5.

5.2.9 Moment of Inertia: In the study of rotatory motion in mechanics, an important role is played by a certain quantity called a moment of inertia which will now be discussed briefly.

We suppose that a particle m at a distance y from the x-axis rotates uniformly about that axis with angular velocity w (that is, it rotates in unit time through an angle w). The kinetic energy of the particle, expressed by half the product of the mass and the square of its velocity, is obviously

We call the coefficient of w²/2, that is the quantity my², the moment of inertia of the particle about the x-axis.

Similarly, if we have n particles with masses m₁, m₂, ··· , m_n and ordinates y₁, y₂, ··· , y_n , we call the expression

the moment of inertia of the system of masses about the x-axis. The moment of inertia is a quantity which belongs to the system of masses itself, without any reference to its state of motion. Its importance lies in the fact that, if the entire system is set in constant rotation about an axis, without change of the distances between pairs of particles, the kinetic energy is obtained by multiplying the moment of inertia about that axis by half the square of the angular velocity. Thus, the moment of inertia about an axis has the same role in rotation about an axis as has mass in rectilinear motion.

Suppose now that we have an arbitrary curve y = f(x), lying between the abscissae x₀and x₁ > x₀, along which a mass is distributed uniformly with unit density. In order to define the moment of inertia of this curve, we just proceed as we did in the 5.2.7; as before, we arrive at an expression for the moment of inertia about the x-axis, namely,

We have for the moment of inertia about the y-axis the corresponding expression

5.3. Examples

The theory of plane curves with its great variety of special forms and properties offers us a rich store of examples of these abstract concepts. But in order to avoid being lost in a mass of detail, we must limit ourselves to a few typical applications.

5.3.1 The Common Cycloid: We obtain at once from the equations x = a(l - cos t), y = a(l - sin t) the equations whence the length of arc is

However, since 1 - cos t = 2sin²t/2, the integrand is equal to 2a sin t/2, whence for 0 £ a £ 2p

In particular, if we consider the length of arc between two successive cusps, we must set a = 2p, since the interval 0 £ t £ 2p of values of the parameter corresponds to one revolution of the rolling circle. We thus obtain the value 8a, that is, the length of arc of the cycloid between successive cusps is equal to four times the diameter of the rolling circle.

Similarly, we calculate the area bounded by one arch of the cycloid and the x-axis:

This area is therefore three times the area of the rolling circle.

For the radius of curvature r = 1/k, we have

at the points t = 0, t = ±2p, ···, this expression has the value zero. These are actually the cusps, where the cycloid meets the x-axis at right angles.

The area of the surface of revolution formed by rotation of an arch of the cycloid about the x-axis is given by our formula (cf. 5.2.8) as

The last integral can be evaluated by means of the substitution cos u = v; we find

As an exercise, the reader should find the height h of the centre of mass of the cycloid above the x-axis as well as its moment of inertia

5.3.2 The Catenary: The length of arc of the catenary has already been found as an example at the end of 5.2.5:

For the area of the surface of revolution obtained by rotating the catenary about the x-axis - the solaced catenoid - , we find

Moreover, this yields the height of the arc's centre of mass between a and b

and, finally, the curvature

5.3.3 The Ellipse and the Lemniscate: The length of arc of these two curves cannot be reduced to elementary functions; they belong to the class of elliptic integrals, referred to in 4.7.1.

For the ellipse , we obtain

where we have set x/a = x, 1 - b²/a² = k². By the substitution x = sin f, this integral becomes

In order to obtain the semi-perimeter of the ellipse, we must let here x cover the interval from -a to +a, which corresponds to the interval

For the lemniscate with the equation in polar co-ordinates r²=2a²cos2t, we obtain in a similar manner

If we introduce the independent variable u = tan t into the last integral, we have

whence

In a complete loop of the lemniscate, u ranges from -1 to +1, whence the length of arc equals

a special elliptic integral which occupied an important place in the research of Gauss.

Exercises 5.2:

1. Calculate the area bounded by the semi-cubical parabola y = x^3/2, the x-axis and the Lines « = a and x = b.

2. Calculate the area of the region bounded by the line y = x and the lower half of the loop of the folium of Descartes. (Use the parametric representation of 5.1.5 , Exercise 7.)

3. Calculate the area of a sector of the Archimedian spiral r = aq (a > 0).

4. Caloulate the area of the cardioid (5.1.5 , Exercise 3.), using polar coordinates

5. Calculate the area of the astroid (5.1.5 , Exercise 6.).

6. Calculate the area of the pedal curve of the circle x² + y² = 1 with respect to a point P (x₀,0} on the x-axis. Show that this area has a minimum when P is at the origin.

