Find exponents with all unknowns

4 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 \(B\) ? \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-4_659_401_269_251} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega _ { \text {rads } } { } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).

3 A small object \(P\) of mass \(m\) is suspended from a fixed point by a light inextensible string of length l. When \(P\) is displaced and released in a certain way, it oscillates in a vertical plane. The time taken for one complete oscillation is called the period and is denoted by \(\tau\). A student is carrying out experiments with \(P\) and suggests the following formula to model the value of \(\tau\). \(\tau = \mathrm { cg } \mathrm { a } ^ { \mathrm { a } } \mathrm { l } _ { \mathrm { m } } { } ^ { \gamma }\) in which

• \(g\) is the acceleration due to gravity,
• \(C\) is a dimensionless constant.

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]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_161_691_1681_246} 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 \(a = \mathrm { km } ^ { \alpha } \mathrm { v } ^ { \beta } \mathrm { h } ^ { \gamma }\), 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]
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 \(N\). [1

4 When two objects are placed a distance apart in outer space each applies a gravitational force to the other. It is suggested that the magnitude of this force depends on the masses of both objects and the distance between them. Assuming that this suggestion is correct, it is further assumed that the magnitude of this force is given by a relationship of the form $$\mathrm { F } = \mathrm { Gm } _ { 1 } ^ { \alpha } \mathrm { m } _ { 2 } ^ { \beta } \mathrm { r } ^ { \gamma }$$ where

• \(F\) is the magnitude of the force
• \(m _ { 1 }\) and \(m _ { 2 }\) are the masses of the two objects
• \(r\) is the distance between the two objects
• \(G\) is a constant.
  1. Using a dimensional argument based on Newton's third law explain why \(\alpha = \beta\).

It is given that the magnitude of the gravitational force is given by such a relationship and that \(G = 6.67 \times 10 ^ { - 11 } \mathrm {~m} ^ { 3 } \mathrm {~kg} ^ { - 1 } \mathrm {~s} ^ { - 2 }\).

Write down the dimensions of \(G\).

By using dimensional analysis, determine the values of \(\alpha , \beta\) and \(\gamma\). You are given that the mass of the Earth is \(5.97 \times 10 ^ { 24 } \mathrm {~kg}\) and that the distance of the Moon from the Earth is \(3.84 \times 10 ^ { 8 } \mathrm {~m}\). You may assume that the only force acting on the Moon is the gravitational force due to the Earth.

By modelling the Earth as stationary and assuming that the Moon moves in a circular orbit around the Earth, determine the period of the motion of the Moon. Give your answer to the nearest day.

4 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 \(\mathrm { N } \mathrm { m } ^ { - 2 } \mathrm {~s}\).

1. By considering its units, find the dimensions of viscosity. A model of the resistive force suggests the following relationship: \(\mathbf { F } = 6 \pi \eta ^ { \alpha } \mathbf { r } ^ { \beta } \mathbf { v } ^ { \gamma }\).
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 \(\mathrm { e } ^ { - \mathrm { kt } } = \frac { \mathrm { g } - \mathrm { kv } } { \mathrm { g } }\) where \(\mathrm { k } = \frac { 6 \pi \eta \mathrm { r } } { \mathrm { m } }\). 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.

1 The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a wave travelling along the surface of a sea is believed to depend on
the depth of the sea, \(d \mathrm {~m}\),
the density of the water, \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\),
the acceleration due to gravity, \(g\), and
a dimensionless constant, \(k\) so that $$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, show that \(\beta = 0\) and find the values of \(\alpha\) and \(\gamma\).

2 The time, \(t\), for a single vibration of a piece of taut string is believed to depend on
the length of the taut string, \(l\),
the tension in the string, \(F\),
the mass per unit length of the string, \(q\), and
a dimensionless constant, \(k\),
such that $$t = k l ^ { \alpha } F ^ { \beta } q ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).

1 A formula for calculating the lift force acting on the wings of an aircraft moving through the air is of the form $$F = k v ^ { \alpha } A ^ { \beta } \rho ^ { \gamma }$$ where \(F\) is the lift force in newtons, \(k\) is a dimensionless constant, \(v\) is the air velocity in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), \(A\) is the surface area of the aircraft's wings in \(\mathrm { m } ^ { 2 }\), and \(\rho\) is the density of the air in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).
By using dimensional analysis, find the values of the constants \(\alpha , \beta\) and \(\gamma\).
[0pt] [6 marks]

