Why conservation of angular momentum is valid about sharp edge

angular momentumrotational-dynamics

Suppose a spherical ball moving on a straight line struck an obstacle of rectangle shape(fixed) , once it tries to go upwards of it (assume it goes ), we say we can conserve angular momentum about that pointy edge about which it tries rotating , but in my opinion why will angular momentum be conserved as such no doubt impulsive torque would be zero about that point edge but what about gravity torque (for full duration from time of rotating and going fully upwards) and what about normal torque from ground at the time of striking ? They dont give zero torque about pointy edge so how is angular momentum conserved about pointy edge ? enter image description here

Best Answer

This setup needs to be thought of in three distinct phases: before the collision, a small time $\Delta t$ around the moment of the collision, and finally the motion after the collision while the ball is rising. Your question is not 100% clear about which phase you're concerned with, so I'll give a brief description of all three.

Before the ball hits the "pointy edge", gravity does indeed exert a torque on the ball if we take the "pointy edge" to be our origin point. So does the normal force from the surface it is rolling on, before it hits the edge. Since these two forces are equal and opposite and act along the same line, they exert opposite torques before the ball hits the pointy edge, and so angular momentum is constant with respect to this point.

In a small amount of time $\Delta t$ around the moment of the collision, the ball may experience an impulsive force from the pointy edge; this exerts no torque. In addition, in this brief amount of time, the torque from gravity will be approximately $m g \ell$, where $\ell$ is the horizontal distance between the edge and the center of the ball. This means that the change in angular momentum of the ball in this time will be approximately $m g \ell \Delta t$, which is negligible if we take the limit $\Delta t \to 0$. Thus, in the collision with the edge, angular momentum is conserved. This means that the angular momentum of the ball immediately after the collision is the same as it was long before the collision. (It is not immediately clear to me whether energy is conserved in the time $\Delta t$ around the collision; I suspect it is not.)

Finally, once the ball has lost contact with the surface below, the normal force vanishes, while gravity still exerts a torque. As before, the contact force exerts no torque on the ball. Thus, there is now a net torque on the ball, and the angular momentum with respect to the pivot point will no longer be constant. This means that we can't use angular momentum conservation to make statements about the motion of the ball while it is "tripping" over the edge. Energy is still conserved from the moment after the collision onwards, though. (This is important for some problems you might ask about this setup.)

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