6.01a Dimensions: M, L, T notation

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

1

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

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

1

1. A particle P is executing simple harmonic motion, and the centre of the oscillations is at the point O . The maximum speed of P during the motion is \(5.1 \mathrm {~ms} ^ { - 1 }\). When P is 6 m from O , its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\). Find the period and the amplitude of the motion.
2. The force \(F\) of gravitational attraction between two objects of 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.
   1. Find the dimensions of \(G\). Three objects, each of mass \(m\), are moving in deep space under mutual gravitational attraction. They move round a single circle with constant angular speed \(\omega\), and are always at the three vertices of an equilateral triangle of side \(R\). You are given that \(\omega = k G ^ { \alpha } m ^ { \beta } R ^ { \gamma }\), where \(k\) is a dimensionless constant.
   2. Find \(\alpha , \beta\) and \(\gamma\). For three objects of mass 2500 kg at the vertices of an equilateral triangle of side 50 m , the angular speed is \(2.0 \times 10 ^ { - 6 } \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   3. Find the angular speed for three objects of mass \(4.86 \times 10 ^ { 14 } \mathrm {~kg}\) at the vertices of an equilateral triangle of side 30000 m .

1

1. 1. Find the dimensions of power. In a particle accelerator operating at power \(P\), a charged sphere of radius \(r\) and density \(\rho\) has its speed increased from \(u\) to \(2 u\) over a distance \(x\). A student derives the formula $$x = \frac { 28 \pi r ^ { 3 } u ^ { 2 } \rho } { 9 P }$$
   2. Show that this formula is not dimensionally consistent.
   3. Given that there is only one error in this formula for \(x\), obtain the correct formula.
2. A light elastic string, with natural length 1.6 m and stiffness \(150 \mathrm { Nm } ^ { - 1 }\), is stretched between fixed points A and B which are 2.4 m apart on a smooth horizontal surface.
   1. Find the energy stored in the string. A particle is attached to the mid-point of the string. The particle is given a horizontal velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to AB (see Fig. 1.1), and it comes instantaneously to rest after travelling a distance of 0.9 m (see Fig. 1.2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_639} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_1128} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Find the mass of the particle.

1

1. 1. Write down the dimensions of the following quantities. \begin{displayquote} Velocity
      Acceleration
      Force
      Density (which is mass per unit volume)
      Pressure (which is force per unit area) \end{displayquote} For a fluid with constant density \(\rho\), the velocity \(v\), pressure \(P\) and height \(h\) at points on a streamline are related by Bernoulli's equation $$P + \frac { 1 } { 2 } \rho v ^ { 2 } + \rho g h = \mathrm { constant } ,$$ where \(g\) is the acceleration due to gravity.
   2. Show that the left-hand side of Bernoulli's equation is dimensionally consistent.
2. In a wave tank, a float is performing simple harmonic motion with period 3.49 s in a vertical line. The height of the float above the bottom of the tank is \(h \mathrm {~m}\) at a time \(t \mathrm {~s}\). When \(t = 0\), the height has its maximum value. The value of \(h\) varies between 1.6 and 2.2.
   1. Sketch a graph showing how \(h\) varies with \(t\).
   2. Express \(h\) in terms of \(t\).
   3. Find the magnitude and direction of the acceleration of the float when \(h = 1.7\).

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. 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 \(\gamma\).

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 \(k\).

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

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

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.

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.

1

1. State the dimensions of force. The force \(F\) required to keep a car moving at constant speed on a circular track is given by the formula $$\mathrm { F } = \frac { \mathrm { mv } ^ { 2 } } { \mathrm { r } }$$ where
   It is proposed that a new unit of force, the trackforce (Tr), should be adopted in motor-racing. 1 Tr is defined as the amount of force required to accelerate a mass of 1 ton at a rate of 1 mile per hour per second. It is given that 1 ton \(= 1016 \mathrm {~kg}\) and 1 mile \(= 1609 \mathrm {~m}\).
2. Determine the number of newtons that are equivalent to 1 Tr .

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.

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