IDNLearn.com makes it easy to find answers and share knowledge with others. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
Sagot :
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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
