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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 10^7 \text{ m s}^{-1}

.

Options

A $0.990 \text{ m}$
B $0.995 \text{ m}$
C $1.010 \text{ m}$
D $1.005 \text{ m}$

Correct answer

B. $0.995 \text{ m}$

Step-by-step solution

Given rest length of the rod,

L_0 = 1 \text{ m}

Speed of the rod,

v = 3 \times 10^7 \text{ m s}^{-1}

Speed of light,

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 10^7}{3 \times 10^8}\right)^2}

L = \sqrt{1 - (0.1)^2}

L = \sqrt{1 - 0.01}

L = (1 - 0.01)^{1/2}

Using the binomial approximation

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

for

x \ll 1

:

L \approx 1 - \dfrac{1}{2}(0.01)

L \approx 1 - 0.005 = 0.995 \text{ m}

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