6.01b Units vs dimensions: relationship

1 Explain why the number 1836.108 for the ratio Rest mass of electron would be suitable for communication with other civilisations whereas neither the rest mass of the proton nor that of the electron would be.

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 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 \(k\).

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 \(T _ { 2 } \mathrm {~N}\).

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.

6 The physical quantity pressure, denoted by \(P\), can be calculated using the formula \(P = \frac { F } { A }\) where \(F\) is a force and \(A\) is an area.

1. Find the dimensions of \(P\). An object of mass \(m\) is moving on a smooth horizontal surface subject to a system of forces which begin to act at time \(t = 0\). The initial velocity of the object is \(u\) and its velocity and acceleration at time \(t\) are denoted by \(v\) and \(a\) respectively. The object exerts a pressure \(P\) on the surface. The total work done by the forces is denoted by \(W\). A Mathematics class suggests three formulae to model the quantity \(W\).
   The first suggested formula is \(W = \frac { 1 } { 2 } m v ^ { 2 } - \frac { 1 } { 2 } m u ^ { 2 } + m P\).
2. Use dimensional analysis to show that this formula cannot be correct. The second suggested formula is \(W = k u ^ { \alpha } v ^ { \beta } t ^ { \gamma }\) where \(k\) is a dimensionless constant.
3. Use dimensional analysis to show that this formula cannot be correct for any values of \(\alpha , \beta\) and \(\gamma\). The third suggested formula is \(W = k u ^ { \alpha } a ^ { \beta } m ^ { \gamma } t ^ { \delta }\) where \(k\) is a dimensionless constant.
4. 1. Explain why it is not possible to use dimensional analysis to determine the values of \(\alpha\), \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
   2. Given that \(\alpha = 3\), use dimensional analysis to determine the values of \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
   3. By considering what the formula predicts for large values of \(t\), explain why the formula derived in part (d)(ii) is likely to be incorrect.

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.

5 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 \(\mathrm { v } ^ { 2 } = \mathrm { pu } ^ { \alpha } + \mathrm { qa } ^ { \beta } \mathrm { s } ^ { \gamma }\) where \(p\) and \(q\) are dimensionless constants.

1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent.
2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\).
3. By considering the case where \(s = 0\), determine the value of \(p\).
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 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.

5 In this question you may assume that if \(x\) and \(y\) are any physical quantities then \(\left[ \frac { \mathrm { dy } } { \mathrm { dx } } \right] = \left[ \frac { \mathrm { y } } { \mathrm { x } } \right]\). A machine drives a piston of mass \(m\) into a vertical cylinder. The equation below is suggested to model the power developed by the machine, \(P\), while it is not doing any other external work. $$\mathrm { P } = \mathrm { k } _ { 1 } \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dt } } + \mathrm { k } _ { 2 } \mathrm { mgv } + \mathrm { k } _ { 3 } \mathrm { E }$$ in which

• \(v\) is the velocity of the piston at a given time,
• \(g\) is the acceleration due to gravity,
• \(E\) is the rate at which heat energy is lost to the surroundings,
• \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are dimensionless constants.

Determine whether the equation is dimensionally consistent. Show all the steps in your argument.

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.

3 A particle \(P\) of mass \(m\) moves on the \(x\)-axis under the action of a force \(F\) directed along the axis. When the displacement of \(P\) from the origin is \(x\) its velocity is \(v\).

1. By using the fact that the dimensions of the derivative \(\frac { d v } { d x }\) are the same as those of \(\frac { v } { x }\), verify that the equation \(\mathrm { F } = \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dx } }\) is dimensionally consistent. It is given that \(\mathrm { v } = \mathrm { km } ^ { - \frac { 1 } { 2 } } \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } }\) where \(a\) and \(k\) are constants.
2. Explain why \([ a ]\) must be the same as \([ x ]\).
3. Deduce the dimensions of \(k\).
4. Find an expression for \(F\) in terms of \(x\) and \(k\).

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 \(\gamma\).

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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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 \(\gamma\).

2 A pile driver of mass \(m _ { 1 }\) falls from a height \(h\) onto a pile of mass \(m _ { 2 }\), driving the pile a distance \(s\) into the ground. The pile driver remains in contact with the pile after the impact. A resistance force \(R\) opposes the motion of the pile into the ground. Elizabeth finds an expression for \(R\) as $$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$ where \(g\) is the acceleration due to gravity.
Determine whether the expression is dimensionally consistent.

2 A car has mass \(m\) and travels up a slope which is inclined at an angle \(\theta\) to the horizontal. The car reaches a maximum speed \(v\) at a height \(h\) above its initial position. A constant resistance force \(R\) opposes the motion of the car, which has a maximum engine power output \(P\). Neda finds a formula for \(P\) as $$P = m g v \sin \theta + R v + \frac { 1 } { 2 } m v ^ { 3 } \frac { \sin \theta } { h }$$ where \(g\) is the acceleration due to gravity.
Given that the engine power output may be measured in newton metres per second, determine whether the formula is dimensionally consistent.

2 A rod, of length \(x \mathrm {~m}\) and moment of inertia \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude \(J\) Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between \(J , x , I\), the acceleration due to gravity \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and a dimensionless constant \(k\), such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
Find the values of \(\alpha , \beta\) and \(\gamma\) for which this relationship is dimensionally consistent.
[0pt] [6 marks]

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 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 \(h = 1.7\).

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.