7. Do the same as in 6. for the ellipse x²/a² + y²/b² = 1.

8. Find the parametric representation of the cardioid when the length of arc is used as parameter.

9. Find the parametric representation of the cycloid when the length of arc is used as parameter.

10. Calculate the length of arc of the semi-cubical parabola y = x^3/2.

11. Calculate the length of the astroid.

12. Calculate the length of arc of:

(a) The Archimmedian spiral r = aq (a > 0).
(b) The logarithmic spiral r = e^m^q.
(c) The cardioid (5.1.5, Exercise 3.).
(d) The curve r = a(q ² - 1).

13. Find the radius of curvature of (a) the parabola y = x²; (6) the ellipse x=acos f, y=bsin f as a function of x and of f, respectively. Find the maxima and minima of the radius of curvature and the points at which these maxima and minima occur.

14. Sketch the curve

and determine its radius of curvature (r).

16. Show that the expression for the curvature of a curve x = x(t), y = y(t) is unaltered by a rotation of axes as well as by the change of parameter t = f(t), where f '(t) > 0.

16. Let r = f(q ) be the equation of a curve in polar co-ordinates. Prove that the curvature is given by

17, Find the volume and surface of a zone of a sphere of radius r, i.e., of the portion of the sphere cut off by two parallel planes distant h₂,h₁ respectively, from the centre.

18. Find the volume and surface area of the torus or anchor ring, obtained by rotating a circle about a line which does not intersect it.

19. Find the area of the catenoid, the surface obtained by rotating an arc of the catenary y = cosh x about the x-axis.

20. Sketch the curve defined by the equation

What is the behaviour of the curve as t ranges from -¥ to +¥? Calculate the curvature k as a function of the length of arc.

21. The curve for which the length of the tangent intercepted between the point of contact and the y-axis is always equal to 1. It is called the tractrix. Find its equation. Show that the radius of curvature at each point of the curve is inversely proportional to the length of the normal intercepted between the point on the curve and the y-axis. Calculate the length of arc of the tractrix and find its parametric equations in terms of the length of arc.

22. Let x = x(t), y = y(t) be a closed curve. A constant length p is measured off along the normal to the curve. The extremity of this segment describes a curve, which is called a parallel curve to the original curve. Find the axes, the length of arc and the radius of curvature of the parallel curve.

23. Find the centre of mass of an arbitrary arc (a) of a circle of radius r, (b) of a catenary.

24. Calculate the moment of inertia about the x-axis of the boundary of the rectangle a £ x £ b, a £ y £ b.

35. Calculate the moment of inertia of an arc of the catenary y = cosh x (a) about the x-axis, (b) about the y-axis.

26. The equation y = f(») + a, a £ x £ b, represent a family of curves, one for each value of the parameter a. Prove that m this family the curve with the smallest moment of inertia about the x-axis is that which has its centre of mass on the x-axis.

Answers and Hints

5.4 Some Very Simple Problems in the Mechanics of a Particle

Next to geometry, the differential and integral calculus are especially indebted to the science of mechanics for their early development. Mechanics rests upon certain basic principles which were first laid down by Newton; the statement of these principles involves the concept of the derivative and their application requires the theory of integration. Without analyzing these basic principles in detail, we shall illustrate by some simple examples how the integral and differential calculus are applied in mechanics.

5.4.1 The Fundamental Hypotheses of Mechanics: We shall restrict ourselves here to the consideration of a single particle, that is of a point at which a mass m is imagined to be concentrated. Moreover, we shall assume that motion can only occur along a certain fixed curve, on which the position of the particle is specified by the length of arc s measured from a fixed point on the curve; in particular, the curve may be a straight line, in which case we shall use the abscissa x as the coordinate of the point instead of s. The motion of the point is determined by expressing the co-ordinate s = f (t) as a function of the time. We shall mean by the velocity of motion the derivative f '(t) or, as we shall also write,

We shall call the second derivative

the acceleration.

In mechanics, we start from the assumption that the motion of a point can be explained by means of forces of definite direction and magnitude. In the case of motion on a given curve, Newton's second fundamental law may be expressed as follows:

The mass multiplied by the acceleration is equal to the force acting on the particle in the direction of the curve; in symbols:

Thus, the direction of the force is always the same as that of the acceleration; its direction is that of increasing values of s, if the velocity in that direction is increasing, otherwise it is opposed to the direction of increasing values of s.

The law of Newton is in the first instance nothing more than a definition of the concept of force. The left hand side of our equation is a quantity, which can be determined by observation of the motion, by means of which we measure the force. But this equation has a far deeper meaning. As a matter of fact, it turns out that in many cases we can determine the acting force from other physical assumptions without any consideration of the corresponding motion. The above fundamental law of Newton is then no longer a definition of the force, but instead it is a relation from which we can draw important conclusions about the motion.