1

1. 1. Write down the dimensions of force. The period, \(t\), of a vibrating wire depends on its tension, \(F\), its length, \(l\), and its mass per unit length, \(\sigma\).
   2. Assuming that the relationship is of the form \(t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\), where \(k\) is a dimensionless constant, use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
   3. Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
2. The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
   1. Find the maximum speed of M .
   2. Find the distance of M from the centre of the motion when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1

1. 1. Write down the dimensions of force and the dimensions of density. When a wire, with natural length \(l _ { 0 }\) and cross-sectional area \(A\), is stretched to a length \(l\), the tension \(F\) in the wire is given by $$F = \frac { E A \left( l - l _ { 0 } \right) } { l _ { 0 } }$$ where \(E\) is Young's modulus for the material from which the wire is made.
   2. Find the dimensions of Young's modulus \(E\). A uniform sphere of radius \(r\) is made from material with density \(\rho\) and Young's modulus \(E\). When the sphere is struck, it vibrates with periodic time \(t\) given by $$t = k r ^ { \alpha } \rho ^ { \beta } E ^ { \gamma }$$ where \(k\) is a dimensionless constant.
   3. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. Fig. 1 shows a fixed point A that is 1.5 m vertically above a point B on a rough horizontal surface. A particle P of mass 5 kg is at rest on the surface at a distance 0.8 m from B , and is connected to A by a light elastic string with natural length 1.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The coefficient of friction between P and the surface is 0.4 , and P is on the point of sliding. Find the stiffness of the string.

1

1. 1. Write down the dimensions of density, kinetic energy and power. A sphere of radius \(r\) is moved at constant velocity \(v\) through a fluid.
   2. In a viscous fluid, the power required is \(6 \pi \eta r v ^ { 2 }\), where \(\eta\) is the viscosity of the fluid. Find the dimensions of viscosity.
   3. In a non-viscous fluid, the power required is \(k \rho ^ { \alpha } r ^ { \beta } v ^ { \gamma }\), where \(\rho\) is the density of the fluid and \(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A rock of mass 5.5 kg is connected to a fixed point O by a light elastic rope with natural length 1.2 m . The rock is released from rest in a position 2 m vertically below O , and it next comes to instantaneous rest when it is 1.5 m vertically above O . Find the stiffness of the rope.

1

1. 1. Write down the dimensions of velocity, acceleration and force. A ball of mass \(m\) is thrown vertically upwards with initial velocity \(U\). When the velocity of the ball is \(v\), it experiences a force \(\lambda v ^ { 2 }\) due to air resistance where \(\lambda\) is a constant.
   2. Find the dimensions of \(\lambda\). A formula approximating the greatest height \(H\) reached by the ball is $$H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } }$$ where \(g\) is the acceleration due to gravity.
   3. Show that this formula is dimensionally consistent. A better approximation has the form \(H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } } + \frac { 1 } { 6 } \lambda ^ { 2 } U ^ { \alpha } m ^ { \beta } g ^ { \gamma }\).
   4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A girl of mass 50 kg is practising for a bungee jump. She is connected to a fixed point O by a light elastic rope with natural length 24 m and modulus of elasticity 2060 N . At one instant she is 30 m vertically below O and is moving vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She comes to rest instantaneously, with the rope slack, at the point A . Find the distance OA .

3

1. 1. Write down the dimensions of velocity, force and density (which is mass per unit volume). A vehicle moving with velocity \(v\) experiences a force \(F\), due to air resistance, given by $$F = \frac { 1 } { 2 } C \rho ^ { \alpha } v ^ { \beta } A ^ { \gamma }$$ where \(\rho\) is the density of the air, \(A\) is the cross-sectional area of the vehicle, and \(C\) is a dimensionless quantity called the drag coefficient.
   2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A light rod is freely pivoted about a fixed point at one end and has a heavy weight attached to its other end. The rod with the weight attached is oscillating in a vertical plane as a simple pendulum with period 4.3 s . The maximum angle which the rod makes with the vertical is 0.08 radians. You may assume that the motion is simple harmonic.
   1. Find the angular speed of the rod when it makes an angle of 0.05 radians with the vertical.
   2. Find the time taken for the pendulum to swing directly from a position where the rod makes an angle of 0.05 radians on one side of the vertical to the position where the rod makes an angle of 0.05 radians on the other side of the vertical.

