Physics

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After burning, heat exchange occurs between helium and atmospheric. Hence, irreversible, isothermal process.

The mass of the cylinder M = ρV = ρ (πR²L), where ρ is the density of the material.
We can express R² in terms of L: R² = M / (πρL).
The moment of inertia is I = M (R²/4 + L²/12).
Substitute R² into the equation for I:
I (L) = M * [ (M / (4πρL) + (L²/12) ]
To find the minimum possible I, we differentiate I with respect to L and set the derivative to zero.
dI/dL = M * [ -M/ (4πρL²) + 2L/12 ] = 0
M/ (4πρL²) = 2L/12 = L/6
M/ (πρ) = (4/6)L³ = (2/3)L³
Since R² = M/ (πρL), we have R²L = M/ (πρ).
Substitute M/ (πρ) into the differentiated equation:
R²L = (2/3)L³

When the two conducting spheres are connected by a wire, they will reach the same electric potential, V.
The total charge Q_total = 12µC + (-3µC) = 9µC. This total charge will redistribute.
Let the final charges be q? and q? + q? = 9µC.
The potential of a sphere is V = kq/r.
V? = V?
k q? /R? = k q? /R? ⇒ q? /R? = q? /R?
q? / (2R/3) = q? / (R/3) ⇒ q? /2 = q? ⇒ q? = 2q?
Substitute this into the charge conservation equation:
2q? + q? = 9µC ⇒ 3q? = 9µC ⇒ q? = 3µC.
Then, q? = 2 * 3µC = 6µC.
The final charges are 6µC and 3µC.

The internal energy of one mole of an ideal gas is given by U = (f/2)RT, where f is the number of degrees of freedom.
For a non-linear triatomic molecule (assumed to be a rigid rotator), there are:

3 translational degrees of freedom.

3 rotational degrees of freedom.
Vibrational modes are generally not considered at temperature T unless specified.
Total degrees of freedom f = 3 + 3 = 6.
U = (6/2)RT = 3RT.

The least count (LC) of the screw gauge is the pitch divided by the number of divisions on the circular scale.
LC = Pitch / N = 0.1 cm / 50 = 0.002 cm.
A measurement taken with this instrument must be a multiple of its least count. We check the options:
(A) 2.121 / 0.002 = 1060.5 (Not a multiple)
(B) 2.123 / 0.002 = 1061.5 (Not a multiple)
(C) 2.124 / 0.002 = 1062 (Is a multiple)
(D) 2.125 / 0.002 = 1062.5 (Not a multiple)
The correctly recorded measurement is 2.124 cm.