The most important example of a known force is given to us by gravity. We know from direct measurements that the force of gravity acting on a mass m is directed vertically downwards and is of magnitude mg, where the constant g, the so-called gravitational acceleration, is approximately equal to 981 if the time is measured in seconds and the lengths in centimetres. If a mass moves along a given curve, we learn by experiments that the force of gravity in the direction of this curve is equal to mgcos a, where a denotes the angle between the vertical and the tangent to the curve at the point under consideration (Fig. 12).

In the case of motion on our given curve, the basic problem of mechanics is: If we know the force acting on the particle (e.g., the force of gravity), we have to determine the position of the point, that is, its co-ordinate s or x, as a function of the time.

If we restrict ourselves to the simplest case, in which this force * mf(s) is known at the outset as a function of the length of arc - so that the force is independent of the time - we shall show how the course of the motion along the curve can be found from the equation

* The separation of the factor m in the expression for the given force is not essential, but makes the formula simpler.

We have to deal here with a differential equation, that is, an equation from which an unknown function - here s(t) - is to be determined and in which the derivative of this function occurs as well as the function itself (3.7.1)

5.4.2 Freely Falling Body. Resistance of Air: In the case of the free fall of a particle along the vertical x-axis, Newton's law yields the differential equation

Hence we have the equation where v₀ is a constant of integration. Its meaning is easily found by setting t = 0. We then find that is, v₀ is the velocity of the particle at the instant from which the time is reckoned - the initial velocity. Another integration yields

where x₀ is also a constant of integration, the value of which is again found by setting t = 0; we thus find that x₀ is the initial position, that is, the co-ordinate of the point at the beginning of the motion.

Conversely, we can choose the initial position x₀ and the initial velocity v₀ arbitrarily and then obtain the complete representation of the motion from the equation

If we wish to take account of the effect of the friction or air resistance acting on the particle, we have to consider this as a force the direction of which is opposite to the direction of motion and concerning which we must make definite physical assumptions.

These assumptions must be chosen to suit the particular system under consideration; for example, the law of resistance for low speeds is not the same as that for high speeds (e.g., bullet velocities).

We shall now work out the results of different physical assumptions: (a) the resistance is proportional to the velocity, being given by an expression of the form where r is a positive constant; (b) the resistance is proportional to the square of the velocity, given as . In accordance with Newton's law, we obtain for the equations of motion

If we at first consider as the function sought, we have so that

Instead of determining by these equations u as a function of t , we determine t as a function of u, writing our differential equations in the form

With the methods, given in the Chapter IV, we can immediately carry out the integrations and obtain

where we have set and t₀ is a constant of integration. Solving these equations for u, we find

These equations at once reveal an important property of the motion. The velocity does not increase with time beyond all bounds, but tends to a definite limit which depends on the mass m. We have for

A second integration, performed on our expressions for by the methods of the Chapter IV yields the results (which may be verified by differentiation)

where c is a new constant of integration. The two constants of integration t₀and c are readily determined, if we know the initial position x(0) = x₀ and the initial velocity of the falling particle.

5.4.3 The Simplest Type of Elastic Vibration: As a second example, we consider the motion of a particle which moves along the x-axis and is pulled back towards the origin by an elastic force. As regards this force, we assume that it is always directed towards the origin and that its magnitude is proportional to the distance from the origin. In other words, we assume the force to be equal to -kx, where the coefficient k is a measure of the stiffness of the elastic link. Since k is assumed to be positive, the force is negative when x is positive and positive when x is negative. Newton's law now says

We cannot expect that this differential equation will determine the motion completely, but it is plausible to assume that for a given instant of time, say t = 0, we can arbitrarily assign the initial position x(0) = x₀ and the initial velocity in physical language, it says that we can launch the particle from an arbitrary position with an arbitrary velocity and that thereafter the motion is determined by the differential equation. Mathematically speaking, this is expressed by the fact that the general solution of our differential equation contains two constants of integration, at first undetermined, the values of which we find by means of the initial conditions. We shall prove this fact immediately.

We can easily state directly such a solution. If we set , we may at once verify by differentiation that our differential equation is satisfied by the above expression. This expression is readily written in the form

we need only write - a sin wd = c₁ and a cos wd = c₂, thus introducing instead of c₁ and c₂the new constants a and d. Motions of this type are said to be sinusoidal or simple harmonic. Such motions are periodic; any state, i.e. position x(t) and velocity , is repeated after the time T=2p /w, which is called the period, since the functions sin w t and cos w t have the period T. The number a is called the maximum displacement or amplitude of the oscillation. The number 1/T = w/2p is called the frequency of the oscillation; it measures the number of oscillations per unit time. We shall return to the theory of oscillations in Chapter XI.

