6.01d Unknown indices: using dimensions

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. Two light elastic strings, each having natural length 2.15 m and stiffness \(70 \mathrm {~N} \mathrm {~m} ^ { - 1 }\), are attached to a particle P of mass 4.8 kg . The other ends of the strings are attached to fixed points A and B , which are 1.4 m apart at the same horizontal level. The particle P is placed 2.4 m vertically below the midpoint of AB , as shown in Fig. 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-2_677_474_482_877} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
   1. Show that P is in equilibrium in this position.
   2. Find the energy stored in the string AP . Starting in this equilibrium position, P is set in motion with initial velocity \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards. You are given that P first comes to instantaneous rest at a point C where the strings are slack.
   3. Find the vertical height of C above the initial position of P .
2. 1. Write down the dimensions of force and stiffness (of a spring). A particle of mass \(m\) is performing oscillations with amplitude \(a\) on the end of a spring with stiffness \(k\). The maximum speed \(v\) of the particle is given by \(v = c m ^ { \alpha } k ^ { \beta } a ^ { \gamma }\), where \(c\) is a dimensionless constant.
   2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

1 The fixed point A is at a height \(4 b\) above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length \(3 b\) and modulus of elasticity \(\lambda\), has one end attached to A and the other end attached to a block of mass \(m\). The block is released from rest at a point B on the surface where \(\mathrm { BC } = 3 b\), as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC . \begin{figure}[h] \end{figure}

1. Show that immediately after release the acceleration of the block is \(\frac { 2 \lambda } { 5 m }\).
2. Show that, when the block reaches C , its speed \(v\) is given by \(v ^ { 2 } = \frac { \lambda b } { m }\).
3. Show that the equation \(v ^ { 2 } = \frac { \lambda b } { m }\) is dimensionally consistent. The time taken for the block to move from B to C is given by \(k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant.
4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
5. Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .

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 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

2

1. Find the dimensions of energy. The moment of inertia, \(I\), of a rigid body rotating about a fixed axis is measured in \(\mathrm { kg } \mathrm { m } ^ { 2 }\).
2. State the dimensions of \(I\). The kinetic energy, \(E\), of a rigid body rotating about a fixed axis is given by the formula \(\mathrm { E } = \frac { 1 } { 2 } \mathrm { I } \omega ^ { 2 }\),
   where \(\omega\) is the angular velocity (angle per unit time) of the rigid body.
3. Show that the formula for \(E\) is dimensionally consistent. When a rigid body is pivoted from one of its end points and allowed to swing freely, it forms a pendulum. The period, \(t\), of the pendulum is the time taken for it to complete one oscillation. A student conjectures the formula \(\mathrm { t } = \left. \mathrm { k } ( \mathrm { mg } ) ^ { \alpha } \mathrm { r } ^ { \beta } \right| ^ { \gamma }\),
   where
   The moment of inertia of a thin uniform rigid rod of mass 1.5 kg and length 0.8 m , rotating about one of its endpoints, is \(0.32 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The student suspends such a rod from one of its endpoints and allows it to swing freely. The student measures the period of this pendulum and finds that it is 1.47 seconds.
4. Using the formula conjectured by the student, determine the value of \(k\).

2 George is investigating the time it takes for a ball to reach a certain height when projected vertically upwards. George believes that the time, \(t\), for the ball to reach a certain height, \(h\), depends on

• the ball's mass \(m\),
• the projection speed \(u\), and
• the height \(h\).

George suggests the following formula to model this situation \(t = k m ^ { \alpha } u ^ { \beta } h ^ { \gamma }\),
where \(k\) is a dimensionless constant.

1. Use dimensional analysis to show that \(t = \frac { k h } { u }\).
2. Hence explain why George's formula is unrealistic. Mandy argues that any model of this situation must consider the acceleration due to gravity, \(g\). She suggests the alternative formula \(t = \frac { u - \sqrt { u ^ { 2 } + g h } } { g }\).
3. Show that Mandy's formula is dimensionally consistent.
4. Explain why Mandy's formula is incorrect.

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 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.

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.

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.

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.

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 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.
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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]
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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 \(\gamma\).

4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
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1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)

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 \(c\).