Answer :
Sure, let's obtain the dimensional formulae for each of the given physical quantities step-by-step:
1. Universal Gravitational Constant (G):
- Dimensional formula: The Universal Gravitational Constant [tex]\( G \)[/tex] is defined in Newton's law of gravitation:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Here, [tex]\( F \)[/tex] is the gravitational force with the dimensional formula: [tex]\([MLT^{-2}]\)[/tex], [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are masses with dimensional formula: [tex]\([M]\)[/tex], and [tex]\( r \)[/tex] is the distance with dimensional formula: [tex]\([L]\)[/tex].
Rewriting the formula for [tex]\( G \)[/tex], we get:
[tex]\[ G = \frac{Fr^2}{m_1 m_2} \][/tex]
Substituting the dimensions, we have:
[tex]\[ [G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = [M^{-1} L^3 T^{-2}] \][/tex]
Hence, the dimensional formula for [tex]\( G \)[/tex] is:
[tex]\[ [M^{-1} L^3 T^{-2}] \][/tex]
2. Coefficient of Viscosity (η):
- Dimensional formula: The coefficient of viscosity [tex]\( η \)[/tex] relates to the force per unit area, acting between fluid layers, with velocity gradient perpendicular to the layers.
[tex]\[ \eta = \frac{F}{A \left(\frac{v}{y}\right)} \][/tex]
Here, [tex]\( F \)[/tex] is force with dimensional formula: [tex]\([MLT^{-2}]\)[/tex], [tex]\( A \)[/tex] is area with dimensional formula: [tex]\([L^2]\)[/tex], [tex]\( v \)[/tex] is velocity with dimensional formula: [tex]\([LT^{-1}]\)[/tex], and [tex]\( y \)[/tex] is distance with dimensional formula: [tex]\([L]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [\eta] = \frac{[MLT^{-2}]}{[L^2] \cdot [LT^{-1}][L^{-1}]} = [M L^{-1} T^{-1}] \][/tex]
Hence, the dimensional formula for [tex]\( η \)[/tex] is:
[tex]\[ [M L^{-1} T^{-1}] \][/tex]
3. Electric Potential (V):
- Dimensional formula: Electric potential [tex]\( V \)[/tex] is defined as the work done per unit charge:
[tex]\[ V = \frac{W}{Q} \][/tex]
Here, [tex]\( W \)[/tex] is work done with dimensional formula: [tex]\([ML^2T^{-2}]\)[/tex] (same as energy) and [tex]\( Q \)[/tex] is charge with dimensional formula: [tex]\([IT]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [V] = \frac{[ML^2T^{-2}]}{[IT]} = [M L^2 T^{-3} I^{-1}] \][/tex]
Hence, the dimensional formula for [tex]\( V \)[/tex] is:
[tex]\[ [M L^2 T^{-3} I^{-1}] \][/tex]
4. Resistance (R):
- Dimensional formula: Electrical resistance [tex]\( R \)[/tex] relates voltage [tex]\( V \)[/tex] to current [tex]\( I \)[/tex] by Ohm's Law:
[tex]\[ R = \frac{V}{I} \][/tex]
Here, [tex]\( V \)[/tex] is electric potential with dimensional formula: [tex]\([M L^2 T^{-3} I^{-1}]\)[/tex] and [tex]\( I \)[/tex] is current with dimensional formula: [tex]\([I]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [R] = \frac{[M L^2 T^{-3} I^{-1}]}{[I]} = [M L^2 T^{-3} I^{-2}] \][/tex]
Hence, the dimensional formula for [tex]\( R \)[/tex] is:
[tex]\[ [M L^2 T^{-3} I^{-2}] \][/tex]
So, the dimensional formulae for the given physical quantities are:
1. Universal gravitational constant ([tex]\(G\)[/tex]): [tex]\([M^{-1} L^3 T^{-2}]\)[/tex]
2. Coefficient of viscosity ([tex]\(η\)[/tex]): [tex]\([M L^{-1} T^{-1}]\)[/tex]
3. Electric potential ([tex]\(V\)[/tex]): [tex]\([M L^2 T^{-3} I^{-1}]\)[/tex]
4. Resistance ([tex]\(R\)[/tex]): [tex]\([M L^2 T^{-3} I^{-2}]\)[/tex]
1. Universal Gravitational Constant (G):
- Dimensional formula: The Universal Gravitational Constant [tex]\( G \)[/tex] is defined in Newton's law of gravitation:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Here, [tex]\( F \)[/tex] is the gravitational force with the dimensional formula: [tex]\([MLT^{-2}]\)[/tex], [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are masses with dimensional formula: [tex]\([M]\)[/tex], and [tex]\( r \)[/tex] is the distance with dimensional formula: [tex]\([L]\)[/tex].
