Path:  physics insights > basics > basic Lagrange > Example:  Sliding and Hanging Weights, and a Ramp

Figure 1 -- Weights and ramp:

This is a very simple example of the use of the Lagrangian formulation of Newtonian mechanics to solve a problem.  We have two weights, m₁ and m₂, attached together with a string of length l  (figure 1).  One is on a ramp; the other is hanging from the string.  The string goes over a pully. Both weights are initially stationary.  We'll start by neglecting friction, then do it again with friction later in the page.

With x units of string between the pulley and weight m₁, it drops  units below the level of the pulley.  So, the (gravitational) potential energy of weight m₁ is

(1)    

When weight m₁ slides x units down the ramp, weight m₂ must rise x units.  So, the (gravitational) potential energy of weight m₂ is

(2)   

The kinetic energy of the system is, of course,

(3)    

So, following equation (lagrange.4), the Lagrangian for the system is:

(4)    

The partial derivatives we need are:

(5)    

As soon as we start to write the equation of motion, we realize this substitution helps a lot:

(5b)    

And with substitions (5b), the equation of motion, following (lagrange.3), is:

(6)    

We can see immediately from (6) that there are three cases.  In case (7a), weight  m₁ slides down the ramp, and weight m₂ rises.

(7a)    

In case (7b), the weights balance, and nothing happens.

(7b)    

And in case (7c), weight  m₁ slides up the ramp, and weight m₂ falls.

(7c)    

We can integrate (6) twice to obtain a closed-form solution.  Assuming the weights are stationary to start with, and m₁ starts at x₀:

(8)

Once Again, with Friction

We'll assume there's friction at work when m₁ slides along the ramp, and we'll use the friction model given in (lagrange.9).  In this case there's just one dimension, so it's very simple:

(9)    

We will assume f₁ is positive for case (7a), and negative for case (7c).  (In case (7b) nothing moves and friction doesn't matter!)   The "cos θ" term is due to the fact that the force of gravity is straight down, which is at angle θ to a line perpendicular to the ramp's surface.  Note that the dimensions of f₂ are time/distance, while f₁ is dimensionless.

Plugging equations (5) and friction force (9) into the equation of motion (lagrange.8), we obtain

(10)    

or, slightly rearranged, and again substituting (5b):

(11)    

We can see a couple of things immediately by looking at (11).  First, if the "constant friction" is too large, the weights won't move at all; the initial acceleration will be zero.  In order for them to move, we must have:

(12)    

As the velocity increases, the viscous friction term becomes larger, and eventually the acceleration again goes to zero; the terminal velocity, when acceleration is zero, must be:

(13)    

Finally, we can solve for the velocity and position.  Let's rearrange (11) a bit:

(14)    

That looks like an exponential.  The coefficient of is positive; the right hand side may be either positive or negative depending on which way the system slides (case (7a) or case (7c)).  Let's substitute a and b for the constants, to reduce clutter, with a>0:

(15)    

A solution to the homogeneous equation (with b=0) is:

(16)    

and with a little fiddling we obtain the particular solution (for b≠0):

(17)    

Adjusting this to match the initial conditions at t=0 (which were x=x₀, =0), we obtain:

(18)    

Taking a firm grip on our pencil and looking back at (14), we substitute the definitions for a and b back into (18), differentiate, and divide out some common factors to obtain the equations for the position and velocity:

(19a)     (19b)

At time t=0, (19b) is certainly zero, and as t->, we see that (19b) approaches the terminal velocity we found in (13), which encourages us to think that these rather ugly equations might actually be correct.  As is often the case, introducing a general friction function made the problem a lot messier.  Keep in mind that, in this case, the sign we assumed for f₁ depends on the ratio of the masses and the angle of the ramp.




Page created on 12/1/06.  Minor corrections, 12/2/06