What are the dimensions of angular momentum?
[M L2 T-1]
Understanding the Dimensions of Angular Momentum
The question asks about the dimensions of angular momentum. Dimensions are fundamental physical quantities like mass (M), length (L), and time (T) that describe the nature of a physical quantity. To find the dimensions of angular momentum, we first need to understand what it represents and how it is defined.
What is Angular Momentum?
Angular momentum is a measure of how much an object is rotating and how difficult it is to stop that rotation. For a single particle, angular momentum (\(L\)) is defined as the cross product of its position vector (\(\mathbf{r}\)) relative to the axis of rotation and its linear momentum (\(\mathbf{p}\)). Mathematically, it is given by:
\( \mathbf{L} = \mathbf{r} \times \mathbf{p} \)
Where linear momentum (\(\mathbf{p}\)) is the product of mass (\(m\)) and velocity (\(\mathbf{v}\)):
\( \mathbf{p} = m \mathbf{v} \)
Substituting the expression for linear momentum into the angular momentum equation:
\( \mathbf{L} = \mathbf{r} \times (m \mathbf{v}) \)
To find the dimensions, we can consider the magnitudes of the quantities (since the cross product operation doesn't change the fundamental dimensions of the quantities involved):
\( L = r \cdot p = r \cdot (m \cdot v) \)
Now let's determine the dimensions of each component:
- Dimension of position vector \(r\) (length): \([L]\)
- Dimension of mass \(m\): \([M]\)
- Dimension of velocity \(v\) (displacement per unit time): \([L T^{-1}]\)
Deriving the Dimensions of Angular Momentum
Using the dimensions of \(r\), \(m\), and \(v\), we can find the dimensions of angular momentum \(L\):
Dimension of \(L\) = Dimension of \(r\) \(\times\) Dimension of \(m\) \(\times\) Dimension of \(v\)
Dimension of \(L\) = \([L] \times [M] \times [L T^{-1}]\)
Combining these dimensions:
Dimension of \(L\) = \([M L^{1+1} T^{-1}]\)
Dimension of \(L\) = \([M L^{2} T^{-1}]\)
So, the dimensions of angular momentum are \([M L^2 T^{-1}]\).
Comparing Dimensions with Options
Let's compare our derived dimensions with the given options:
- Option 1: \([M L^{2} T^{-2}]\)
- Option 2: \([M L^{2} T^{-3}]\)
- Option 3: \([M L^{2} T^{-1}]\)
- Option 4: \([M L T^{-1}]\)
Our derived dimension \([M L^2 T^{-1}]\) matches Option 3.
Summary of Dimensions
The fundamental dimensions of angular momentum are Mass to the power of 1, Length to the power of 2, and Time to the power of -1.
| Quantity | Symbol | Formula (Example) | Dimensions |
|---|---|---|---|
| Position | \(r\) | Length | \([L]\) |
| Mass | \(m\) | Mass | \([M]\) |
| Velocity | \(v\) | \( \Delta x / \Delta t \) | \([L T^{-1}]\) |
| Linear Momentum | \(p\) | \(m v\) | \([M L T^{-1}]\) |
| Angular Momentum | \(L\) | \(r \times p\) | \([M L^2 T^{-1}]\) |
Revision Table: Key Physical Quantities and Dimensions
| Quantity | Dimensions |
|---|---|
| Length | \([L]\) |
| Mass | \([M]\) |
| Time | \([T]\) |
| Velocity | \([L T^{-1}]\) |
| Acceleration | \([L T^{-2}]\) |
| Force | \([M L T^{-2}]\) |
| Work/Energy | \([M L^2 T^{-2}]\) |
| Power | \([M L^2 T^{-3}]\) |
| Linear Momentum | \([M L T^{-1}]\) |
| Angular Momentum | \([M L^2 T^{-1}]\) |
| Torque | \([M L^2 T^{-2}]\) |
Additional Information: Angular Momentum Conservation and Torque
Angular momentum is a very important concept in physics, especially in rotational motion. Similar to how linear momentum is conserved in the absence of external forces, angular momentum is conserved in the absence of external torques.
Torque (\(\mathbf{\tau}\)) is the rotational equivalent of force and is defined as the cross product of the position vector and the force (\(\mathbf{\tau} = \mathbf{r} \times \mathbf{F}\)). Let's quickly check its dimensions:
Dimension of \(\tau\) = Dimension of \(r\) \(\times\) Dimension of \(F\)
Dimension of \(r\) = \([L]\)
Dimension of \(F\) = \([M L T^{-2}]\) (from \(F = ma\))
Dimension of \(\tau\) = \([L] \times [M L T^{-2}] = [M L^2 T^{-2}]\)
Note that the dimensions of torque \([M L^2 T^{-2}]\) are the same as those of work or energy. However, torque and work are fundamentally different physical quantities despite having the same dimensions. Angular momentum has different dimensions, \([M L^2 T^{-1}]\).
Understanding the dimensions helps in checking the consistency of equations in physics. If an equation is correct, the dimensions on both sides of the equation must be the same.
Important Questions from Conservation of Linear Momentum
- A particle of mass $6m$ at rest suddenly breaks on its own into three fragments.
Two fragments of mass $m$ and $2m$ move along mutually perpendicular directions with speeds $2v$ and $v$ respectively.
The energy released during the process is, - Body A of mass $m$ moving with speed $u$ collides with another body B of mass $3m$, at rest. The collision is head-on and elastic in nature. After the collision, the fraction of energy lost by the colliding body A is:
The propulsion of a rocket is based on which of Newton's laws of motion?
For a system of interacting particles, which condition is fundamental for the conservation of its total linear momentum $\vec{P}$?
The total momentum of a system of masses (i.e. moving bodies) in any one direction remains constant, unless acted upon by an external force in that direction. This statement is called-