5.4.4 Motion on a Given Curve: Finally, we shall discuss the most general form of the problem stated above, namely, the problem of the motion along a given curve under an arbitrary pre-assigned force mf(s).

The objective here is the determination of the function s(t) as a function of t by means of the differential equation

where f(s) is a given function. This differential equation in s can be solved completely by the following device.

We begin by considering any primitive function F(s) of f(s), so that F '(s)=f(s), and multiply both sides of the equation by We can then write the left hand side in the form as we see at once by differentiating the expression however, he right hand side is the derivative of F(s) with respect to the time t, if we regard in F(s) the quantity s as a function of t. Hence, we have immediately

or, by integration,

where s denotes a constant yet to be determined.

Let us write this equation in the form We see that we cannot find s immediately from this equation as a function of t by integration. However, we arrive at a solution of the problem, if we at first content ourselves with finding the inverse function t(s), that is, the time taken by the particle to reach a definite position s. We have for this the equation

thus, the derivative of the function t(s) is known and we have

where c₁ is another constant of integration. As soon as we have performed the last integration, we have solved the problem, for which we have not the position s as a function of t; we have found inversely the time t as a function of the position s. The fact that the two constants of integration c and c₁are still available, allows us to fit the general solution to special initial conditions.

In the above example of elastic motion, we have to identify x with s; we have f(s) = -w ²s and correspondingly, say, F(s) =-w ²s ². We therefore obtain

whence

However, this Integral is easily evaluated by introducing as a new variable; we thus obtain

or, forming the inverse function

Thus, we are led to exactly the same statement of the solution as before.

From this example, we also see what the constants of integration mean and how they are to be determined. For example, if we require that at time t = 0 the particle shall be at the point s = 0 and at that instant have the velocity we obtain the two equ»tions

from which we find that the constants have the values c₁ = 0, c = ½. The constants of integration c and c₁can be determined in exactly the same way when the initial position s₀ and the initial velocity are prescribed arbitrarily.

Exercises 5.3:

1. A point A moves at constant velocity 1 along a circle with radius r and centre at the origin. The point A is linked to a point B by a line of constant length l (>r); B is constrained to move on the x-axis (like a crank, connecting-rod and piston of a steam engine). Calculate the velocity and acceleration of B as functions of the time.

2. A particle starts from the origin at velocity 4 and slides under the influence of gravity down a straight wire until it reaches the vertical line x = 2. What must be the slope of the path in order that the point may reach the vertical line in the shortest time?

3. A particle moves along a straight line subject to a resistance producing the retardation ku³, where u is the velocity and k a constant. Find expressions for the velocity (u) and the time (i) in terms of s, the distance from the initial position, and v₀, the initial velocity.

4. A particle of unit mass moves along the x-axis and is acted upon by a force f(x) = - sin x.
(a) Determine the motion of the point if it is at time t = 0 at the point x = 0 and has the velocity v₀ = 2. Show that as t ® ¥, the particle approaches a limiting position and find this position.
(b) If the conditions are the same except that v₀ may have any value, show that, if v₀ > 2, the point moves to an infinite distance as a t ® ¥ and, if v₀ < 2, the point oscillates abont the origin.

5. Choose axes with their origin at the centre of the earth with radius R. By Newton's law of gravitation, a particle of unit mass lying on the y-axis is attracted by the earth with a force mM/r², where m is the gravitational constant and M is the mass of the earth.
(a) Calculate the motion of the particle after it has been released at the point y₀(> R), i.e., if at time t = 0, it is at the point y = y₀ and has the velocity v₀= 0. (b) Find the velocity with which the particle in (a) strikes the earth.
(c) Using the result of (b), calculate the velocity of a particle falling to the earth from infinity.
This is the same velocity at which a projectile would have to be launched in order that it should leave the earth and never return (escape velocity).

6.* A particle of mass m moves along the ellipse r = k/(l - e cos q). The force on the particle is cm/r² and directed towards the origin. Describe the motion of the particle, find its period and show that the radius vector to the particle sweeps out equal areas in equal times.

Answers and Hints

5.5 Further Applications. A Particle sliding down a Curve

5.5.1 General Remarks: The case of a particle sliding along a frictionless curve under the influence of gravity can be treated very simply by the method just described. We shall first discuss the motion in general and then with special reference to the case of the ordinary pendulum and the cycloidal pendulum. We choose axes in such a way that the y-axis points vertically upwards, i.e., opposite to the direction of the force of gravity, and consider the curve as given in terms of a parameter q by the parametric equations

x = f(q) = x(q), y = y(q) = y(q).