Rewriting the formula for [tex]\( G \)[/tex], we get:
[tex]\[ G = \frac{Fr^2}{m_1 m_2} \][/tex]
Substituting the dimensions, we have:
[tex]\[ [G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = [M^{-1} L^3 T^{-2}] \][/tex]
Hence, the dimensional formula for [tex]\( G \)[/tex] is:
[tex]\[ [M^{-1} L^3 T^{-2}] \][/tex]
2. Coefficient of Viscosity (η):
- Dimensional formula: The coefficient of viscosity [tex]\( η \)[/tex] relates to the force per unit area, acting between fluid layers, with velocity gradient perpendicular to the layers.
[tex]\[ \eta = \frac{F}{A \left(\frac{v}{y}\right)} \][/tex]
Here, [tex]\( F \)[/tex] is force with dimensional formula: [tex]\([MLT^{-2}]\)[/tex], [tex]\( A \)[/tex] is area with dimensional formula: [tex]\([L^2]\)[/tex], [tex]\( v \)[/tex] is velocity with dimensional formula: [tex]\([LT^{-1}]\)[/tex], and [tex]\( y \)[/tex] is distance with dimensional formula: [tex]\([L]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [\eta] = \frac{[MLT^{-2}]}{[L^2] \cdot [LT^{-1}][L^{-1}]} = [M L^{-1} T^{-1}] \][/tex]
Hence, the dimensional formula for [tex]\( η \)[/tex] is:
[tex]\[ [M L^{-1} T^{-1}] \][/tex]
3. Electric Potential (V):
- Dimensional formula: Electric potential [tex]\( V \)[/tex] is defined as the work done per unit charge:
[tex]\[ V = \frac{W}{Q} \][/tex]
Here, [tex]\( W \)[/tex] is work done with dimensional formula: [tex]\([ML^2T^{-2}]\)[/tex] (same as energy) and [tex]\( Q \)[/tex] is charge with dimensional formula: [tex]\([IT]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [V] = \frac{[ML^2T^{-2}]}{[IT]} = [M L^2 T^{-3} I^{-1}] \][/tex]
Hence, the dimensional formula for [tex]\( V \)[/tex] is:
[tex]\[ [M L^2 T^{-3} I^{-1}] \][/tex]
4. Resistance (R):
- Dimensional formula: Electrical resistance [tex]\( R \)[/tex] relates voltage [tex]\( V \)[/tex] to current [tex]\( I \)[/tex] by Ohm's Law:
[tex]\[ R = \frac{V}{I} \][/tex]
Here, [tex]\( V \)[/tex] is electric potential with dimensional formula: [tex]\([M L^2 T^{-3} I^{-1}]\)[/tex] and [tex]\( I \)[/tex] is current with dimensional formula: [tex]\([I]\)[/tex].
Substituting the dimensions, we have:
[tex]\[ [R] = \frac{[M L^2 T^{-3} I^{-1}]}{[I]} = [M L^2 T^{-3} I^{-2}] \][/tex]
Hence, the dimensional formula for [tex]\( R \)[/tex] is:
[tex]\[ [M L^2 T^{-3} I^{-2}] \][/tex]
So, the dimensional formulae for the given physical quantities are:
1. Universal gravitational constant ([tex]\(G\)[/tex]): [tex]\([M^{-1} L^3 T^{-2}]\)[/tex]
2. Coefficient of viscosity ([tex]\(η\)[/tex]): [tex]\([M L^{-1} T^{-1}]\)[/tex]
3. Electric potential ([tex]\(V\)[/tex]): [tex]\([M L^2 T^{-3} I^{-1}]\)[/tex]
4. Resistance ([tex]\(R\)[/tex]): [tex]\([M L^2 T^{-3} I^{-2}]\)[/tex]