1

1. A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   1. Find the tension in the string.
   2. Find the angle that the string makes with the vertical.
2. A particle Q of mass \(m\) moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length \(d\) and stiffness \(k\). With the string at its natural length, Q is set in motion with initial speed \(u\) perpendicular to the string. In the subsequent motion, the maximum length of the string is \(2 d\), and the string first returns to its natural length after time \(t\). You are given that \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) and \(t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }\), where \(A\) is a dimensionless constant.
   1. Show that the dimensions of \(k\) are \(\mathrm { MT } ^ { - 2 }\).
   2. Show that the equation \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) is dimensionally consistent.
   3. Find \(\alpha , \beta\) and \(\gamma\). You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness \(60 \mathrm { Nm } ^ { - 1 }\).
   4. Find the initial speed \(u\), and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.

1

1. The speed \(v\) of sound in a solid material is given by \(v = \sqrt { \frac { E } { \rho } }\), where \(E\) is Young's modulus for the material and \(\rho\) is its density.
   1. Find the dimensions of Young's modulus. The density of steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the speed of sound in steel is \(6100 \mathrm {~ms} ^ { - 1 }\).
   2. Find Young's modulus for steel, stating the units in which your answer is measured. A tuning fork has cylindrical prongs of radius \(r\) and length \(l\). The frequency \(f\) at which the tuning fork vibrates is given by \(f = k c ^ { \alpha } E ^ { \beta } \rho ^ { \gamma }\), where \(c = \frac { l ^ { 2 } } { r }\) and \(k\) is a dimensionless constant.
   3. Find \(\alpha , \beta\) and \(\gamma\).
2. A particle P is performing simple harmonic motion along a straight line, and the centre of the oscillations is O . The points X and Y on the line are on the same side of O , at distances 3.9 m and 6.0 m from O respectively. The speed of P is \(1.04 \mathrm {~ms} ^ { - 1 }\) when it passes through X and \(0.5 \mathrm {~ms} ^ { - 1 }\) when it passes through Y.
   1. Find the amplitude and the period of the oscillations.
   2. Find the time taken for P to travel directly from X to Y .

1

1. In an investigation, small spheres are dropped into a long column of a viscous liquid and their terminal speeds measured. It is thought that the terminal speed \(V\) of a sphere depends on a product of powers of its radius \(r\), its weight \(m g\) and the viscosity \(\eta\) of the liquid, and is given by $$V = k r ^ { \alpha } ( m g ) ^ { \beta } \eta ^ { \gamma } ,$$ where \(k\) is a dimensionless constant.
   1. Given that the dimensions of viscosity are \(\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }\) find \(\alpha , \beta\) and \(\gamma\). A sphere of mass 0.03 grams and radius 0.2 cm has a terminal speed of \(6 \mathrm {~ms} ^ { - 1 }\) when falling through a liquid with viscosity \(\eta\). A second sphere of radius 0.25 cm falling through the same liquid has a terminal speed of \(8 \mathrm {~ms} ^ { - 1 }\).
   2. Find the mass of the second sphere.
2. A manufacturer is testing different types of light elastic ropes to be used in bungee jumping. You may assume that air resistance is negligible. A bungee jumper of mass 80 kg is connected to a fixed point A by one of these elastic ropes. The natural length of this rope is 25 m and its modulus of elasticity is 1600 N . At one instant, the jumper is 30 m directly below A and he is moving vertically upwards at \(15 \mathrm {~ms} ^ { - 1 }\). He comes to instantaneous rest at a point B , with the rope slack.
   1. Find the distance AB . The same bungee jumper now tests a second rope, also of natural length 25 m . He falls from rest at A . It is found that he first comes instantaneously to rest at a distance 54 m directly below A .
   2. Find the modulus of elasticity of this second rope. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_559_705_262_680} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
      \end{figure} The region R shown in Fig. 2.1 is bounded by the curve \(y = k ^ { 2 } - x ^ { 2 }\), for \(0 \leqslant x \leqslant k\), and the coordinate axes. The \(x\)-coordinate of the centre of mass of a uniform lamina occupying the region R is 0.75 .
      1. Show that \(k = 2\). A uniform solid S is formed by rotating the region R through \(2 \pi\) radians about the \(x\)-axis.
      2. Show that the centre of mass of S is at \(( 0.625,0 )\). Fig. 2.2 shows a solid T made by attaching the solid S to the base of a uniform solid circular cone C . The cone \(C\) is made of the same material as \(S\) and has height 8 cm and base radius 4 cm . \begin{figure}[h]
         \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_455_794_1521_639} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
         \end{figure}
      3. Show that the centre of mass of T is at a distance of 6.75 cm from the vertex of the cone. [You may quote the standard results that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) and its centre of mass is \(\frac { 3 } { 4 } h\) from its vertex.]
      4. The solid T is suspended from a point P on the circumference of the base of C . Find the acute angle between the axis of symmetry of T and the vertical. \begin{figure}[h]
         \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-4_668_262_255_904} \captionsetup{labelformat=empty} \caption{Fig. 3}
         \end{figure} One end of a light elastic string, of natural length 2.7 m and modulus of elasticity 54 N , is attached to a fixed point L . The other end of the string is attached to a particle P of mass 2.5 kg . One end of a second light elastic string, of natural length 1.7 m and modulus of elasticity 8.5 N , is attached to P . The other end of this second string is attached to a fixed point M , which is 6 m vertically below L . This situation is shown in Fig. 3. The particle P is released from rest when it is 4.2 m below L . Both strings remain taut throughout the subsequent motion. At time \(t \mathrm {~s}\) after P is released from rest, its displacement below L is \(x \mathrm {~m}\).
         1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 10 ( x - 4 )\).
         2. Write down the value of \(x\) when P is at the centre of its motion.
         3. Find the amplitude and the period of the oscillations.
         4. Find the velocity of P when \(t = 1.2\).

