Solving a lever problem using Newton's laws, without energy conservation?

Question

Context: I have no background in physics, and I'm trying to understand the role of the concepts of energy, momentum and other invariants/symmetries in physics. In particular, I'm trying to get a feel for why reasoning based on forces is/can be less powerful than reasoning based on invariants like energy.

In the Feynman lectures, he gives an example of a lever problem where it is helpful to solve it with the principle of conservation of energy:

The problem is to find a weight $W$ that would keep the rod horizontal. He uses the definition of the potential energy of the three weights, and the condition that energy is conserved which implies that the potential energy must not change by a rotation of the rod.

Generalizing this, we can derive the general rule "the law of the lever" from energy conservation. I am wondering whether we can find the solution merely using Newton's laws, without the concept of energy (and of course without already using the law of the lever).

Answer 1

If the pulley is friction-less and has no mass, and if the string also has no mass, you can simply set the torques on both sides equal to each other (hence, no net torque, so no net acceleration).

If this problem meets the criteria I outlined above for the string and pulley, we know that the tensions on both sides of the rope are equal, so we can simply set the torque due to the (upward) tension on right side of (connected to the end of the lever arm), equal to the net torque due to the 2 masses and the lever arm (if the lever arm has mass).

By doing this, we are able to determine the tension on the rope, and using this, the weight and mass of the block connected to the pulley.

Answer 2

Yes, assuming that you include rotational motion as part of "Newton's Laws". If so, this is a bog-standard torque problem. Using $\sum \tau = I \alpha$, and the fact that the rod does not rotate ($\alpha = 0$), you can calculate the torques exerted by the 100-kg weight and the 60-kg weight about the pivot point. The torque exerted by the unknown weight is then equal in magnitude to the sum of the other two torques, and since it's applied at a known position, we can find the unknown weight $W$.

The question is then how rotational motion relates to Newton's famous three laws. Briefly, they can be thought of as a consequence of Newton's Laws when applied to a system of particles — for example, all of the atoms in a body such as the rod in your diagram. In addition, we need to make two additional assumptions:

• The force between any two particles always points along the line between those two particles (i.e., either directly attractive or directly repulsive.)
• The body is rigid: it doesn't bend or warp in any way, so it can only rotate.

With these assumptions, you can derive the equation $\sum \tau = I \alpha$, where $\tau$ is the torque exerted, $I$ is the moment of inertia of the body (which depends both on the mass and how it's distributed), and $\alpha$ is the angular acceleration of the body (how fast its angular velocity is changing.) If you view Newton's Laws as fundamental, and you're willing to grant the additional two assumptions above, then the rotational laws of motion follow straightforwardly, and you don't need to explicitly invoke energy or momentum conservation.

[As an aside, what I've described is the simplest version of "the rotational laws of motion". There are more general forms of the above equation that apply to bodies that don't have a fixed axis of rotation, but they're trickier to understand and I didn't want to get overcomplicated.]

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Finally: I would say that the problem posed is not a good example of how reasoning from invariants is more powerful than reasoning from forces. A better example is a roller coaster: a car rolls down a frictionless track with some crazy shape. It starts at rest at $(x,y)$, and finishes at some final point $(0,0)$. What is its final velocity? If you try to solve this with forces, you quickly find that the problem is intractable: the normal force exerted on the car is constantly changing direction and magnitude, and it depends both on the speed of the car and its position on the track. So while the equation $F = ma$ does, in principle, determine the final velocity of the car, in practice it won't be possible to find the answer.

But by energy conservation, we know that $\frac{1}{2} m v_f^2 = mgy$ and we're done.