6.01b Units vs dimensions: relationship

1 The specific energy of a substance has SI unit \(\mathrm { J } \mathrm { kg } ^ { - 1 }\) (joule per kilogram).

1. Determine the dimensions of specific energy. A particular brand of protein powder contains approximately 345 Calories (Cal) per 4 ounce (oz) serving. An athlete is recommended to take 40 grams of the powder each day. You are given that \(1 \mathrm { oz } = 28.35\) grams and \(1 \mathrm { Cal } = 4184 \mathrm {~J}\).
2. Determine, in joules, the amount of energy in the athlete's recommended daily serving of the protein powder.

2

1. Write down the dimensions of pressure. The SI unit of pressure is the pascal (Pa). 15 Pa is equivalent to \(Q\) newtons per square centimetre.
2. Find the value of \(Q\). Simon thinks the speed, \(v\), of sound in a gas is given by the formula \(v = k P ^ { x } d ^ { y } V ^ { z }\),
   where \(P\) is the pressure of the gas, \(d\) is the density of the gas, \(V\) is the volume of the gas, \(k\) is a dimensionless constant.
3. Use dimensional analysis to

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 \(\gamma\).

1

1. State the dimensions of the following quantities.
   A student investigating the drag force \(F\) experienced by an object moving through air conjectures the formula \(\mathrm { F } = \mathrm { ku } ^ { 2 } \left( \rho \mathrm {~m} ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\),
   where
   The student carries out experiments in an airflow tunnel. When the air density is doubled, the drag force is found to double as well, with all other conditions remaining the same.
2. Show that the student's formula is inconsistent with the experimental observation. The student's teacher suggests revising the formula as \(\mathrm { F } = \mathrm { k } \rho ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { A } ^ { \gamma }\) where \(m\) has been replaced by \(A\), the cross-sectional area of the object. The constant \(k\) is still dimensionless.
3. Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).

2

1. State the dimensions of force. Use the following metric-imperial conversion factors for the rest of this question.
   A unit of force used in the imperial system is the pound-force (lbf). 1 lbf is defined as the gravitational force exerted on 1 lb on the surface of the Earth.
2. Show that 1 lbf is approximately equal to 4.45 N . The pascal (Pa) is a unit of pressure equivalent to 1 Newton per square metre. Pressure can also be measured in pound-force per square inch (psi). A diver, at a depth of 40 m , experiences a typical pressure of \(5 \times 10 ^ { 5 } \mathrm {~Pa}\).
3. Determine whether this is greater or less than the pressure in a bicycle tyre of 80 psi . In various physical contexts, energy density is the amount of energy stored in a given region of space per unit volume.
4. Show that energy density and pressure are dimensionally equivalent.

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 Ns
2.4 N s 2 In this question One of these formulae is dimensionally consistent.
Circle your answer.
[0pt] [1 mark] $$T = 2 \pi \sqrt { \frac { a } { l } } \quad v ^ { 2 } = \frac { 2 a l } { T } \quad F l = m v ^ { 2 } \quad F T = m \sqrt { a }$$ Turn over for the next question

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 \(G\).

5 A simple pendulum consists of a small sphere of mass \(m\) connected to one end of a light rod of length \(h\). The other end of the rod is freely hinged at a fixed point. When the sphere is pulled a short distance to one side and released from rest the pendulum performs oscillations. The time taken to perform one complete oscillation is called the period and is denoted by \(P\).

1. Assuming that \(P = k m ^ { \alpha } h ^ { \beta } g ^ { \gamma }\), where \(g\) is the acceleration due to gravity and \(k\) is a dimensionless constant, find the values of \(\alpha , \beta\) and \(\gamma\). A student conducts an experiment to investigate how \(P\) varies as \(h\) varies. She measures the value of \(P\) for various values of \(h\), ensuring that all other conditions remain constant. Her results are summarised in the table below.

