Match the items of List I with the items of List II and choose the correct answer from the code given below. List I List II (a) Dimension of Torque (i) ML-1T-2 (b) Dimension of Impulse (ii) ML2T-2 (c) Dimension of Pressure (iii) M-1L3T-2 (d) Dimension of Gravitational Constant (iv) MLT-1
Understanding Dimensions of Physical Quantities
In physics, the dimension of a physical quantity is the expression of its dependence on the fundamental quantities like mass (M), length (L), and time (T). Dimensional analysis is a useful tool for checking the consistency of equations and understanding the relationship between different physical quantities. Let's determine the dimensions for each quantity given in List I and match them with List II.
Calculating Dimensions for List I Quantities
We will determine the dimensions for each item in List I based on their definitions and fundamental units.
-
(a) Dimension of Torque:
Torque (\(\tau\)) is the rotational equivalent of force. It is defined as the product of force and the perpendicular distance from the axis of rotation to the point where the force is applied.
Formula: \(\tau = r \times F\)
Dimensions of Force (\(F\)): \([MLT^{-2}]\)
Dimensions of distance (\(r\)): \([L]\)
Dimension of Torque: \([\tau] = [L] \times [MLT^{-2}] = [ML^{2}T^{-2}]\) -
(b) Dimension of Impulse:
Impulse (\(J\)) is the change in momentum of an object. It is also defined as the product of force and the time interval over which the force acts.
Formula: \(J = F \times \Delta t\)
Dimensions of Force (\(F\)): \([MLT^{-2}]\)
Dimensions of time interval (\(\Delta t\)): \([T]\)
Dimension of Impulse: \([J] = [MLT^{-2}] \times [T] = [MLT^{-1}]\) -
(c) Dimension of Pressure:
Pressure (\(P\)) is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed.
Formula: \(P = \frac{F}{A}\)
Dimensions of Force (\(F\)): \([MLT^{-2}]\)
Dimensions of Area (\(A\)): \([L^2]\)
Dimension of Pressure: \([P] = \frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}]\) -
(d) Dimension of Gravitational Constant:
The Universal Gravitational Constant (\(G\)) appears in Newton's Law of Universal Gravitation, which describes the force between two masses.
Formula: \(F = G \frac{m_1 m_2}{r^2}\)
Rearranging for \(G\): \(G = \frac{F r^2}{m_1 m_2}\)
Dimensions of Force (\(F\)): \([MLT^{-2}]\)
Dimensions of distance (\(r\)): \([L]\), so \(r^2\) is \([L^2]\)
Dimensions of mass (\(m_1, m_2\)): \([M]\), so \(m_1 m_2\) is \([M^2]\)
Dimension of Gravitational Constant: \([G] = \frac{[MLT^{-2}] [L^2]}{[M^2]} = \frac{[ML^3T^{-2}]}{[M^2]} = [M^{-1}L^{3}T^{-2}]\)
Matching List I with List II Dimensions
Based on our calculations, let's match the quantities from List I with the dimensions from List II.
| List I (Quantity) | Calculated Dimension | Matches List II |
|---|---|---|
| (a) Dimension of Torque | \([ML^{2}T^{-2}]\) | (ii) \([ML^2T^{-2}]\) |
| (b) Dimension of Impulse | \([MLT^{-1}]\) | (iv) \([MLT^{-1}]\) |
| (c) Dimension of Pressure | \([ML^{-1}T^{-2}]\) | (i) \([ML^{-1}T^{-2}]\) |
| (d) Dimension of Gravitational Constant | \([M^{-1}L^{3}T^{-2}]\) | (iii) \([M^{-1}L^3T^{-2}]\) |
The correct matching is: (a) - (ii), (b) - (iv), (c) - (i), (d) - (iii).
Revision Table: Dimensions of Physical Quantities
| Physical Quantity | Common Formula | Dimension | SI Unit |
|---|---|---|---|
| Torque | \(r \times F\) | \([ML^2T^{-2}]\) | Newton-meter (Nm) |
| Impulse | \(F \times \Delta t\) or \(\Delta p\) | \([MLT^{-1}]\) | Newton-second (Ns) or kg m/s |
| Pressure | \(F / A\) | \([ML^{-1}T^{-2}]\) | Pascal (Pa) or N/m\(^2\) |
| Gravitational Constant (\(G\)) | \(G = \frac{F r^2}{m_1 m_2}\) | \([M^{-1}L^3T^{-2}]\) | Nm\(^2\)/kg\(^2\) |
| Force (\(F\)) | \(ma\) | \([MLT^{-2}]\) | Newton (N) |
| Energy / Work | \(F \times d\) | \([ML^2T^{-2}]\) | Joule (J) |
Additional Information on Dimensional Analysis
Dimensional analysis is a powerful technique in physics used for several purposes:
- Checking Consistency of Equations: In any physically correct equation, the dimensions of all terms on both sides of the equation must be the same. This is based on the principle of homogeneity of dimensions. For example, in the equation \(v = u + at\), the dimensions must be \([LT^{-1}] = [LT^{-1}] + [LT^{-2}][T] = [LT^{-1}]\).
- Deriving Relationships: Sometimes, the relationship between different physical quantities can be derived if we know which quantities they depend on. For instance, the time period (\(T\)) of a simple pendulum might depend on its mass (\(m\)), length (\(l\)), and acceleration due to gravity (\(g\)). By assuming \(T \propto m^a l^b g^c\) and matching dimensions on both sides, we can find the values of \(a, b, c\).
- Changing Units: Dimensions help in converting a physical quantity from one system of units (like SI) to another (like CGS).
Fundamental dimensions are typically taken as Mass (M), Length (L), Time (T), Electric Current (A), Thermodynamic Temperature (K), Amount of Substance (mol), and Luminous Intensity (cd). Most mechanical quantities can be expressed in terms of M, L, and T.
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