Using Work Energy theorem to find acceleration

Question

Here's a sample problem:

A block of mass $m$ is free to move vertically on a wedge of mass $M$ and angle of inclination $\theta$ that rests on flat ground. If all surfaces are frictionless, then find the magnitude of acceleration of the block and the wedge when the system is released from rest.

The usual approach would be to mark all the force vectors, balance the forces, and then use the constraint relation to find the acceleration. However, I found that the same result can be achieved by using Work-every theorem, and with much less effort. Here's my solution:

I take the horizontal displacement of the wedge to be $x_1$ and the vertical displacement of the block to be $x_2$. It's easy to figure that

$$ x_2 = x_1 \tan \theta. $$

Now, applying work-energy theorem on the system, I can write

$$ mgx_2 = \frac12 m\dot{x}_2^2 + \frac12 M\dot{x}_1^2 \\ \implies 2mgx_1 \tan \theta = m\dot{x}_1^2 \tan^2 \theta + M\dot{x}_1^2 \\ \implies \dot{x}_1^2 = \frac{2mg \tan \theta}{M + m\tan^2 \theta}x_1 $$

I don't need to worry about constraint forces because they don't do any work here. Differentiating with respect to $x_1$, I finally have the acceleration of the wedge as

$$ \ddot{x}_1 = \frac{mg \tan \theta}{M + m\tan^2 \theta}. $$

Hence, the acceleration of the block would be

$$ \ddot{x}_2 = \frac{mg \tan^2 \theta}{M + m\tan^2 \theta}. $$

See how short and efficient this technique is? I have found this technique especially useful in solving problems involving accelerated pure rolling.

My questions are:

$(1)$ Is the vectorial analysis technique more widely used solely because it gives one a feel for the problem?

$(2)$ Why couldn't I find much documentation for the technique I present here, on the internet? If this technique is a direct application of D'Alembert's Principle, as mentioned in a comment, I request a detailed explanation for why and how so.

I request the admins not to close this question as off-topic, because this might just become some meaningful conversation.

Answer 1

You seem to have found a special applied case of the Lagrangian mechanics. I wonder whether you just had an experimental physics lecture about mechanics and not yet any about theoretical physics.

If you want to read up on that, look at Lagrangian mechanics. That is based on the potential and kinetic energy and will give you a straightforward way to compute the equations of motions. It is powerful enough to solve the double pendulum within one or two pages of writing.

Answer 2

(1) The reasons why one technique is more widely used or preferred rather than any other is not a question about physics but about problem-solving and teaching. Whether or not one technique "gives you a feel for the problem" is subjective. These issues are off topic for this site.

(2) No you are not applying D'Alembert's Principle. If you want to find out what that Principle is there are plenty of internet sites which will teach you. Your "technique" is just to apply the conservation of energy (or the work-energy theorem) which is used throughout physics and is extensively documented.

The relation $\tan\theta=x_2/x_1$ is a simple geometrical constraint, it is not an application of D'Alembert's Principle.

You seem to be promoting your "technique" as "short and efficient" but you have not compared it with any other "technique" such as applying $F=ma$, to show that your technique uses much less effort.

Using the normal force $N$ between the block and the wedge, applying $F=ma$ to each rigid body we have
$mg-N\cos\theta=m\ddot x_2$
$N\sin\theta = M\ddot x_1$.
From the equation of geometrical constraint we have $\ddot x_2=\ddot x_1\tan\theta$. Combining these equations and eliminating $N$ we get your results, with little difference in the amount of effort required. The only "inefficiency" has been the use of an unknown normal reaction force $N$ which we did not need to know.