A portion of the curve, for which the motion will be studied, is shown in Fig. 13. At every point of the curve, the force of gravity acts downwards (i.e., in the direction of decreasing y) on the particle with magnitude mg. If we denote the angle between the negative y-axis and the tangent to the curve by a, then, according to the hypothesis in 5.4.1, the force acting along the direction of the curve is

where

(Note that here the inverted comma ' denotes the derivative with respect to q and not with respect to x.) In particular, if we introduce the length of arc s as parameter in place of q, we obtain the expression - mgdy/ds for the force along the curve. Hence, by Newton's law, the function s(t) satisfies the differential equation

The right hand side of this equation is a known function of s, since we know the curve and must therefore regard the quantities x and y to be known functions of s.

As in 5.4.4, we multiply both sides of this equation by The left hand side then becomes the derivative of with respect to t. If we regard in the function y(s) the variable s to be a function of t, the right hand side of our equation is the derivative of -gy with respect to t. Hence we find on integrating

where c is a constant of integration. In order to fix the meaning of this constant, we assume that at the time t = 0 our particle is at the point of the curve for which the value of the parameter is q₀ and the co-ordinates are x=f(q₀), y₀=y(q₀), and that at this instant its velocity is zero, i.e., Then, setting t = 0, we find immediately whence

Now, instead of regarding s as a function of t, we shall consider the inverse function t(s). Then we obtain at once

which is equivalent to

where c₁ is a new constant of integration. As regards the sign of the square root, which is the same as the sign of , we note that, if the particle moves along an arc, which is lower than y₀ everywhere except at the ends, the sign cannot change. In fact, the sign of can change only when = 0, that is, where y - y₀ = 0. The integrand on the right hand side is known in terms of the parameter q, since the curve is known. Introducing q as independent variable, we obtain

where the functions x' = f '(q ), y' = y '(q ), y = y (q ) are known. In order to determine the constant of integration c₁, we note that for t = 0 the value of the parameter must be q₀. This yields immediately our solution in the form

When integrated, the equation represents the time taken by the particle to move from the parameter value q₀ to the parameter value q. The inverse function q(t) of this function t(q ) enables us to describe the motion completely, because we can determine at each instant t the point x = f {q (t)}, y=y{q (t)}, which the particle is then passing.

5.5.2 Discussion of the Motion: We can deduce the general nature of the motion by simple intuitive reasoning from the equations just found, without an explicit expression for the result of the integration. We assume that our curve is of the type shown in Fig. 14, i.e., that it consists of an arc which is convex downwards; we take s as increasing from the left hand side to the right hand side. If we initially release the particle at the point A with the co-ordinates x=x₀, y=y₀, corresponding to q = q₀, the velocity increases, because the acceleration is positive. The particle travels from A to the lowest point with increasing velocity. However, after it reaches the lowest point, the acceleration is negative, since the right hand side -gdy/ds of the equation of motion is negative. The velocity therefore decreases. From the equation s² = - 2g(y — y₀), we see at once that the velocity reaches the value 0 when the particle reaches the point B, the height of which is the same as that of the initial position A. Since the acceleration is still negative, the motion of the particle must be reversed at this point, so that the particle will swing back to the point A; this action will repeat itself indefinitely. (The reader will recall that friction has been disregarded!) In this oscillatory motion, the time which the point takes to return from B to A must obviously be the same as the time taken to move from A to B. If we denote the time required for a complete journey from A to B and back by T, the motion will obviously be periodic with the period T. If q₀ and q₁ are the values of the parameter corresponding to the points A and B, respectively, the half-period is given by

If q₂is the value of the parameter corresponding to the lowest point of the curve, the time which the particle takes to fall from A to this lowest point is

5.5.3 The Ordinary Pendulum:. The simplest example is given by the so-called ordinary pendulum, when the curve under consideration is a circle of radius l:

where the angle q is measured in the positive sense from the position of rest. We obtain at once from the general expression above

where a (0 < a < p) denotes the amplitude of oscillation of the pendulum, i.e., the angular position from which the particle is released at time t=0 at velocity 0. The substitution

yields for the period of oscillation of the pendulum

We have thus expressed the period of oscillation of the pendulum by an elliptic integral. If we assume that the amplitude of the oscillation is small, so that we may with sufficient accuracy replace the second factor under the square root by 1, we obtain

as an approximation for the period of oscillation. We can evaluate this last integral by using 13. in our table of integrals and obtain for T the approximate expression

5.5.4 The Cycloidal Pendulum: The fact that the period of oscillation of the ordinary pendulum is not strictly independent of the amplitude of oscillation caused Christian Huygens, in his prolonged efforts to construct accurate clocks, to look for a curve for which the period of oscillation is strictly independent of the particular position on the curve at which the oscillating particle begins its motion; the oscillations are then said to be isochronous. Huygens discovered that the cycloid is such a curve.

