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Concepts Of Physics MCQ Edition [Volume 2]

Exam: Concepts Of Physics MCQ Edition [Volume 2]
Subject: Physics
Chapter: The Special Theory of Relativity

Concepts Of Physics MCQ Edition [Volume 2] · Physics · The Special Theory of Relativity

Measured at rest, a rod has an exact length of 1 m . Determine its...

Measured at rest, a rod has an exact length of

1 \text{ m}

. Determine its length when it travels at a speed of

3 \times 10^5 \text{ m s}^{-1}

.

Options

A $0.9999990 \text{ m}$
B $1.0000010 \text{ m}$
C $1.0000005 \text{ m}$
D $0.9999995 \text{ m}$

Correct answer

D. $0.9999995 \text{ m}$

Step-by-step solution

The rest length of the rod is

L_0 = 1 \text{ m}

.

The speed of the rod is

v = 3 \times 10^5 \text{ m s}^{-1}

.

The speed of light is

c = 3 \times 10^8 \text{ m s}^{-1}

.

Using the length contraction formula:

L = L_0 \sqrt{1 - \dfrac{v^2}{c^2}}

Substituting the given values:

L = 1 \times \sqrt{1 - \left(\dfrac{3 \times 10^5}{3 \times 10^8}\right)^2}

L = \sqrt{1 - (10^{-3})^2}

L = (1 - 10^{-6})^{1/2}

Using the binomial approximation

(1 - x)^n \approx 1 - nx

for

x \ll 1

:

L \approx 1 - \dfrac{1}{2} \times 10^{-6}

L \approx 1 - 0.0000005

L = 0.9999995 \text{ m}

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