\(\mathrm{T}=2 \pi \sqrt{\frac{m}{k}}\)
when the spring is cut into two equal halves, than the force constant of each part is 2k. When the mass is suspended from one of the halves, new time period is \(T^{\prime}=2 \pi \sqrt{\frac{m}{2 k}}=\frac{T}{\sqrt{2}}\)

Average K.E of each degree of freedom
\(=\frac{1}{2} k T=\frac{1}{2} N T[k=N]\)
N - no. of moles
T - temperature

Moment of Inertia = MK²
K-Radius of gyration
During heating K would be increased.
Hence moment of inertia will increase

\(\frac{F}{A}=Y \frac{\Delta l}{l} \quad F=Y \frac{\Delta l}{l} \times A\)
\(A_{1}=\mathrm{A} \quad \mathrm{A}_{2}=2 \mathrm{~A}\)
\(\therefore F \propto \Delta l \times A\)
\(\mathrm{F} \propto \Delta l_{1} A_{1}\)
\(\mathrm{F}_{1} \propto 2 \Delta l_{2} \times A_{2}\)
\(\frac{F}{F_{1}}=\frac{\Delta l A}{2 \Delta l \times 2 A}\)
\(\therefore F_{1}=4 \mathrm{~F}\)

T² = C (R + h)³
\(\therefore T \alpha (R_E)^\frac{3}{2}\)
R_E = 2R_E
\(\therefore T \alpha (2R_E)^\frac{3}{2}\)
Time period increases by \(2^\frac{2}{3}\)= 2\(\sqrt 2\) 
No. of days in a year = (365.4) \(\times\)2\(\sqrt 2\)
= 1032

Velocity and acceleration are in the same direction: So speed increases.

Dimensional formula of Planck's
\(\text { constant }=\frac{\text { Energy }}{\text { Frequency }}=\frac{\mathrm{ML}^{2} \mathrm{~T}^{-2}}{\mathrm{~T}^{-1}}\)
\(=M L^{2} T^{-2+1}=M L^{2} T^{-1}\)