Lecture 14

Elasticity

Whenever a force is applied to an object which is constrained not to move, the effect of the force will be to cause a deformation. This effect is in general not very visible, since the deformations are very small, especially if the object is very rigid, i.e. not very elastic (a perfectly rigid body is defined as one where the distance between any two points does not change, elastic is exactly the opposite).

Given a body of somewhat regular shape, one can think of applying forces perpendicular to one surface (stretching or compressing), parallel to it (shear) or somewhat uniformly to the whole bulk of the body. In general, one says that a certain stress causes a certain strain.

All of these situations are handled in a similar way, by realizing that the deformation in the object under stress is, over a fairly wide range, proportional to the force applied. More specifically, one describes the effects in terms of fractional (or percentage) deformation, i.e., for e.g. the case of stretching:

$F=YA\Delta L/L$

where F is the applied force, L is the unstretched length of the body, A its cross section, and Y (Young's modulus) is a constant, characterisitc of the material under exam. Given that the deformation caused by ordinary forces are very small, the values of Y (measured in N/m²) are very large (Y=1 would imply that the force of 1 N stretches a sample of unit cross section (1 m²) by 100%, hardly an everyday occurrrence)

Obviously an expression like the one above cannot be valid for any arbitrary value of applied force : when stretching an object with ever increasing force, we will say that we are in the elastic regime if, after the force ceases, the objects regains its original length. But even when in the elastic regime, the proportionality described by the above formula will be valid only up to a certain point.

When the force exceeds the elastic limit, the body will undergo permanent deformation (this is the "plastic regime", think of of putty or of pizza dough ....). If we keep increasing the force, we will eventually reach the breaking point.

Similar considerations also apply to shear and bulk deformations, which are described by similar expressions:

$F/A=S~\Delta X/L$ $F/A=-B~\Delta V/V$,

with S and B respectively the Shear and the Bulk moduli. Notice that the relevant quantity is not just the force, but the force per unit surface, F/A. This quantity is referred to as either stress or pressure, and is measured in pascal= newton/m² (stress is more commonly used for solids, pressure for liquids or gases.

Problem 10.14 Springs.

In many cases, we want to deal with objects that exhibit substantial (elastic) deformation when subjected to even ordinary forces : this is e.g. the case of springs. When dealing with springs, it is customary to think in terms of absolute rather than relative elongation. In other words, if applying a force F causes the spring to lengthen (or shorten) by an amount x, we write :

F=kx

where k is the spring constant. The bigger the value of the spring constant, the stiffer is the spring. It is easy to see the relation between spring constant and Young modulus. From

F=kx=YAx/L, we get k=AY/L

from which we can see, in a much more direct way than the one followed in example 5, that if we cut a given spring into two pieces, the pieces are twice as stiff as the original spring.

When dealing with a stretched or compressed spring, we can think either in terms of the force required to compress or stretch it by an amount x (F=kx) or of the force exerted by the spring itself upon you (F=-kx, by Newton's third Law).

Knowing the force, we should be able to study the motion of a mass driven by a spring:

$F = -kx = ma = m\Delta v/\Delta t = m \Delta (\Delta x/\Delta t)/\Delta t$

which tells us that the problem is not too simple, since the force depends on the position, but the position changes all the time, and therefore so does the force (a rigorous solution requires elementary calculus, see the Appendix).

But even if we do not know what the solution will look like, direct experience and thought can give us a fairly good idea : if we attach a mass m to a spring, stretch it and let it go, the spring will tend to return to its equilibrium value and, in doing so it will give some acceleration to the mass. When the system reaches the spring's position of equilibrium, the mass has a certain non-zero velocity (therefore some kinetic energy); due to inertia, the mass will tend to continue its motion and compress the spring, until all of its energy is used and it comes to rest. But now the spring will want to go back to its equlibrium position, in doing so it will put the mass back in motion .... etc. etc. The resulting motion is that of a periodic oscillation. The mathematical solution would confirm this result : the motion of a mass attached to a spring is properly described by the cosine function:

$x(t)~=~x_{max}\cos (2\pi t/T)$

where x_max is the amount by which the string was stretched before setting the mass in motion. Why does the expression in parenthesis (the "argument" of the cosine) look the way we have written it? Here is the reasoning :

at t=0 the mass is at x=x_max, this is in agreement with the formula above, since $\cos (0) =1$. We have seen the the motion is a periodical oscillation. If T is the period (i.e. the time it takes to perform a full oscillation) then at time t=T the mass must be back at x_max. And this is in fact what the equation says, since $\cos (2\pi)=1$, in fact $\cos (n\times 2\pi) =1$, the oscillation keeps repeating over and over.... A motion described by the above expression is called simple harmonic motion.

Circular vs. Harmonic Motion

Harmonic motion can be investigated more in detail by realizing that, if an object is moving with uniform circular motion (i.e moving in a circle with constant angular velocity $\omega$), then its projection on an arbitrary axis will move with simple harmonic motion. Specifically, if r is the circle's radius,

$x(t)~=~ r\cos (\omega t)~=~r\cos 2\pi t/T~=~r\cos 2\pi ft$

where we have remembered the relation between angular velocity $\omega$, period T and frequency f=1/T.

It's important to notice that a given harmonic circular motion is fully described by two independent quantities, the amplitude A (i.e what in the previous examples we called either x_max or r) and ..... either $\omega$ or T or f, take your pick.

