NEET-XII-Physics
47: The Special Theory of Relativity
- #3The length of a rod is exactly 1 m when measured at rest. What will be its length when it moves at a speed of (a) 3 × 105 m s-1, (b) 3 × 106 m s-1 and (c) 3 × 107 m s-1? (a) 3 × 105 m s-1, (b) 3 × 106 m s-1 and (c) 3 × 107 m s-1?Ans : Given:
Proper length of the rod, L = 1 m
If v is the velocity of the rod, then the moving length of the rod is given by
`` L\text{'}=L\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}`` (a) Here,
v = 3 × 105 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{10}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-6}}=0.9999995\,\mathrm{\,m\,}`` (b) Here,
v = 3 × 106 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{12}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-4}}``
`` =0.99995\,\mathrm{\,m\,}`` (c) Here,
v = 3 × 107 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{14}}{9\times {10}^{16}}}``
`` =1\sqrt{1-{10}^{-2}}``
`` =0.995\,\mathrm{\,m\,}``
Page No 458: (a) Here,
v = 3 × 105 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{10}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-6}}=0.9999995\,\mathrm{\,m\,}`` (b) Here,
v = 3 × 106 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{12}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-4}}``
`` =0.99995\,\mathrm{\,m\,}`` (c) Here,
v = 3 × 107 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{14}}{9\times {10}^{16}}}``
`` =1\sqrt{1-{10}^{-2}}``
`` =0.995\,\mathrm{\,m\,}``
Page No 458:
- #3-a3 × 105 m s-1,Ans : Here,
v = 3 × 105 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{10}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-6}}=0.9999995\,\mathrm{\,m\,}``
- #3-b3 × 106 m s-1 andAns : Here,
v = 3 × 106 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{12}}{9\times {10}^{16}}}``
`` =\sqrt{1-{10}^{-4}}``
`` =0.99995\,\mathrm{\,m\,}``
- #3-c3 × 107 m s-1?Ans : Here,
v = 3 × 107 m/s
`` L\text{'}=1\times \sqrt{1-\frac{9\times {10}^{14}}{9\times {10}^{16}}}``
`` =1\sqrt{1-{10}^{-2}}``
`` =0.995\,\mathrm{\,m\,}``
Page No 458: