With mass M, the time period of the spring is \(T=2\pi \sqrt { \cfrac { M }{ k } } \)
With mass M + m, the time period becomes 
\(\cfrac { 5T }{ 3 } =2\pi \sqrt { \cfrac { M+m }{ k } } \cfrac { 5 }{ 3 } \times 2\pi \sqrt { \cfrac { M }{ k } } =2\pi \sqrt { \cfrac { M+m }{ k } } \)
\(\cfrac { 25 }{ 9 } M=M+m;\cfrac { 16 }{ 9 } M\) or \(\cfrac { m }{ M } =\cfrac { 16 }{ 9 } \)

Find the rms velocity of its molecules in ms^-1
T₁= 27°C + 273 = 300 K, T₂ = 127°e + 273 = 400 K
rms speed of molecules is \({ v }_{ rms }=\sqrt { \cfrac { 3RT }{ M } } \)
So it depends only on temperature,
v_rms ∝ \(\sqrt { T } \)
\(\cfrac { { v }_{ 1 } }{ { v }_{ 2 } } =\sqrt { \cfrac { { T }_{ 1 } }{ { T }_{ 2 } } } \Rightarrow \cfrac { 200 }{ { V }_{ 2 } } =\sqrt { \cfrac { 300 }{ 400 } } ;\cfrac { 200 }{ { v }_{ 2 } } =\cfrac { \sqrt { 3 } }{ 2 } \)
\({ v }_{ 2 }=\cfrac { 400 }{ \sqrt { 3 } } \)
v₂ = 231ms^-1

As heat engine, let Q₁ be the heat energy absorbed from the source and Q₂ be the energy rejected to the sink. Then
Q₁ = W + Q₂ = 10 + Q₂
\(\eta =1-\cfrac { { Q }_{ 2 } }{ { Q }_{ 1 } } =\cfrac { 1 }{ 10 } \)
\(\cfrac { { Q }_{ 2 } }{ { Q }_{ 1 } } =1-\cfrac { 1 }{ 10 } =\cfrac { 9 }{ 10 } \)
\({ Q }_{ 1 }=\cfrac { 10 }{ 9 } { Q }_{ 2 }\)
From equation (1) and (2),
\(10+{ Q }_{ 2 }=\cfrac { 10 }{ 9 } \)
\(10=\left( \cfrac { 10 }{ 9 } -1 \right) { Q }_{ 2 }\)
\(\therefore\) Q₂ = 90 J

Weight of liquid A displaced is \({ W }_{ A }=\cfrac { 1 }{ 2 } { V }{ \rho }_{ A }g\)
Weight of liquid B displaced is \({ W }_{ B }=\cfrac { 2 }{ 3 } V\rho _{ B }g\)
According to Archimede's principle W_A = W_B
\(\cfrac { 1 }{ 2 } { V }{ \rho }_{ A }g=\cfrac { 2 }{ 3 } V\rho _{ B }g\)
\(\cfrac { { \rho }_{ A } }{ { \rho }_{ B } } =\cfrac { 4 }{ 3 } \)

We know that \(g=\cfrac { GM }{ 2 } \)
\(g'=g\left( \cfrac { R }{ R+h } \right) ^{ 2 }=g\left( \cfrac { R }{ R+\frac { R }{ 2 } } \right) ^{ 2 }=g\left( \cfrac { 2R }{ 3R } \right) ^{ 2 }\)
\(g'=\cfrac { 4 }{ 9 } g\)
Weight \(W'=\cfrac { 4 }{ 9 } W=\cfrac { 4 }{ 9 } \left( 72 \right) \)
W'=32N

Radial Component.

Yes. When two bodies move in opposite directions, the relative velocity of each is greater than the individual velocity of either body.

Work done is more, while moving up a hill.

Mean Absolute error, 
\(\triangle a_m={{1}\over{n}}\sum^{n}_{i=1}{|\triangle a_i|}\)

Jumping on a concrete cemented floor is more dangerous than jumping on the sand. Sand brings the body to rest slowly than the concrete floor, so that the average force experienced by the body will be lesser.