In order that a particle may actually be able to oscillate on a cycloid, its cusps must point in the direction opposite to that of the force of gravity, i.e., we must rotate the cycloid, considered earlier (5.1.2) about the x-axis (Fig. 15), whence we write the equations of the cyloid in the form

which also involves a translation of the curve by a distance 2a in the positive y-direction. The time which the particle takes to travel from a point at the height

down to the lowest point is, by the formula worked out in 5.5.1,

We now use the equation

it yields

We then work out the definite integral, employing the substitution

whence

The period of oscillation T is therefore actually independent of the amplitude.

5.6 Work

5.6.1 General Remarks: The concept of work throws new light on the considerations of the last section and on many other questions of mechanics and physics.

Let us think again of a particle, moving on a curve under the influence of a force acting along the curve, and assume that its position is specified by the length of the arc measured from any fixed initial point. The force itself will then, as a rule, be a function of s. We assume that it is a continuous function f(s) of the length of arc. This function will have positive values, where the direction of the force is the same as the direction of increasing values of s, and negative values where the direction of the force is opposite to that of increasing value of s.

If the magnitude of the force is constant along the path, we mean by the work done by the force the product of the force by the distance (s₁ - s₀) traversed, where s₁ denotes the final point and s₀ the initial point of the motion. If the force is not constant, we define the work by means of a limiting process. We subdivide the interval from s₀ to s₁ into n equal or unequal sub-intervals and note that, if the sub-intervals are small, the force in each one is nearly constant; if s_n is an arbitrarily chosen point in the n-th subinterval, then throughout this subinterval the force will be approximately f(s_n). If the force throughout the n-th subinterval were exactly f(s_n), the work done by our force would be exactly

which we naturally call the work done by the force.

If the direction of the force and that of the motion on the curve are the same, the work done by the force is positive; we then say that the force does work. On the other hand, if the direction of the force and that of the motion are opposed, the work done by the force is negative; we then say that work is done against the force.

Note that we must here carefully distinguish the forces uder consideration. For example, in lifting a weight, the work done by the force of gravity is negative: Work is done against gravity. But from the point of view of the person doing the lifting, the work done is positive, for the person must exert a force opposed to gravity.

If we regard the co-ordinate of the position s as a function of the time t, so that the force f(s) = p is also a function of t, then we can plot the point in a plane with rectangular co-ordinates s and p with the co-ordinates s = s(t), p = p(t) as a function of the time. This point will describe a curve, which may be called the work diagram of the motion. If we are dealing with a periodic motion, as in the case of any machine, then, after a certain time T (one period), the moving point s = s(t), p = p(t) will return to the same position, i.e., the work diagram will he a closed curve. In this case, the curve may consist simply of one and the same arc, traversed first forwards and then backwards, as happens, for instance, in elastic oscillations. But it is also possible for the curve to be a more general closed curve, enclosing an area, for example, in the case of machines in which the pressure on a piston is not the same during the forward and backward stroke. The work done in one cycle, i.e., in time T, will then be given simply by the negative area of the work diagram or, in other words, by the integral

where the interval of time from t₀ to t₀ + T represents exactly one period of the motion. If the boundary of the area is positively traversed, the work done is negative, if negatively traversed, it is positive. If the curve consists of several loops, some traversed positively and others negatively, the work done is given by the sum of the areas of the loops, each with its sign changed.

These considerations are illustrated in practice by the indicator diagram of a steam engine. By a suitably designed mechanical vice, a pencil is made to move over a sheet of paper; the horizontal motion of the pencil relative to the paper is proportional to the distance s of the piston from its extreme position, while the vertical motion is proportional to the steam pressure, and hence proportional to the total force p of the steam on the piston. The piston therefore describes the work diagram for the engine on a known scale. The area of this diagram is measured (usually by means of a planimeter) and the work done by the steam on the piston is thus found. Here we also see that our convention for the sign of an area, as discussed in 5.2.2, is not only of theoretical interest. In fact, it does happen sometimes, when an engine is running light, that the highly expanded steam at the end of the stroke has a pressure lower than that required to expel it on the return stroke; on the diagram, this is shown by a positively traversed loop; the engine itself is drawing energy from the flywheel instead of providing energy.

