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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 10^6 \text{ m s}^{-1}

.

Options

A $1.00010 \text{ m}$
B $0.99990 \text{ m}$
C $1.00005 \text{ m}$
D $0.99995 \text{ m}$

Correct answer

D. $0.99995 \text{ m}$

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 10^6 \text{ m s}^{-1}

.

The speed of light is

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

.

According to the special theory of relativity, the contracted length

L

is given by:

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

Substituting the given values:

L = 1 \times \sqrt{1 - \left(\dfrac{3 \times 10^6}{3 \times 10^8}\right)^2}

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

L = \sqrt{1 - 10^{-4}}

Using the binomial approximation

(1 - x)^{1/2} \approx 1 - \dfrac{x}{2}

for

x \ll 1

:

L \approx 1 - \dfrac{10^{-4}}{2}

L \approx 1 - 0.00005

L = 0.99995 \text{ m}

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