A uniform cylindrical rod of length \(L\) and radius \(r,\) is made from a material whose Young’s modulus of Elasticity equals \(Y.\) When this rod is heated by temperature \(T\) and simultaneously subjected to a net longitudinal compressional force \(F,\) its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to:

When the temperature of a metal wire is increased from \(0^\circ ~\mathrm{C}\) to \(10^\circ ~\mathrm{C}\), its length increases by \(0.02\%\). The percentage change in its mass density will be closest to:
1. \(0.008\)%
2. \(0.06\)%
3. \(0.8\)%
4. \(2.3\)%

Two different wires having lengths \(L_1\) and \(L_2\), and respective temperature coefficient of linear expansion \(\alpha_1\) and \(\alpha _2\), are joined end-to-end. Then the effective temperature coefficient of linear expansion is:
1. \( 4 \frac{\alpha_1 \alpha_2}{\alpha_1+\alpha_2} \frac{L_2 L_1}{\left(L_2+L_1\right)^2} \)
2. \( 2 \sqrt{\alpha_1 \alpha_2} \)
3. \( \frac{\alpha_1+\alpha_2}{2} \)
4. \( \frac{\alpha_1 L_1+\alpha_2 L_2}{L_1+L_2}\)

A cube is constructed from a metal sheet such that each of its sides has a length \(a\) at room temperature \(T.\) The coefficient of linear expansion of the metal is \(\alpha.\) The entire cube is then heated uniformly so that its temperature increases by a small amount \(\Delta T,\) making the new temperature \(T+\Delta T.\) Assuming the expansion is small and isotropic, what is the cube's volume increase due to this heating?