In a purely harmonic motion, e.g. a mass on a spring, T and f have an immediate physical interpretation, while this is not the case for $\omega$, as this is the angular velocity of a fictitious circular motion associated with the harmonic motion at hand. Still, interpreting harmonic motion as the projection of a uniform circular motion is extremely useful since, with simple steps (no need for calculus) it allows to derive the proper expressions for both velocity and acceleration (see derivations in the book):

$x(t)~=~A\cos (\omega t)$

$v(t)~=~-A\omega\sin (\omega t)$

$a(t)~=~-A\omega^{2}\cos (\omega t)$

You are not expected to memorize these expressions, but it is very important that you are aware of their existence, and that you have a good feeling for what they tell us and how to interpret them (e.g. what are velocity and acceleration when x = 0, or what is x when the acceleration is at a maximum, etc. etc., see for instance Conceptual Question 10.8 or Problems 10.41, 10.42).

Combining the expressions above with F = ma alows us to get a very important result about the period of an oscillating mass-spring system. We have seen that

$x~=~A\cos (2\pi t/T)$

and

$a~=~-A\omega^{2}\cos\omega t~=~-A(2\pi/T)^{2}\cos (2\pi t/T)$

But F = -kx = ma, therefore

$-kA\cos (2\pi t/T)~=~-mA(2\pi/T)^{2}\cos (2\pi t/T)$

$T~=~2\pi\sqrt{m/k}$

i.e. the period of oscillation is determined uniquely by the values of the mass and of the spring constant. This result is a special case of an extremely important property of vibrating or oscillating bodies, and is exploited in countless applications. The rule can be roughly summarized as follows:

when a given body is made to oscillate or vibrate, the frequency of vibration is determined uniquely by the body's mass/shape/size. In other words,

any object vibrates at its own natural frequency.

A related situation is represented by the pendulum, whose period of oscillation (for small enough amplitudes, i.e. less than $\approx 10^{0}$) can be shown to be

$T_{pendulum} = 2\pi\sqrt{L/g}$

where L is the pendulum length (in general, the distance between the point of suspension and the center of gravity) and g is the usual acceleration of gravity. This result has obviously been exploited in numerous applications, going from grandfather clocks to mapping of the earth's interior.

Energy and Harmonic Motion

Nobody would argue that to compress or stretch a spring requires a certain amount of work, or alternatively, that a stretched or compressed spring has the ability of doing work, therefore it possesses a certain amount of energy. These ideas can be quantified in the following way : suppose you stretch a spring, i.e. you move one of its ends from its initial position x=0 to a final position x₀. The work required to do this is W = F x₀ (there is no $cos\theta$term since force and displacement are parallel). But the problem is that the force is not constant, in fact at any position x the force is given by F=kx. We could get around this complication if we knew what the average force is; in this case it is easy to estimate that: since the force varies linearly with position, the average force is given by the value it has at the average position x₀/2 . In conclusion, the work to stretch a spring by x₀ is given by

W = 1/2kx₀²

And, by the work energy theorem,this represents the (elastic, potential) energy of a compressed or stretched spring. It turns out that forces due to springs are conservative, therefore, when springs are involved, one can thing in terms of conservation of mechanical energy. Putting it all together, the total amount of mechanical energy a system could have is given by

$E_{mech}~=~1/2(mv^{2}~+~I\omega^{2}~+~kx^{2})~+~mgh$

and, when no non-conservative forces are present (i.e. if the system is subjected only to gravity and springs) this quantity is conserved.

Probably you will never encounter problems involving at the same time rotations and springs. For simpler situations, like e.g. spring-driven motion on a horizontal plane, the expression simplifies to

mv²+kx² = constant

This, combined with the results we have obtained earlier for the harmonic motion, allows to solve apparently intractable problems.... like e.g. 10.84 and 10.87

Damped and Driven Harmonic Motions

Under ideal conditions, if a system is put into harmonic motion under the influence of a force (e.g. the restoring force for a spring, or gravity for a pendulum) it will keep oscillating forever at a constant frequency and amplitude. In reality, in any realistic situation the effects of friction, air resistance, etc. will gradually remove energy from the oscillating system and will bring it to a gradual stop. In such instances we say that the oscillation was damped. And obviously the time (and the number of oscillations) it takes to bring it to a halt will depend upon the strength of the damping agent.

In many cases (automobile shock absorber, swinging door, high tech downhill skis, etc.) we want to be able to dissipate the oscillation energy as fast as possible, in order to bring the system to the equilibrium position in the shortest possible way. The response of a system to damping will depend on the relative values of its natural frequency and the damping force. In general, we distinguish the following situations:

• when the system comes to rest after a few oscillation, we say it is underdamped
• the system does not perform even a single oscillation, but it takes a very long time to come back to equilibrium, this is the condition of overdamping
• an optimal choice of parameters would cause the system to reach equilibrium in the shortest possible time (and with no oscillations); this is the case of critical damping
  
  

In the opposite scenario, we can have a force acting onto a system and adding energy to it. If the force is acting on an oscillating system in a random way, it will probably not have much of an effect. Think of a swing : if you give it random pushes, sometimes you will push it when it comes towards you, sometimes when it's moving away, and the swing will not swing very widely. But if you synchronize your pushes with the swing's natural oscillation frequency (a swing is like a pendulum, it's frequency of oscillation is uniquely determined by the location of its center of mass), then you will achieve large amplitude of oscillations. This effect is called resonance, and its consequences can be quite spectacular (we will see even more examples when dealing with sound, remember that sound is the audible result of a vibration).