Torqued Pulley

  Consider a variation of Atwood's machine with a mass hanging from a string wrapped around a pulley.

The expressions for the height of the mass and balance of forces are the same as before. For the pulley we need to balance the torque, the product of tension and its distance from the center of the pulley, with the inertial force due to the acceleration of the pulley. It is assumed that the quantity of motion for the pulley is proportional to the rate of change of the angle θ or P=Iω=Iθ̇ where I is the moment of inertia. Note that the resistance to angular acceleration is directed upwards. This time the inertial factor, m+I/r², includes the inertia of the pulley.

And the tension in the string is again less than the gravitational force acting on the mass m. Barton's Analytic Mechanics derives the formulas for Atwood's machine including the inertia of the pulley.

Atwood's machine

  Atwood's machine gives a unique insight into forces acting on a body. One can find a description of it's use in Comstock's System of Natural Philosophy. The machine consists of two weights hanging from a pulley. One only needs one variable, s, to represent the position of the apparatus.

The value s is the distance along the perimeter of the pulley corresponding to its rotation through an angle θ. From the formulas for the positions we can express the acceleration of the masses and ignoring the inertia of the pulley we can look at the balance of forces acting on the masses due to gravity, the tension in the string and the "inertial force" associated with resistance of a mass to a change in its state of motion. After solving for the acceleration of the pulley we can obtain the tension in the string.

The balance forces acting on the pulley gives us the tension in the line holding it in position. The net force acting on the apparatus is the difference in gravitational force and an inertial factor equal to the sum of the masses and more generally including the moment of inertia of the pulley. The tension in the string depends on the distribution of the masses and is maximum when both masses are equal and the system is in equilibrium.

The units of the vertical scale are T/g and the horizontal scale is the ratio of one mass to their sum.

A difference in tension would be needed to accelerate the pulley so in general one cannot say that the tension in the string on both sides of the pulley would be the same.

Another Look at Lagrange Multipliers

  Lagrange introduced the Method of Multipliers in his Analytical Mechanics of 1811 but when one first encounters this method it's not very clear why it works. One can deduce the procedure starting with a set of condition equations, Φ, and use least squares.

One ends up with a linear combination of the condition equations with arbitrary coefficients, the dΦ, set equal to zero. Division by dΦ₁ removes some of the arbitrariness since the linear combination equals a constant, -Φ₁.

Edit (Mar 20): Add dΦ and last sentence.

A Correction Equation Failure

  The gradient correction equation procedure is not free from error and can be forced to fail as seen in the following example.

The procedure works correctly for an initial rough estimate of x=2, y=6 for the zeroes of the Condition Equations Φ.

But the procedure fails along the x=y diagonal.

Least Squares Parameter Fit for the Catenary

 This iterative correction for the fit of the parameters for the catenary using the least squares gradient formula works quite well too. The equations of condition for x_min and L are fairly smooth over a wide range as these grid plots show.

And the iteration of the corrections converges rapidly to the optimal values of x_min and L using an  initial rough estimate of their values.

Using the formulas for the catenary we can now compute exact values for y_min and y(b).

Proof the Equilibrium Curve is a Catenary

  Using s_min and σ=A/(gλ) to rewrite the parametric equations for x and y one can eliminate the mass density λ and show that the curve is in fact a catenary.

Supplemental (Mar 16): One can also show that the tension in the string is,

Is the Equilibrium Curve a Catenary?

  Although the parametric equations for the equilibrium curve involve the linear mass density, the fits for A and B don't appear to be affected by changes in the value for the linear mass density, λ. All the  values for the minimum agree to 15 decimal places.

At the minimum gλs+B=0 so B=-g λs_min making the argument of the sinh^-1u equal to u=gλ(s-s_min)/A so there are two parameters, A/gλ and s_min appear to be independent.

What we've been calling the equilibrium equations may in fact be the parametric equations for a catenary.

A Convergent Method for Fitting a Suspended Line

  The relative linearity of the equations of condition for the fit of the equilibrium curve allows us to use numerical methods for its solution. We can start with a rough estimate and improve on it by using the estimate to evaluate the condition equations and their derivatives and estimating the necessary correction.

This method works quite well.

Cobwebs vs Itercepts for Solving Condition Equations

  Cobwebs can be used to solve a pair of simultaneous equations but they are notorious for the problems associated with them. In the preceding blog I estimated the intersection point by using a formula for computing where two straight lines will cross.

In the figure below the two lines were fitted using least squares to obtain the coefficients for the linear equations. The values a and a' above are the points where the lines intersect the y-axis with b and b' being the slopes of the lines. The distance of the intersection point from the origin is x.

A simple explanation of the formula is the distance of the intersection point from the origin is the separation of the lines there, Δa, divided by their rate of approach, Δb. As one gets closer to the intersection point the data becomes relatively more linear but eventually one sees more and more relative error in the estimate as one zooms in.

Supplemental (Mar 15): The equilibrium price is a topic discussed in Economics. See the Wikipedia articles on Economic equilibrium and Bayesian game. One can explain the divergent cobwebs as due to the incomplete information that results from basing estimates on past history in a changing market. It's an example of a flaw in the use of Bayes' theorem where current statistics differ from those of the past. Basically what works here doesn't always work there and the situation spirals out of control.

The Curve for a Suspended Line

 A while ago I did some posts on the equilibrium position of a weight hanging from zip line and some similar problems. We can also take a look at the curve for a cord or chain hanging freely. Lagrange states that the center of gravity is a minimum when the cord is in equilibrium and the curve is a catenary. Galileo also mentions the hanging line and notes that it closely approximates a parabola. I used the Calculus of Variations to derive a general equation for the catenary assuming the curve has minimal potential energy for a cord with fixed length L.

One can also try to find the curve for a cord in equilibrium by doing a derivation similar to that found in Lamb's Statics.

A comparison shows that the catenary and equilibrium curve appear identical although the equilibrium curve is dependent on the cord's linear mass density λ.

The difficulty in doing the comparison is fitting the curves exactly so the endpoints match up with the chosen positions for the desired length. In each case one has two condition equations that need to be solved for the unknown parameters. For the catenary one can do a two dimensional search for the values of x_min and A but the surfaces F and G do not intersect in a straight line. The Excel plot below shows the rms deviation of the functions F and G for chosen values of the parameters. It's difficult but one can zoom in on the parameters for the minimum.

The parametric equations are a little easier to work with since the surfaces involved are more nearly planar.

One can see that the curves for Δx=0 and Δy=0 are nearly linear and when plotted as points in the A,B plane appear to be two lines crossing each other. A simultaneous solution for the two condition equations requires that the values of A and B be identical for both lines.

In the plot above the center and widths were chosen and Δx and Δy were computed for a grid of points in the A,B plane. Interpolation was used to estimate the positions of the zeros to get the two lines then the set of points for each line were fit to find a linear equation for each line and their intersection was estimated (shown as a grey circle above). By shifting to the estimated intersection point and changing the widths one can more easily zoom in on the intersection point.

One can do a "notch fit" to show how the distance between the two lines varies with the value of B by estimating the intersection of the two segments confirming the "zeros."

The computed minima for both curves agree to 5 decimal places but the parametric equations are easier to work with.