6.01d Unknown indices: using dimensions

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]

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.

The gravitational attraction \(F\) N between two point masses \(m_1\) kg and \(m_2\) kg at a distance \(x\) m apart is given by \(F = \frac{km_1m_2}{x^2}\), where \(k\) is a constant. Given that a small body of mass \(1\) kg experiences a force of \(g\) N at the surface of the Earth, which has radius \(R\) m and mass \(M\) kg,

1. show that \(k = \frac{gR^2}{M}\). [2 marks]

A small communications satellite of mass \(m\) kg is put into a circular orbit of radius \(r\) m around the Earth. Modelling the Earth as a particle of mass \(M\) kg, and using the value of \(k\) from (a),

1. prove that the period of rotation, \(T\) s, of the satellite is given by \(T = \frac{2\pi}{R}\sqrt{\frac{r^3}{g}}\). [4 marks]

To cover transmission to any point on the Earth, three small satellites \(X\), \(Y\) and \(Z\), each of mass \(m\) kg, are placed in a common circular orbit of radius \(r\) and form an equilateral triangle as shown. \includegraphics{figure_6}

1. Show on a copy of the diagram the direction of the three forces acting on \(X\). [1 mark]
2. State, with a reason, the direction of the resultant force on \(X\). [2 marks]
3. Show that the period of rotation of \(X\) is given by \(T\sqrt{\frac{3M}{2M + m\sqrt{3}}}\) s, where \(T\) s is the period found in (b). [7 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]

Kepler's Third Law of planetary motion for the period of a circular orbit around the Earth is given by the formula, $$t = 2\pi\sqrt{\frac{r^3}{Gm}}$$ where, \(t\) is the time taken for one orbit \(r\) is the radius of the circular orbit \(m\) is the mass of the Earth \(G\) is a gravitational constant. Use dimensional analysis to determine the dimensions of \(G\) [4 marks]

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]

\includegraphics{figure_2} A particle is projected with speed \(v\) from a point O on horizontal ground. The angle of projection is \(\theta\) above the horizontal. The particle passes, in succession, through two points A and B, each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram. You are given that \(d^2 = \frac{v^\alpha \sin^2 2\theta}{g^\beta} - \frac{8h^2 \cos^2 \theta}{g}\). Use dimensional analysis to find \(\alpha\) and \(\beta\). [4]

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

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

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

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

One end of a light spring is attached to a fixed point. A mass of 2 kg is attached to the other end of the spring. The spring hangs vertically in equilibrium. The extension of the spring is 0.05 m.

1. Find the stiffness of the spring. [2]
2. Find the energy stored in the spring. [2]
3. Find the dimensions of stiffness of a spring. [1]

A particle P of mass \(m\) is performing complete oscillations with amplitude \(a\) on the end of a light spring with stiffness \(k\). The spring hangs vertically and the maximum speed \(v\) of P is given by the formula $$v = Cm^{\alpha}a^{\beta}k^{\gamma},$$ where C is a dimensionless constant.

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

A student conducts an experiment by first stretching a length of wire and fixing its ends. The student then plucks the wire causing it to vibrate. The frequency of these vibrations, \(f\), is modelled by the formula $$f = kC^\alpha l^\beta \sigma^\gamma,$$ where \(C\) is the tension in the wire, \(l\) is the length of the stretched wire, \(\sigma\) is the mass per unit length of the stretched wire and \(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha\), \(\beta\) and \(\gamma\). [5]

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

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

\includegraphics{figure_6}

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

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

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

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

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

A 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 $$E = km^ar^b\omega^c$$ where \(a\), \(b\) and \(c\) are constants and \(k\) is a dimensionless constant. Find the values of \(a\), \(b\) and \(c\). [3 marks]

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

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

One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac{1}{2}I\omega^2\), where \(I\) is a quantity called the moment of inertia of the rod.

1. Deduce the dimensions of \(I\). [3]
2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. [3]

A student notices that the formula \(E = \frac{1}{2}I\omega^2\) looks similar to the formula \(E = \frac{1}{2}mv^2\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac{d\omega}{dt}\)) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I\alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.

1. Use dimensional analysis to show that the student's suggestion is incorrect. [2]
2. State the dimensions of a quantity \(x\) for which the equation \(Fx = I\alpha\) would be dimensionally consistent. [1]
3. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct. [1]