(n\) is the number of moles of gas present and \(R\) is a quantity called the Universal Gas Constant. The value of \(R\), correct to 3 significant figures, is \(8.31 \mathrm { JK } ^ { - 1 }\).

1. Show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\) and \([ R ] = \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 }\).
2. Hence show that \(\alpha = 1\) and \(\beta = 1\). 5 moles of gas are present in the container which initially has volume \(0.03 \mathrm {~m} ^ { 3 }\) and which is maintained at a temperature of 300 K .
3. Find the pressure of the gas, as predicted by the model. An improved model of the relation is given by \(\left( P + \frac { a n ^ { 2 } } { V ^ { 2 } } \right) ( V - n b ) = n R \theta\), where \(a\) and \(b\) are constants.
4. Determine the dimensions of \(b\) and \(a\). The values of \(a\) and \(b\) (in appropriate units) are measured as being 0.14 and \(3.2 \times 10 ^ { - 5 }\) respectively.
5. Find the pressure of the gas as predicted by the improved model. Suppose that the volume of the container is now reduced to \(1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 3 }\) while keeping the temperature at 300 K .
6. By considering the value of the pressure of the gas as predicted by the improved model, comment on the validity of this model in this situation.

A lunar mapping satellite of mass \(m_1\) measured in kg is in an elliptic orbit around the moon, which has mass \(m_2\) measured in kg. The effective potential, \(E\), of the satellite is given by $$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G\) Nm\(^2\)kg\(^{-2}\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:

1. \(E\), [3 marks]
2. \(K\). [3 marks]

A formula for the elastic potential energy, \(E\), stored in a stretched spring is given by $$E = \frac{kx^2}{2}$$ where \(x\) is the extension of the spring and \(k\) is a constant. Use dimensional analysis to find the dimensions of \(k\). [3 marks]

Kepler's Third Law of planetary motion for the period of a circular orbit around the Earth is given by the formula, $$t = 2\pi\sqrt{\frac{r^3}{Gm}}$$ where, \(t\) is the time taken for one orbit \(r\) is the radius of the circular orbit \(m\) is the mass of the Earth \(G\) is a gravitational constant. Use dimensional analysis to determine the dimensions of \(G\) [4 marks]

The Reynolds number, \(R\), is an important dimensionless quantity in fluid dynamics; it can be used to predict flow patterns when a fluid is in motion relative to a surface. The Reynolds number is defined as $$R = \frac{\rho ul}{\mu},$$ where \(\rho\) is the density of the fluid, \(u\) is the velocity of the fluid relative to the surface, \(l\) is the distance travelled by the fluid and \(\mu\) is the viscosity of the fluid. Find the dimensions of \(\mu\). [4]

One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac{1}{2}I\omega^2\), where \(I\) is a quantity called the moment of inertia of the rod.

1. Deduce the dimensions of \(I\). [3]
2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. [3]

A student notices that the formula \(E = \frac{1}{2}I\omega^2\) looks similar to the formula \(E = \frac{1}{2}mv^2\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac{d\omega}{dt}\)) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I\alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.

1. Use dimensional analysis to show that the student's suggestion is incorrect. [2]
2. State the dimensions of a quantity \(x\) for which the equation \(Fx = I\alpha\) would be dimensionally consistent. [1]
3. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct. [1]