5.6.2 The Mutual Attraction of Two Masses: Let a particle attract another particle according to Newton's law of attraction; as a first example, we shall consider the work done by the force of attraction as the second particle moves along the line joining the two particles. By Newton's law of gravitation, the attracting force is inversely proportional to the square of the distance. If we imagine the first particle to be at rest at the origin and the second particle at the distance r from the origin, the attracting force is given by

where m is a positive constant. Hence the work done by this force, as the particle moves from the distance r to the distance r₁ > r, is positive and equal to the integral

If the particle is moved further away from the origin from r to the distance r₁ > r by means of an opposing force, the work done by the force of attraction will, of course, still be given by this integral (now negative). The work done by the opposing force has the same numerical value, but the opposite sign, whence it is equal to m (1/r - 1/r₁). If we think of the final position as being chosen further and further away, this approaches the limiting value m/r, which we may call the work which must be done against the force of attraction in order to move the particle from the distance r to infinity. This important expression is called the mutual potential of the two particles. Hence, the potential here is defined as the work required to separate completely two attracting masses; for example, it is the work required to tear an electron completely away from an atom (ionization potential)

5.6.3 The Stretching of a Spring: As a second example, we consider the work done in stretching a spring. Usually, in the theory of elasticity, we assume (5.4.2) that the force needed to stretch the spring is proportional to x - the increase in the length of the spring, i.e., p = kx, where h is a constant. The work which must be done in order to stretch the spring from the unstressed position x = 0 to the final position x = x₁ is therefore given by the integral

5.6.4 The Charging of a Condenser:. The concept of work in other branches of physics can be treated in a similar manner. For example, we may consider the charging of a condenser. If we denote the quantity of electricity in the condenser by Q, its capacity by C and the potential difference (voltage) across the condenser by V, then we know from physics that Q = CV. Moreover, the work done in moving a charge Q through a potential difference V is equal to QV. Since during the charging of the condenser the difference of potential V is not constant, but increases with Q, we perform a passage to the limit exactly analogously to that in 5.6.1 and obtain for the work done during charging a condenser

where Q₁ is the total quantity of electricity fed into the condenser and V₁is the difference of potential across the condenser at the end of the charging process.

Appendix to Chapter V

A5.1 Properties of the Evolute

The parametric equations

for the evolute of a given curve x = x(t), y = y(t) (5.2.6) enable us to derive some interesting geometrical relations between it and the given curve. For the sake of convenience, we use the length of arc s as parameter, so that

We thus have the equations

on differentiation, they yield

whence

Since the direction cosines of the normal to the curve are given by -, it follows that the normal to the curve is tangent the evolute at the centre of curvature, or, the tangents to the evolute are the normals of the given curve or the evolute is the envelope of the normals (Fig. 16).

Moreover, if we denote the length of arc of the evolute, measured from an arbitrarily fixed point, by s, we have

Since we obtain from the above formulae

so that, if we choose the direction in which s is measured in a suitable way, then

provided that s ¹ 0, or on integration

Hence the length of arc of the evolute between two points is equal to the difference of the corresponding radii of curvature, provided that does not vanish for the arc under consideration.

This last condition is not superfluous, because, if changes sign, according to the length of arc, on passing the corresponding point of the evolute, has a maximum or minimum, i.e., on passing this point, we do not simply continue to reckon s onward, but must reverse the sense in which it is measured. If we wish to avoid this, we must on passing such a point change the sign in the above formula, i.e., set

It may also be noted that the centres of curvature, which correspond to maxima or minima of the radius of curvature, are cusps of the evolute. (The proof will not be given here.)

The geometrical relationship, which we have just found, can be expressed in yet another way. If we imagine a flexible, inextensible thread laid along an arc of the evolute and stretched so that a part of it extends away from the curve tangentially to it, and if, in addition, the end-point Q of this thread lies on the original curve C, then, as we unwind the thread, the point Q will describe the curve C. This accounts for the name evolute (Latin: evolvere, unwind). The curve C is called an involute. On the other hand, we may start with an arbitrary

The

curve E and construct its involute C by this unwinding process. We then see that E conversely is the evolute of C.

In order to prove this, consider the curve E, which is now the given curve, as given in the form x=x(s), h=h(s), where the current rectangular co-ordinates are denoted by x and h, and the parameter s is the length of arc. The winding is done as indicated in Fig. 17; when the thread is completely wound on to the evolute E, its end Q coincides with the point A of E, corresponding to the length of arc a. If the thread is now unwound until it is tangent to the evolute at the point P, corresponding to the length of arc s £ a, the length of the segment PQ will be (a - s) and its direction cosines will be where the dot denotes differentiation with respect to s. Thus, for the co-ordinates x, y of the point Q, we obtain the expressions

which yield the equations for the involute, described by the point Q in terms of the parameter s. Differentiation with respect to s yields

Since , we find at once that

which shows that the line PQ is normal to the involute C. We can therefore state that the normals to the curve C are tangent to the curve E. But this is the characteristic property of E - the evolute of C. Hence every curve is the evolute of all its involutes.

We consider as a particular case the evolute of the cycloid x=t-sin t, y=t-cost. By (5.2.6),

hence we obtain the evolute in the form x = t + sin t, h = -t + cos t. If we set t=t+p, then
x - p = t - sin t, h - 2 = 1 - cos t and these equations show that the evolute is itself a cycloid which is similar to the original curve and can be obtained from it by translation, as is indicated in Fig. 18.

