The Relativity of Length MCQ Quiz - Objective Question with Answer for The Relativity of Length - Download Free PDF

Concept:

L = L₀ √(1 - v²/C²)

• In special relativity, an object moving at velocity v undergoes **length contraction** along the direction of motion.
• The contracted length L is given by the formula:
• The length component along the y-axis remains unchanged.
• We resolve the length components along x and y and apply length contraction to the x-component.

Calculation:

Length of the meter stick in its rest frame, L₀ = 1 m

Angle with the x-axis, θ = 45°

Velocity of the rod, v = (1/√2) C

Speed of light, C

⇒ Components of length in rest frame:

L₀ₓ = L₀ cos(θ) = 1 × cos 45° = 1/√2

L₀ᵧ = L₀ sin(θ) = 1 × sin 45° = 1/√2

⇒ Length contraction in the x-direction:

Lₓ = L₀ₓ √(1 - v²/C²)

⇒ Lₓ = (1/√2) √(1 - (1/2))

⇒ Lₓ = (1/√2) × (√1/√2) = 1/2

⇒ Total length in frame S:

L = √(Lₓ² + L₀ᵧ²)

⇒ L = √((1/2)² + (1/√2)²)

⇒ L = √(1/4 + 1/2)

⇒ L = √(3/4)

⇒ L = √3 / 2

∴ The length of the rod in frame S is √3/2 meters.

Solution:

In this solution, primed quantities () refer to the moving frame, while unprimed quantities () refer to the stationary frame.

Let be the rest frame of the rod, as shown in the diagram. In its rest frame, the rod forms an angle with the -axis. The lower end of the rod is at the origin of the frame, while the upper end is at:

In the stationary frame (), the rod moves to the right with velocity . The measurements are made at , when the origins of the two frames coincide.

To calculate the position of the rod in the frame, it is necessary to use the full Lorentz transformations:

Step 1: Lower end of the rod

At , the lower end of the rod is at:

Step 2: Upper end of the rod

At , the upper end of the rod is at:

Step 3: Orientation of the rod

The angle of the rod in the frame is given by:

For , , so .

Step 4: Length of the rod

The length of the rod in the frame is:

For , , and:

The correct option is 3).

Using work energy theorem.
Work done = Fx = change in K.E.
F_x = K.E_f - K.E_i
⇒ Fx = (γ - 1) m₀c² - 0

Using the work energy theorem again.
2F_x = K.E_f – K.E_i = K.E_f

The velocity components of particle with respect to lab frame

The velocity components with respect to moving frame is

Concept:

If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length L₀ that would be measured if the object were stationary. 

At relativistic speeds, Close to the speed of light, distances measured are not the same when measured by different observers.

Length Contraction:

Length contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object.

It is given by 

where L₀ is the length of the object in its rest frame, and L is the length in the frame moving with velocity V;

Calculation:

Given L = 98 cm; L₀ = 1m = 100 cm;

V = 0.199 C

CONCEPT:

• According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant.
• However special theory of relativity leads us to the concept of variation of mass with velocity.
• It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest.
• Some Interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations.

Time Dilation: 

• According to classical physics, time is an absolute quantity. But according to the special theory of relativity, Time is not an absolute quantity. It depends upon the motion of the frame of reference.
• If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by

This means that t’ has increased or dilated. In other words, the clock will go slow.

Length Contraction:

• The distance from the earth to a star measured by an observer in a moving spaceship would seem smaller than the distance measured by an observer on earth. i.e: (i-e S’

L'

Variation of mass:

• The mass is also not invariant.
• If a body at rest has a mass m0 it's mass when it moves with a velocity v, increases to m given by:

CALCULATION:

Given - Length of the cube = L0

• The volume of the cube is 

• If the cube moves with velocity V parallel to its one edge then its volume becomes

CONCEPT:

• According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant.
• However special theory of relativity leads us to the concept of variation of mass with velocity.
• It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest.
• Some Interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations.

Length Contraction:

The distance from the earth to a star measured by an observer in a moving spaceship would seem smaller than the distance measured by an observer on earth. i.e: (i-e S’

L'

Calculation:

Given - Length of the arrow = 1 m and v = 0.8c​

The length of the arrow due to Lorentz-Fitzgerald contraction as measured by an observer is 

Important Points

Variation of mass:

The mass is also not invariant.

If a body at rest has a mass m0 it's mass when it moves with a velocity v, increases to m given by:

Time Dilation: 

• According to classical physics, time is an absolute quantity. But according to the special theory of relativity, Time is not an absolute quantity. It depends upon the motion of the frame of reference.
• If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by

This means that t’ has increased or dilated. In other words, the clock will go slow.

