Verify dimensional consistency

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.

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.

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 \(k\).

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.

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. 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 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).

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 .

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.

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 \(\gamma\).

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 .

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]