As a farther example we shall work out the equation for the involute of the circle. We begin with the circle x = cos t, h =sin t and unwind the tangent, as is shown in Fig. 19. The involute of the circle is then given in the form

Finally, we shall determine the evolute of the ellipse x = a cos t, y = bsin t. We find immediately

and

which is a parametric representation of the evolute. If we eliminate from these equations in the usual way, we obtain the equation of the evolute in the non-parametric form

This curve is called an astroid (Fig.20). By means of the parametric equation, we may readily convince ourselves that the centres of curvature corresponding to the vertices of the ellipse are actually the cusps of the astroid.

Exercises 5.4:

1. Show that the evolute of an epicycloid (Exercise 2 above) is another epicyoloid, which can be obtained from the first by rotation and contraction.

2. Show that the evolute of a hypocycloid (Exercise 4 above) is another hypocycloid which can be obtained from the first by rotation and expansion.

No Answers or Hints

A5.2 Areas bounded by closed curves

We have seen in 5.2.2 that the area bounded by a closed curve x=x(t), y=y(t), which nowhere intersects itself (a so-called simple closed curve) is given by the integral

where the value obtained is positive or negative according to whether the boundary is described in the positive or negative sense. We shall now extend this result to more general curves. Suppose that the curve C, given by the equations x=x(t), y=y(t) intersects itself at a finite number of points, thus dividing the plane into a finite number of sections R₁, R₂, ··· . Moreover, assume that the derivatives are continuous, except perhaps at a finite number of jump discontinuities and that ¹ 0 except perhaps at a finite number of values of t which may correspond to corners. Finally, also assume that the curve has a finite number of lines of support .

We then assign to each region R_i an index m_i, which is defined as follows: Select an arbitrary point Q in R_i, which does not lie on any line of support, and erect the line extending from Q upwards in the direction of the positive y-axis. . We count the number of times which the curve C crosses the half-line from right to left and subtract the number of times it crosses from left to right; the difference is the index m_i.

For example, the interior of the curve in Fig. 6 has the index m = +1; in Fig. 21 above, the regions R₁, . . . , R₅ have the indices m₁=-l, m₂=+l, m₃=+2, m₄=-2, m_{5
= -1}. These numbers m_i actually depend on the region R_i and not on the particular point Q chosen in R_i, as we readily see in the following manner. We choose any other point Q' in R_i, not on a line of support, and join Q to Q' by a broken line lying entirely in the region R_i. As we proceed along this broken line from Q to Q', the number of right-to-left crossings minus the number of left-to-right crossings is constant; in fact, the number of crossings between lines of support of either type is unchanged, while on crossing a line of support either the number of crossings of both types increases by one or else both the numbers decrease by one; in either case, the difference is unaltered. In the case where the line of support meets the curve at several different points, say A, B, ··· , H, we consider it as several different lines of support FA, FB, ··· , FH, where F is the point of the x-axis vertically below all the points A, B, ··· , H. Our argument then applies to each of these lines. Hence the number m_i has the same value whether we use Q or Q' in determining it.

In particular, if our curve does not intersect itself, the interior of the curve consists of a single region R the index of which is +1 or -1 according to whether the sense in which the boundary is described is positive or negative. In order to see this, we draw any vertical line (not a line of support) intersecting the curve; on this line, we find the highest point of intersection (P) with the curve and choose in R a point Q below P so near to it that no point of intersection lies between P and Q. Then there lies above Q one crossing of the curve, which, if the curve is traversed positively, must be a right-to-left crossing, so that m= +1, while otherwise m_i= -1. As we have just seen, this same value of m holds for every other point of R. For such a curve, and, in fact, for all closed curves, one of the regions, the outside of the curve, extends unboundedly in all directions; we see immediately that this region has the index 0, whence we neglect it.

Our theorem about the area is now as follows: The value of the integral is equal to the sum of the absolute areas of the regions R_i, each area R_i being counted m_i times, in symbols

The proof is simple. We assume, as we are entitled to do, that the entire curve lies above the x-axis. The lines of support cut R_i into a finite number of sections; let r be one of them. Then, on taking the integral for each single-valued branch of the curve, we find that the absolute area of r is counted +1 times for each right-to-left branch over r and -1 times for each left-to-right branch over r, altogether m_i times. The same is true for every other portion of R_i, whence R_i is counted m_i times. Thus, the integral around the complete curve has the value as stated. This formula agrees with what we have found for simple closed curves, as we recognize from the discussion of the values of m for such curves.

The definition given for the index m_i has the disadvantage of being stated in terms of a particular co-ordinate system. However, as a matter of fact, it can be shown that the value of m_i is independent of the co-ordinate system and depends solely on the curve; however, we will not prove this here.

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