   \(h ( \mathrm {~m} )\)	0.40	2.50	3.60
   \(P ( \mathrm {~s} )\)	1.27	2.17	3.81

2. Show that these results are not consistent with the answers to part (i).
3. The student later realises that she has recorded one of her values of \(P\) incorrectly.
   • Identify the incorrect value.
   • Estimate the correct value that she should have recorded.

3 A student is investigating fluid flowing through a pipe.
In her first model she assumes a relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma }\) where \(\rho\) is the density of the fluid, \(h\) is the length of the pipe, \(P\) is the pressure difference between the ends of the pipe, \(g\) is the acceleration due to gravity and \(S\) is a dimensionless constant. You are given that \(\rho\) is measured in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).

1. Use the fact that pressure is force per unit area to show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\).
2. Find the values of \(\alpha , \beta\) and \(\gamma\). The density of the fluid the student is using is \(540 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). In her experiment she finds that when the length of the pipe is 1.40 m the pressure difference between the ends of the pipe is \(3.25 \mathrm { Nm } ^ { - 2 }\).
3. Find the length of the pipe for which her first model would predict a pressure difference between the ends of the pipe of \(4.65 \mathrm { Nm } ^ { - 2 }\). In an alternative model the student suggests a modified relationship of the form \(P = S \rho ^ { \alpha } g ^ { \beta } h ^ { \gamma } + \frac { 1 } { 2 } h v ^ { 2 }\), where \(v\) is the average velocity of the fluid in the pipe.
4. Use dimensional analysis to assess the validity of her alternative model.

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.

3 The speed, \(v\), of a particle moving in a horizontal circle is given by the formula \(v = r \omega\) where: \(v =\) speed \(r =\) radius \(\omega =\) angular speed.
Show that the dimensions of angular speed are \(T ^ { - 1 }\) [0pt] [2 marks]

2 A solenoid is a device formed by winding a wire tightly around a hollow cylinder so that the wire forms (approximately) circular loops along the cylinder (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{91098ecb-fb4a-44aa-9e59-6c6fe3704966-02_164_697_1484_233} When the wire carries an electrical current a magnetic field is created inside the solenoid which can cause a particle which is moving inside the solenoid to accelerate. A student is carrying out experiments on particles moving inside solenoids. His professor suggests that, for a particle of mass \(m\) moving with speed \(v\) inside a solenoid of length \(h\), the acceleration \(a\) of the particle can be modelled by a relationship of the form \(a = k m ^ { \alpha } v ^ { \beta } h ^ { \gamma }\), where \(k\) is a constant. The professor tells the student that \([ k ] = \mathrm { MLT } ^ { - 1 }\).

1. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. The mass of an electron is \(9.11 \times 10 ^ { - 31 } \mathrm {~kg}\) and the mass of a proton is \(1.67 \times 10 ^ { - 27 } \mathrm {~kg}\). For an electron and a proton moving inside the same solenoid with the same speed, use the model to find the ratio of the acceleration of the electron to the acceleration of the proton.
3. The professor tells the student that \(a\) also depends on the number of turns or loops of wire, \(N\), that the solenoid has. Explain why dimensional analysis cannot be used to determine the dependence of \(a\) on \(N\).

3 The resistive force, \(F\), on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, \(r\), the velocity of the sphere, \(v\), and the viscosity of the fluid, \(\eta\). You are given that \(\eta\) is measured in \(\mathrm { Nm } ^ { - 2 } \mathrm {~s}\).

1. By considering its units, find the dimensions of viscosity. A model of the resistive force suggests the following relationship: \(F = 6 \pi \eta ^ { \alpha } r ^ { \beta } v ^ { \gamma }\).
2. Explain whether or not it is possible to use dimensional analysis to verify that the constant \(6 \pi\) is correct.
3. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\). A sphere of radius \(r\) and mass \(m\) falls vertically from rest through the fluid. After a time \(t\) its velocity is \(v\).
4. By setting up and solving a differential equation, show that \(\mathrm { e } ^ { - k t } = \frac { g - k v } { g }\) where \(k = \frac { 6 \pi \eta r } { m }\). As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.
5. Find, in terms of \(g\) and \(k\), the terminal velocity of the sphere. In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.
6. Describe how, according to the model, the terminal velocity of the sphere changes as the viscosity of the fluid through which it falls increases.

