According to Newton, time and space are absolute. The velocity of observer has no effect on it. But, according to special theory of relativity Newton?s laws are true, as long as we are dealing with velocities which are small compare to velocity of light. Hence the time and space measured by two observers in relative motion are not same. Some conclusions drawn by the special theory of relativity about mass, time and distance which are as follows :
(1) Let the length of a rod at rest with respect to an observer is \[{{L}_{0}}.\] If the rod moves with velocity \[\upsilon \] w.r.t. observer and its length is L, then \[L={{L}_{0}}\sqrt{1-{{v}^{2}}/{{c}^{2}}}\]
where, c is the velocity of light.
Now, as\[\upsilon \] increases L decreases, hence the length will appear shrinking.
(2) Let a clock reads \[{{T}_{0}}\] for an observer at rest. If the clock moves with velocity \[\upsilon \] and clock reads T with respect to observer, then \[T=\frac{{{T}_{0}}}{\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}}\]
Hence, the clock in motion will appear slow.
(3) Let the mass of a body is \[{{m}_{0}}\] at rest with respect to an observer. Now, the body moves with velocity \[\upsilon \] with respect to observer and its mass is m, then \[m=\frac{{{m}_{0}}}{\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}}\]
\[{{m}_{0}}\] is called the rest mass.
Hence, the mass increases with the increases of velocity.
Note :
- If \[v<<c,\] i.e., velocity of the body is very small w.r.t. velocity of light, then \[m={{m}_{0}}.\] i.e., in the practice there will be no change in the mass.
- If \[\upsilon \] is comparable to c, then \[m>{{m}_{0}}\] i.e., mass will increase.
- If \[v=c,\] then \[m=\frac{{{m}_{0}}}{\sqrt{1-\frac{{{v}^{2}}}{{{v}^{2}}}}}\] or \[m=\frac{{{m}_{0}}}{0}=\infty .\] Hence, the mass becomes infinite, which is not possible, thus the speed cannot be equal to the velocity of light.
- The velocity of particles can be accelerated up to a certain limit. Even in cyclotron the speed of charged particles cannot be increased beyond a certain limit.
\[\tan \theta =\frac{{{F}_{l}}}{R}\]
\[\therefore \,\,\tan \theta ={{\mu }_{0}}\]
[As we know \[\frac{{{F}_{l}}}{R}={{\mu }_{s}}\] ] or \[\theta ={{\tan }^{-1}}({{\mu }_{L}})\]
Hence coefficient of static friction is equal to tangent of the angle of friction. 
(iii) Brake works on the basis of friction.
(iv) Writing is not possible without friction.
(v) The transfer of motion from one part of a machine to other part through belts is possible by friction.
(2) Disadvantages of friction
(i) Friction always opposes the relative motion between any two bodies in contact. Therefore extra energy has to be spent in over coming friction. This reduces the efficiency of machine.
(ii) Friction causes wear and tear of the parts of machinery in contact. Thus their lifetime reduces.
(iii) Frictional force result in the production of heat, which causes damage to the machinery. 
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Normal reaction R = mg cos \[\theta \] + mb sin \[\theta \]
and ma = mg sin \[\theta \] - mb cos \[\theta \]
\[\therefore \] a = g sin \[\theta \] - b cos \[\theta \]
Note :
