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The pendulum

Page history last edited by Joe Redish 12 years ago

Class content > Oscillations and Waves > Harmonic Oscillation > Mass on a spring

 

Pre-requisites:

 

Setting up the pendulum

Another nice example of harmonic (or nearly harmonic) motion is the long pendulum. Imagine hanging a small sphere from a long thin rod from a vertical support and allowing it to swing back and forth. If the swing is small, the back and forth motion looks a lot like a mass on a spring. It also provides a nice example of how an energy analysis can give us insight into the motion of an oscillator.

 

In order to make our first model as simple as possible, we will assume:

  • The sphere is small enough that we can ignore its extent.
  • The rod is light enough (compared to the sphere) that we can ignore its mass.
  • The rod is rigid enough that we can ignore any bending of the rod.
  • Any friction in the pivot is small enough that it can be ignored.
  • The effect of air drag and viscosity on the mass can be ignored.

Of course some of these effects build up, so it might be okay to ignore them for a few dozen swings (or even hundred in a good case).

 

 

The energy of the pendulum

Let's look at the free-body diagram for the sphere. Since we're ignoring air drag, the only forces acting on the sphere is the pull of the earth, its weight, and the tension force from the rod. The rod constrains the sphere to move along a circular path, so its change of position is always perpendicular to the tension force from the rod. This means that the tension force does no work on the sphere -- it only acts to change its direction of motion, not its speed. This, together with the fact that we have decided we can ignore any resistive forces, tells us that the only work we need to consider is that done by the force of gravity. So the kinetic energy plus the gravitational potential energy should be conserved. 

 

The result is that the speed of the sphere is related to how high up it is by

 

E0 = ½mv2 + mgh

 

where we have written E0 for the (constant) value of the energy (and to indicate that we mean a particular value, not a variable). If we take the 0 of h to be at the lowest position of the pendulum, then a little trigonometry shows us that

 

h = L - L cos θ = L (1 - cos θ)

 

so we get

 

E0 = ½mv2 + mgL (1 - cos θ)

 

If we take the starting angle of the pendulum (when v = 0) to be θ0 then we get

 

 mgL (1 - cos θ0) = ½mv2 + mgL (1 - cos θ)

 

The mgL term on both sides cancels giving

 

mgL (cos θ - cos θ0) = ½mv2

 

or

v2 = 2 gL (cos θ - cos θ0).

 

In principle, this solves everything. We could take the square root of both sides, write v as the derivative of position (Lθ with θ in radians) and then integrate the RHS over time giving the angle as a function of time. This is a fair mess with the result being something called an "elliptic integral" -- something rarely seen outside of the advanced math Tripos exams at Cambridge University (UK). But if we're willing to accept a useful and important result from a Calc III class, we can make the connection with the harmonic oscillator.

 

The small angle approximation

One of the most important results of calculus is the Taylor series expansion. This says that if you know a function only in a small region, but you know its derivatives in that region, then you can use those derivatives to extrapolate beyond that region. You can imagine that this is an important result in scientific applications of mathematics where we rarely know something exactly, but we might have a pretty good idea how it behaves for limited values of a variable. This math lets us make the next step.

 

One of the results from this math is the description of trig functions for small angles. (See The small angle approximation for more details.) These results turn out to be of great important in the study of light, both in the theory of lenses, and in the wave theory of interference. But for now, let's just look at the results.

 

The result of the advanced math analysis of trig functions of an angle θ given in radians (this is necessary), is that the sine, cosine, and tangent have an expression in terms of powers of the angle:

 

                         Formula

                         Formula

                         Formula

These are pretty scary! They go on forever (but it's pretty clear what happens next for each case) so what good are they? Let's just look at the sin first. The first term is just θ. If θ is small -- say 0.1 radians (about 6 degrees) then the second term is θ3/6 (since 3! = 3 x 2 x 1 = 6) which has a value of 0.00017. This is pretty small, even compared to 0.1. The next term, θ5/120 (since 5! = 5 x 4 x 3 x 2 x 1 = 120) only has a value of 0.000000083. So the first three terms are about 10-1, 10-4, and 10-7. Unless we want very high accuracy, the first term or two will suffice. (And if you're a mathematician you can even prove that the infinite sum has a finite result and can get an upper bound on the corrections, showing that they are small. This is why some of us really like math!)

 

So for many applications in physics -- where we can keep the angles less than about 20o, we can use the small angle approximations:

 

 

These are the small angle approximations for the trig functions. You will be expected to know and be able to use these. Let's apply them to the energy of the pendulum.

 

The pendulum as SHM

If we only let our pendulum do small angle oscillations, the cosine in the energy becomes 1 - θ2/2 and the 1 cancels giving

E0 = ½mv2 + mgL (1 - cos θ) = ½mv2 + ½mgLθ2

 

but since Lθ~ L sin θ = x, we can write our energy as

 

E0 = ½mv2 + ½(mg/L)x2

 

If we write k = mg/L for our combination of constants, then the energy becomes just the same as the energy for a mass on a spring

 

E0 = ½mv2 + ½k x2

 

If the relation between the velocity and the position is the same for two systems they have to move in the same way. So our pendulum will oscillate back and forth just like a harmonic oscillator -- as long as the angle doesn't get too big.

 

We can even figure out the period by using our analogy. We know for SHM the period is 2π/ω0 or

 

 

Note that the mass has dropped out! The period of an (ideal) pendulum is independent of the mass. We probably shouldn't be surprised since we know that when gravity is the only force, the mass cancels out and all objects accelerate the same way. Here, only the gravitational force is changing the sphere's speed so we get the same kind of result as for free fall.

 

The pendulum has some valuable applications. It was important in the design of the first clocks and it plays a role in the analysis of the gaits of animals. The way the animals arms and legs swing naturally have a lot to do with how efficient motion is patterned.

 

Joe Redish 3/15/12

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