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Isn't the force calculated twice here? For example, the force along AB is at first calculated for the resultant force along OB, then for the resultant force along AC. I think the compression and tensile stress should be ##\frac{F}{2a}##.

How can a permeable piston be adiabatic? If substances can go in and out of the cylinder and the substances have heat energy, heat can be exchanged through a permeable piston. I came across this term in the book, but cannot understand.

Thanks. That's exactly what I wanted to know.

We know, $$dU=TdS-PdV$$ ##\int PdV## can be calculated if the equation of state is given. I tried to express ##S## as a function of ##P ,V## or ##T## (any two of those). $$dS=\left(\frac{\partial S}{\partial V}\right)_T dV+\left(\frac{\partial S}{\partial T}\right)_V dT$$ $$=\left(\frac{\partial...

That was not my point. ##d \vec r## represents infinitesimal change in position vector, while ##\vec r## represents position vector. Could you please give me a practical example where the net torque is calculated by ##\int \vec F \times d \vec r## ?

Isn't torque defined as ##\vec r \times \vec F## ?

In the vector calculus course, I calculated integrals like, ##\int \vec F \times \vec{dr} ## Does this kind of integrals have physical significance or practical application other than Biot-Savart's Law?

To prove the wye-delta transformation formula, it is said 'If the two circuits are to be equivalent, the total resistance between any two terminals must be the same.' But why ? I can't convince myself that it is sufficient condition for the equivalence of circuits.

We can mathematically derive the equation of wave, \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}, where v is the velocity of wave propagation. Can we prove this equation physically (not just taking derivatives of the equation of wave, but making physical meaning in...