JEE Main & Advanced Physics NLM, Friction, Circular Motion Modification of Newton's Laws of motion

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.