6.01a Dimensions: M, L, T notation

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.

2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.

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.

2 \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-02_810_743_831_644} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m . The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. Show that \(v ^ { 2 } = 33.32 - 15.68 \cos \theta\).
2. Prove that the bead is never at rest.
3. Find the maximum value of \(v\).
4. Write down the dimension of density. The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \mathrm {~m} ^ { 2 }\) and the density of the oil is \(920 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) then the period of oscillation of the pump is 0.7 s .
   A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C \rho ^ { \alpha } g ^ { \beta } A ^ { \gamma }\) where \(C\) is a dimensionless constant.
5. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
6. Hence give the value of \(C\) to 3 significant figures.
7. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho , g\) and \(A\). A car of mass 1250 kg experiences a resistance to its motion of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). At a point \(A\) on the road the car's speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At a point \(B\) on the road the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
8. Find the values of \(k\) and \(P\). The power is increased to 15 kW .
9. Calculate the maximum steady speed of the car on a straight horizontal road.

3 The kinetic energy, \(E\), of a compound pendulum is given by $$E = \frac { 1 } { 2 } I \omega ^ { 2 }$$ where \(\omega\) is the angular speed and \(I\) is a quantity called the moment of inertia.
3

1. Show that for this formula to be dimensionally consistent then \(I\) must have dimensions \(M L ^ { 2 }\), where \(M\) represents mass and \(L\) represents length.
   [0pt] [2 marks]
   3
2. The time, \(T\), taken for one complete swing of a pendulum is thought to depend on its moment of inertia, \(I\), its weight, \(W\), and the distance, \(h\), of the centre of mass of the pendulum from the point of suspension. The formula being proposed is $$T = k I ^ { \alpha } W ^ { \beta } h ^ { \gamma }$$ where \(k\) is a dimensionless constant. Determine the values of \(\alpha , \beta\) and \(\gamma\).

5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, \(F\), between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5

1. Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
   Note that there are 1600 metres in one mile.
   5
2. Gary assumes that on a wet day typical values for friction, \(F\), are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.

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]

1 State the dimensions of power.
Circle your answer. \(M L ^ { 2 } T ^ { - 3 }\) \(M L ^ { 3 } T ^ { - 3 }\) \(M L ^ { 3 } T ^ { - 2 }\) \(M L ^ { 2 } T ^ { - 2 }\)

2 A particle moves in a straight line with constant acceleration. Its initial and final velocities are \(u\) and \(v\) respectively and at time \(t\) its displacement from its starting position is \(s\). An equation connecting these quantities is \(s = k \left( u ^ { \alpha } + v ^ { \beta } \right) t ^ { \gamma }\), where \(k\) is a dimensionless constant.

1. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
2. By considering the case where the acceleration is zero, determine the value of \(k\).

2 A student is studying the speed of sound, \(u\), in a gas under different conditions.
He assumes that \(u\) depends on the pressure, \(p\), of the gas, the density, \(\rho\), of the gas and the wavelength, \(\lambda\), of the sound in the relationship \(u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)

1. Use the fact that density is mass per unit volume to find \([ \rho ]\).
2. Given that the units of \(p\) are \(\mathrm { Nm } ^ { - 2 }\), determine the values of \(\alpha , \beta\) and \(\gamma\).
3. Comment on what the value of \(\gamma\) means about how fast sounds of different wavelengths travel through the gas. The student carries out two experiments, \(A\) and \(B\), to measure \(u\). Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of \(u\) in experiment \(B\) is double the value in experiment \(A\).
4. By what factor has the density of the gas in experiment \(A\) been multiplied to give the density of the gas in experiment \(B\) ? Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-03_211_795_285_244} \(A\) collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\). \(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.
   1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
   2. Find the coefficient of restitution for the collision between \(B\) and the wall.

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]

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]

State the dimensions of force. Circle your answer. [1 mark] \(MLT\) \(ML^2T\) \(MLT^{-1}\) \(MLT^{-2}\)

State the dimensions of impulse. Circle your answer. \(MLT^{-2}\) \quad\quad \(MLT^{-1}\) \quad\quad \(MLT\) \quad\quad \(MLT^2\) [1 mark]

A spring has stiffness \(k\)

1. Determine the dimensions of \(k\) [1 mark]
2. One end of the spring is attached to a fixed point. A particle of mass \(m\) kg is attached to the other end of the spring. The particle is set into vertical motion and moves up and down, taking \(t\) seconds to complete one oscillation. A possible model for \(t\) is $$t = pm^a g^b k^c$$ where \(p\) is a dimensionless constant and \(g \text{ m s}^{-2}\) is the acceleration due to gravity. Find the values of \(a\), \(b\) and \(c\) for this model to be dimensionally consistent. [3 marks]

When a sphere of radius \(r\) metres is falling at \(v\) m s\(^{-1}\) it experiences an air resistance force \(F\) newtons. The force is to be modelled as $$F = kr^\alpha v^\beta$$ where \(k\) is a constant with units kg m\(^{-2}\)

1. State the dimensions of \(F\) [1 mark]
2. Use dimensional analysis to find the value of \(\alpha\) and the value of \(\beta\) [3 marks]

1. Write down the dimension of density. [1]

The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \, \text{m}^2\) and the density of the oil is \(920 \, \text{kg m}^{-3}\) then the period of oscillation of the pump is 0.7 s. A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C\rho^{\alpha} g^{\beta} A^{\gamma}\) where \(C\) is a dimensionless constant.

1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
2. Hence give the value of \(C\) to 3 significant figures. [2]
3. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho\), \(g\) and \(A\). [2]

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]

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]