3 Jodie is doing an experiment involving a simple pendulum. The pendulum consists of a small object tied to one end of a piece of string. The other end of the string is attached to a fixed point O and the object is allowed to swing between two fixed points A and B and back again, as shown in Fig. 3. \begin{figure}[h] \end{figure} Jodie thinks that \(P\), the time the pendulum takes to swing from A to B and back again, depends on the mass, \(m\), of the small object, the length, \(l\), of the piece of string, and the acceleration due to gravity \(g\). She proposes the formula \(P = k m ^ { \alpha } l ^ { \beta } g ^ { \gamma }\).

1. What is the significance of \(k\) in Jodie's formula?
2. Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Jodie finds that when the mass of the object is 1.5 kg and the length of the string is 80 cm the time taken for the pendulum to swing from A to B and back again is 1.8 seconds.
3. Use Jodie's formula and your answers to part (ii) to find each of the following.
   (A) The value of \(k\) (B) The time taken for the pendulum to swing from A to B and back again when the mass of the object is 0.9 kg and the length of the string is 1.4 m
4. Comment on the assumption made by Jodie that the formula for the time taken for the pendulum to swing from A to B and back again is dependent on \(m , l\) and \(g\).

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

2 The speed of propagation, \(c\), of a soundwave travelling in air is given by the formula \(c = k p ^ { \alpha } d ^ { \beta }\),
where

• \(p\) is the air pressure,
• \(d\) is the air density,
• \(k\) is a dimensionless constant.
  1. Use dimensional analysis to determine the values of \(\alpha\) and \(\beta\).

During a series of experiments the speed of propagation of soundwaves travelling in air is initially recorded as \(340 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a later time it is found that the air pressure has increased by \(1 \%\) and the air density has fallen by \(0.5 \%\).

Determine, for the later time, the speed of propagation of the soundwaves.

3

1. Find the dimensions of
   • density and
   • pressure (force per unit area).
   The frequency, \(f\), of the note emitted by an air horn is modelled as \(f = k s ^ { \alpha } p ^ { \beta } d ^ { \gamma }\), where
   • \(s\) is the length of the horn,
   • \(\quad p\) is the air pressure,
   • \(d\) is the air density,
   • \(k\) is a dimensionless constant.
   • Determine the values of \(\alpha , \beta\) and \(\gamma\).
   A particular air horn emits a note at a frequency of 512 Hz and the air pressure and air density are recorded. At another time it is found that the air pressure has fallen by \(2 \%\) and the air density has risen by \(1 \%\). The length of the horn is unchanged.
2. Calculate the new frequency predicted by the model.

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.

1 A ball of mass \(m\) is travelling vertically downwards with speed \(u\) when it hits a horizontal floor. The ball bounces vertically upwards to a height \(h\). It is thought that \(h\) depends on \(m , u\), the acceleration due to gravity \(g\), and a dimensionless constant \(k\), such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).

3 A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\).
A possible model for the kinetic energy \(E\) of the disc is $$E = k m ^ { a } r ^ { b } \omega ^ { c }$$ where \(a , b\) and \(c\) are constants and \(k\) is a dimensionless constant.
Find the values of \(a , b\) and \(c\).

4

1. State the dimensions of force. 4
2. The velocity of an object in a circular orbit can be calculated using the formula $$v = G ^ { a } m ^ { b } r ^ { c }$$ where: \(G =\) Universal constant of gravitation in \(\mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }\) \(m =\) Mass of the Earth in kg \(r =\) Radius of the orbit in metres
   Use dimensional analysis to find the values of \(a , b\) and \(c\) [0pt] [4 marks]