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.

Briefly define the following terms used in modelling in Mechanics:

1. lamina,
2. uniform rod,
3. smooth surface,
4. particle.

[4 marks]

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]

Materials have a measurable property known as the Young's Modulus, \(E\). If a force is applied to one face of a block of the material then the material is stretched by a distance called the extension. Young's modulus is defined as the ratio \(\frac{\text{Stress}}{\text{Strain}}\) where Stress is defined as the force per unit area and Strain is the ratio of the extension of the block to the length of the block.

1. Show that Strain is a dimensionless quantity. [1]
2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). [2]

It is suggested that the speed of sound in a material, \(c\), depends only upon the value of Young's modulus for the material, \(E\), the volume of the material, \(V\), and the density (or mass per unit volume) of the material, \(\rho\).

1. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E\), \(V\) and \(\rho\). [5]
2. The speed of sound in a certain material is \(500\) m s\(^{-1}\).
   1. Use your formula from part (c) to predict the speed of sound in the material if the value of Young's modulus is doubled but all other conditions are unchanged. [1]
   2. With reference to your formula from part (c), comment on the effect on the speed of sound in the material if the volume is doubled but all other conditions are unchanged. [1]
3. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c). [1]

A particle P of mass \(m\) kg is projected with speed \(u \text{ m s}^{-1}\) along a rough horizontal surface. During the motion of P, a constant frictional force of magnitude \(F\) N acts on P. When the velocity of P is \(v \text{ m s}^{-1}\), it experiences a force of magnitude \(kv\) N due to air resistance, where \(k\) is a constant.

1. Determine the dimensions of \(k\). [3]

At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is $$T = \frac{mu}{F} - \frac{kmu^2}{2F^2} + \frac{1}{3}k^2m^{\alpha}u^{\beta}F^{\gamma}.$$

1. Use dimensional analysis to find \(\alpha\), \(\beta\) and \(\gamma\). [4]

Fig. 6 shows a pendulum which consists of a rod AB freely hinged at the end A with a weight at the end B. The pendulum is oscillating in a vertical plane. The total energy, \(E\), of the pendulum is given by $$E = \frac{1}{2}I\omega^2 - mgh\cos\theta,$$ where

• \(\omega\) is its angular speed
• \(m\) is its mass
• \(h\) is the distance of its centre of mass from A
• \(\theta\) is the angle the rod makes with the downward vertical
• \(g\) is the acceleration due to gravity
• \(I\) is a quantity known as the moment of inertia of the pendulum.

\includegraphics{figure_6}

1. Use the expression for \(E\) to deduce the dimensions of \(I\). [4]

It is suggested that the period of oscillation, \(T\), of the pendulum is given by \(T = kI^\alpha(mg)^\beta h^\gamma\), where \(k\) is a dimensionless constant.

1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [5]

A class experiment finds that, when all other quantities are fixed, \(T\) is proportional to \(\frac{1}{\sqrt{m}}\).

1. Determine whether this result is consistent with your answer to part (ii). [1]

A particle of mass \(m\) moves in a straight line with constant acceleration \(a\). Its initial and final velocities are \(u\) and \(v\) respectively and its final displacement from its starting position is \(s\). In order to model the motion of the particle it is suggested that the velocity is given by the equation $$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$ where \(p\) and \(q\) are dimensionless constants.

1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent. [2]
2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\). [4]
3. By considering the case where \(s = 0\), determine the value of \(p\). [1]
4. By multiplying both sides of the equation by \(\frac{1}{2}m\), and using the numerical values of \(\alpha\), \(\beta\) and \(\gamma\), determine the value of \(q\). [2]