Concept:

• According to Einstein’s Theory of relativity, Length contraction is the phenomenon in which the length of an object is measured to shorten than its proper length, which is the length as measured in the rest frame of the object.
• It is also known as Lorentz contraction and mainly noticeable when the inertial frame of reference moving with a substantial fraction of the speed of light.
• If L0 is rest length or the length measured by a stationary frame of reference the rod then the length of the rod measured in the frame moving with the velocity v is given as​

  ...(i)

Explanation:

• Length contraction is only observed in the direction in which the body is travelling.
• However, there is no change in the length in the perpendicular direction of the motion.

Concept:

If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length L₀ that would be measured if the object were stationary. 

At relativistic speeds, Close to the speed of light, distances measured are not the same when measured by different observers.

Length Contraction:

Length contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object.

It is given by 

where L₀ is the length of the object in its rest frame, and L is the length in the frame moving with velocity V;

Calculation:

Given L = 98 cm; L₀ = 1m = 100 cm;

V = 0.199 C

Concept:

L = L₀ √(1 - v²/C²)

• In special relativity, an object moving at velocity v undergoes **length contraction** along the direction of motion.
• The contracted length L is given by the formula:
• The length component along the y-axis remains unchanged.
• We resolve the length components along x and y and apply length contraction to the x-component.

Calculation:

Length of the meter stick in its rest frame, L₀ = 1 m

Angle with the x-axis, θ = 45°

Velocity of the rod, v = (1/√2) C

Speed of light, C

⇒ Components of length in rest frame:

L₀ₓ = L₀ cos(θ) = 1 × cos 45° = 1/√2

L₀ᵧ = L₀ sin(θ) = 1 × sin 45° = 1/√2

⇒ Length contraction in the x-direction:

Lₓ = L₀ₓ √(1 - v²/C²)

⇒ Lₓ = (1/√2) √(1 - (1/2))

⇒ Lₓ = (1/√2) × (√1/√2) = 1/2

⇒ Total length in frame S:

L = √(Lₓ² + L₀ᵧ²)

⇒ L = √((1/2)² + (1/√2)²)

⇒ L = √(1/4 + 1/2)

⇒ L = √(3/4)

⇒ L = √3 / 2

∴ The length of the rod in frame S is √3/2 meters.

CONCEPT:

• According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant.
• However special theory of relativity leads us to the concept of variation of mass with velocity.
• It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest.
• Some Interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations.

Time Dilation: 

• According to classical physics, time is an absolute quantity. But according to the special theory of relativity, Time is not an absolute quantity. It depends upon the motion of the frame of reference.
• If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by

This means that t’ has increased or dilated. In other words, the clock will go slow.

Length Contraction:

• The distance from the earth to a star measured by an observer in a moving spaceship would seem smaller than the distance measured by an observer on earth. i.e: (i-e S’

L'

Variation of mass:

• The mass is also not invariant.
• If a body at rest has a mass m0 it's mass when it moves with a velocity v, increases to m given by:

CALCULATION:

Given - Length of the cube = L0

• The volume of the cube is 

• If the cube moves with velocity V parallel to its one edge then its volume becomes

The speed of light in moving frame

Therefore, in the lab frame the velocity components of light can be written as

Correct Answer: 4)  

Explanation:

Given Data
A rod of length lies in the x'y' plane of its rest frame, making an angle of with the x' axis.
The rod moves to the right with velocity v relative to the lab frame x, y .
We are to find the length and orientation of the rod in the lab frame.

Since the rod is moving relative to the lab frame, only the component of the rod's length along the direction of motion (i.e., along the x axis in the lab frame) will experience Lorentz contraction.
The component of the length perpendicular to the direction of motion (along the y axis) will remain unchanged.

The rod makes an angle of with the x' axis in its own frame, so we can split the length into x' and y' components:
     
   
     
     
     

The x' component of the length will be contracted in the lab frame due to its motion along the x axis.
    According to Lorentz contraction, the contracted length in the lab frame is given by:

     
    The y component of the length remains unchanged in the lab frame, as it is perpendicular to the direction of motion:
     

     
    The total length L of the rod in the lab frame can be calculated by combining the x and y components:
     

     
     Substituting the values:
     
   
     Simplifying this expression gives:
     
     
     

CONCEPT:

• According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant.
• However special theory of relativity leads us to the concept of variation of mass with velocity.
• It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest.
• Some Interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations.

Length Contraction:

The distance from the earth to a star measured by an observer in a moving spaceship would seem smaller than the distance measured by an observer on earth. i.e: (i-e S’

L'

Calculation:

Given - Length of the arrow = 1 m and v = 0.8c​

The length of the arrow due to Lorentz-Fitzgerald contraction as measured by an observer is 

Important Points

Variation of mass:

The mass is also not invariant.

If a body at rest has a mass m0 it's mass when it moves with a velocity v, increases to m given by:

Time Dilation: 

• According to classical physics, time is an absolute quantity. But according to the special theory of relativity, Time is not an absolute quantity. It depends upon the motion of the frame of reference.
• If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by

This means that t’ has increased or dilated. In other words, the clock will go slow.