Dimensional Analysis

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]

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 \(g\).

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 .

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]

1 A tank containing a liquid has a small hole in the bottom through which the liquid escapes. The speed, \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at which the liquid escapes is given by $$u = C V \rho g$$ where \(V \mathrm {~m} ^ { 3 }\) is the volume of the liquid in the tank, \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) is the density of the liquid, \(g\) is the acceleration due to gravity and \(C\) is a constant. By using dimensional analysis, find the dimensions of \(C\).
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6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
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1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
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2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).

A ball is thrown vertically upwards with speed \(u\) so that at time \(t\) its displacement \(s\) is given by the formula $$s = ut - \frac{gt^2}{2}$$ Use dimensional analysis to show that this formula is dimensionally consistent. Fully justify your answer. [4 marks]

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 \(h = 1.7\).

3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to \(\mathbf { B }\).

1. Find the elastic energy stored in the string. A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
2. Show that the mass of P is 15 kg .
3. Verify that P can rest in equilibrium when it is 1.0 m vertically below M . In general, a light elastic string, with natural length \(a\) and modulus of elasticity \(\lambda\), has its ends attached to fixed points which are a distance \(d\) apart on the same horizontal level. A particle of mass \(m\) is attached to the midpoint of the string, and in the equilibrium position each half of the string has length \(h\), as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-4_280_755_1064_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula $$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$
4. Show that \(\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }\) is dimensionless.
5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
6. Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).

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]

3 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 \(k\).

6 A particle moves in a straight line with constant acceleration \(a\). Its initial velocity is \(u\) and at time \(t\) its velocity is \(v\). It is assumed that \(v\) depends only on \(u , a\) and \(t\).

1. Assuming that this dependency is of the form \(\mathrm { u } ^ { \alpha } \mathrm { a } ^ { \beta } \mathrm { t } ^ { \gamma }\), use dimensional analysis to find \(\alpha\) and \(\gamma\) in terms of \(\beta\).
2. By noting that the graph of \(v\) against \(t\) must be a straight line, determine the possible values of \(\beta\). You may assume that the units of the given quantities are the corresponding SI units.
3. By considering \(v\) when \(t = 0\) seconds and when \(t = 1\) second, derive the equation of motion \(\mathrm { v } = \mathrm { u } + \mathrm { at }\), explaining your reasoning.

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

1 The specific energy of a substance has SI unit \(\mathrm { J } \mathrm { kg } ^ { - 1 }\) (joule per kilogram).

1. Determine the dimensions of specific energy. A particular brand of protein powder contains approximately 345 Calories (Cal) per 4 ounce (oz) serving. An athlete is recommended to take 40 grams of the powder each day. You are given that \(1 \mathrm { oz } = 28.35\) grams and \(1 \mathrm { Cal } = 4184 \mathrm {~J}\).
2. Determine, in joules, the amount of energy in the athlete's recommended daily serving of the protein powder.

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 tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, \(v\), at which liquid exits the tank. David thinks that \(v\) will depend on the height of the liquid in the tank, \(h\), the acceleration due to gravity, \(g\), and the density of the liquid, \(\rho\), such that \(v \propto g ^ { a } h ^ { b } \rho ^ { c }\) where \(a\), \(b\) and \(c\) are constants. Sarah thinks that \(v\) will not depend on the density of the liquid and suggests the model \(v \propto g ^ { a } h ^ { b }\) 3

1. By considering dimensions, explain which student's model should be rejected.
   [0pt] [2 marks]
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2. Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
   [0pt] [2 marks]

3 The speed, \(v\), of a particle moving in a horizontal circle is given by the formula \(v = r \omega\) where: \(v =\) speed \(r =\) radius \(\omega =\) angular speed.
Show that the dimensions of angular speed are \(T ^ { - 1 }\) [0pt] [2 marks]

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.

1 The time \(T\) taken for a simple pendulum to make a single small oscillation is thought to depend only on its length \(l\), its mass \(m\) and the acceleration due to gravity \(g\). By using dimensional analysis:

1. show that \(T\) does not depend on \(m\);
2. express \(T\) in terms of \(l , g\) and \(k\), where \(k\) is a dimensionless constant.