Question

$$ \begin{aligned} &\text { Match the column }\\\ &\begin{array}{|l|l|} \hline \text { Column - I } & \text { Column - II } \\ \hline \text { (1) space is isotropic } & \begin{array}{l} \text { (P) conservation of linear } \\ \text { momentum } \end{array} \\ \hline \text { (2) space is homogeneous } & \text { (Q) conservation of energy } \\ \hline \text { (3) Time is homogeneous } & \begin{array}{l} \text { (R) conservation of charge still } \\ \text { not known } \end{array} \\ \hline \text { (4) Time is isotropic } & \begin{array}{l} \text { (S) conservation of } \\ \text { angular momentum } \end{array} \\ \hline \end{array} \end{aligned} $$ (a) \(1-(\mathrm{S}), 2-(\mathrm{P}), 3-(\mathrm{R}), 4-(\mathrm{Q})\) (b) \(1-(\mathrm{S}), 2-(\mathrm{P}), 3-(\mathrm{Q}), 4-(\mathrm{R})\) (c) \(1-(\mathrm{P}), 2-(\mathrm{S}), 3-(\mathrm{R}), 4-(\mathrm{Q})\) (d) \(1-(\mathrm{R}), 2-(\mathrm{Q}), 3-(\mathrm{P}), 4-(\mathrm{S})\)

Short answer

The short answer is: (b) \(1-(\mathrm{S}), 2-(\mathrm{P}), 3-(\mathrm{Q}), 4-(\mathrm{R})\).

Step-by-step solution

1. Space is isotropic

Isotropic space means that the properties of space do not depend on the direction. As a result, angular momentum does not change when the direction changes. This isotropy of space corresponds to the conservation of angular momentum. So, (1) matches with (S).

2. Space is homogeneous

Homogeneous space means that the properties of space do not depend on the position. Thus, linear momentum will remain conserved since it does not depend on a specific location. So, (2) matches with (P).

3. Time is homogeneous

Homogeneous time means that the properties of time do not depend on a specific moment. In this case, the total energy of a system will remain conserved since it does not depend on a specific point in time. So, (3) matches with (Q).

4. Time is isotropic

Isotropic time means the properties of time do not depend on the direction of time. This property still hasn't been proven, and it is not widely known regarding its conservation law. So, (4) matches with (R). Based on the matches above, the correct option is: (b) \(1-(\mathrm{S}), 2-(\mathrm{P}), 3-(\mathrm{Q}), 4-(\mathrm{R})\)

Key concepts

Isotropic Space

In physics, when we say that space is isotropic, we mean that space looks the same in all directions. There is no preferred direction; no matter where you look or how you orient yourself, the properties remain constant. This trait implies that natural laws remain unchanged regardless of direction. This isotropy is fundamental in physics because it leads to an important conservation law: the conservation of angular momentum.
Angular momentum is a detailed measure of rotational motion. When space is isotropic, the absence of any directional preference ensures that the angular momentum of isolated systems does not change. This concept can be observed in astronomical systems and everyday occurrences such as a spinning figure skater pulling in her arms to spin faster.

• Isotropy removes directional bias.
• Preserves rotational motion consistency.
• Conserves angular momentum in isolated systems.

Homogeneous Space

The concept of homogeneous space implies that the characteristics and laws of space remain identical throughout different locations. Homogeneity assures that nothing fundamentally changes when you move from one point in space to another.
In terms of conservation laws, it is closely tied to the conservation of linear momentum. Since physical processes do not change across different positions, the linear momentum of a system is conserved. Linear momentum relates to the motion in a straight line and is an essential concept in Newtonian physics. This concept manifests in real life when you push a box across a frictionless surface—the box continues to move unless acted on by another force.

• Space's properties remain consistent throughout.
• Key to the conservation of linear momentum.
• Ensures processes remain uniform across positions.

Conservation of Angular Momentum

Conservation of angular momentum is a crucial principle that emerges from isotropic space. It states that in the absence of an external torque, the angular momentum of a system remains constant over time. This principle applies both to microscopic systems, like electrons in atoms, and to macroscopic systems, such as galaxies.
Mathematically, it can be expressed as \( L = I \omega \), where \( L \) is angular momentum, \( I \) is the moment of inertia, and \( \omega \) is angular velocity. For any closed system, if there is no net external torque, \( \Delta L = 0 \), meaning there is no change in angular momentum. This is seen when an ice skater extends her arms to slow her rotation and pulls them in to speed up—changing her moment of inertia but keeping her angular momentum constant.

• An outcome of isotropic space.
• No change in angular momentum without external torque.
• Fundamental in understanding rotational dynamics.

Conservation of Linear Momentum

The conservation of linear momentum stems from the homogeneity of space. According to this law, in an isolated system with no external forces, the total linear momentum remains constant. Linear momentum, defined as \( p = mv \) for mass \( m \) and velocity \( v \), is a vector quantity meaning it has both magnitude and direction.
To see it in action, consider a game of pool. When the cue ball strikes another ball, momentum is transferred from the cue ball to the target ball, but the total momentum of the two-ball system remains constant.

• Arises from uniform space properties.
• Unchanging without external forces.
• Illustrated in interactions and collisions.