367370
Identify the pair of physical quantities which have different dimensions
1 wave number and Rydberg's constant
2 Stress and Coefficient of elasticity
3 Coercivity and Magnetisation
4 Specific heat capacity and Latent heat
Explanation:
For choice (4) We use \((K)\) for symbol of temperature in dimensional analysis as 'kelvin' is \(SI\) unit of temperture Denote: Specific heat capacity \( = s\) Latent heat \( = {L_{{\rm{heat }}}}\quad \) (Note: \(L\) for length) Their defining formulae are \(Q=m s(\Delta \theta)\) \(Q = m{L_{{\rm{heat}}}}.\) Here \(Q = \) energy (in form of heat) \(m = {\rm{mass}}\) \(\Delta \theta=\) change in temperature \(s=\dfrac{Q}{m(\Delta \theta)}\) \(=\dfrac{M L^{2} T^{-2}}{M(K)}\) \(=\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\). \(L_{\text {heat }}=\dfrac{M L^{2} T^{-2}}{M}\) \( = {L^2}\;{T^{\, - 2}}.\) The pair in choice (4) has different dimensions. For choice (1) \(\bar{v}=\) number of waves per \(cm.\) \(=\left[L^{-1}\right]\) \(R_{H}=\dfrac{1}{\left[\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right] \lambda}\) (where \({n_1},\,\,{n_2}\) are serial numbers of Bohr orbits) Both \(\bar{v}\) and \(R_{H}\) have same dimensional formula \(\operatorname{viz}:\left[L^{-1}\right]\) For choice (2): Stress \(=\dfrac{F}{A}=\dfrac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2}\) Coefficient of elasticity is elastic modulus such as Youngs modulus \((Y)\) \(Y=\dfrac{\text { stress }}{\text { strain }}\) \(=\dfrac{M L^{-1} T^{-2}}{\text { strain }}=M L^{-1} T^{-2}\) \(\left(\because \operatorname{strain}=\dfrac{\Delta l}{l}=\dfrac{[L]}{\left[L^{-1}\right]}=L^{0}\right)\) For choice (3): We know \(B=\mu_{0}(H+M)\) As 'similar quantities' only can be added, the magnetisation \(M\) has same unit as \(H\) (\(i.e.,\) Tesla)
This hystersis graph indicates that 'coercivity' is a particular value of \(H\) for a given magnetic material.
JEE - 2022
PHXI02:UNITS AND MEASUREMENTS
367371
The dimension of the ratio of angular momentum to linear momentum is
1 \({{L}^{0}}\)
2 \({{L}^{1}}\)
3 \({{L}^{2}}\)
4 \(MLT\)
Explanation:
Angular momentum \({L=m v r}\) Linen momentum \({=p=m v}\) \({\dfrac{[L]}{[p]}=\dfrac{\left[{ML}^{2} {~T}^{-1}\right]}{\left[{MLT}^{-1}\right]}=\left[{M}^{0} {~L}^{1} {~T}^{0}\right]}\). So correct option is (2).
PHXI02:UNITS AND MEASUREMENTS
367372
The dimensional formula for impulse is
1 \(ML{T^{ - 1}}\)
2 \(M{L^{ - 1}}\)
3 \({M^{ - 1}}L{T^{ - 1}}\)
4 \(M{L^{ - 1}}{T^{ - 1}}\)
Explanation:
Impulse \(=\) force \(×\) time \( = [ML{T^{ - 2}} \times T] = [ML{T^{ - 1}}]\)
KCET - 2007
PHXI02:UNITS AND MEASUREMENTS
367373
The dimensions of emf in \(M K S\) system of unit is:
367374
If \(G\) be the gravitational constant and \(u\) be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{u G}\).
1 Gravitational potential.
2 Pressure gradient per unit mass.
3 Energy per unit mass.
4 Force per unit mass.
Explanation:
Dimensions of gravitational constant and energy density are given as \([G]=\left[M^{-1} L^{3} T^{-2}\right]\) \([u]=\dfrac{[E]}{[V]}=\dfrac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}=\left[M L^{-1} T^{-2}\right]\) Therefore, dimensions of \(\sqrt{u G}\), \(\begin{aligned}& {[\sqrt{u G}]=\left[\sqrt{\left[M L^{-1} T^{-2}\right] \times\left[M^{-1} L^{3} T^{-2}\right]}\right]} \\& \left.=\left[\sqrt{L^{2} T^{-4}}\right]\right]=\left[L T^{-2}\right]\end{aligned}\) The dimension of \(\sqrt{u G}\) is same as the dimension of Force per unit mass. So, option (4) is correct.
367370
Identify the pair of physical quantities which have different dimensions
1 wave number and Rydberg's constant
2 Stress and Coefficient of elasticity
3 Coercivity and Magnetisation
4 Specific heat capacity and Latent heat
Explanation:
For choice (4) We use \((K)\) for symbol of temperature in dimensional analysis as 'kelvin' is \(SI\) unit of temperture Denote: Specific heat capacity \( = s\) Latent heat \( = {L_{{\rm{heat }}}}\quad \) (Note: \(L\) for length) Their defining formulae are \(Q=m s(\Delta \theta)\) \(Q = m{L_{{\rm{heat}}}}.\) Here \(Q = \) energy (in form of heat) \(m = {\rm{mass}}\) \(\Delta \theta=\) change in temperature \(s=\dfrac{Q}{m(\Delta \theta)}\) \(=\dfrac{M L^{2} T^{-2}}{M(K)}\) \(=\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\). \(L_{\text {heat }}=\dfrac{M L^{2} T^{-2}}{M}\) \( = {L^2}\;{T^{\, - 2}}.\) The pair in choice (4) has different dimensions. For choice (1) \(\bar{v}=\) number of waves per \(cm.\) \(=\left[L^{-1}\right]\) \(R_{H}=\dfrac{1}{\left[\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right] \lambda}\) (where \({n_1},\,\,{n_2}\) are serial numbers of Bohr orbits) Both \(\bar{v}\) and \(R_{H}\) have same dimensional formula \(\operatorname{viz}:\left[L^{-1}\right]\) For choice (2): Stress \(=\dfrac{F}{A}=\dfrac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2}\) Coefficient of elasticity is elastic modulus such as Youngs modulus \((Y)\) \(Y=\dfrac{\text { stress }}{\text { strain }}\) \(=\dfrac{M L^{-1} T^{-2}}{\text { strain }}=M L^{-1} T^{-2}\) \(\left(\because \operatorname{strain}=\dfrac{\Delta l}{l}=\dfrac{[L]}{\left[L^{-1}\right]}=L^{0}\right)\) For choice (3): We know \(B=\mu_{0}(H+M)\) As 'similar quantities' only can be added, the magnetisation \(M\) has same unit as \(H\) (\(i.e.,\) Tesla)
This hystersis graph indicates that 'coercivity' is a particular value of \(H\) for a given magnetic material.
JEE - 2022
PHXI02:UNITS AND MEASUREMENTS
367371
The dimension of the ratio of angular momentum to linear momentum is
1 \({{L}^{0}}\)
2 \({{L}^{1}}\)
3 \({{L}^{2}}\)
4 \(MLT\)
Explanation:
Angular momentum \({L=m v r}\) Linen momentum \({=p=m v}\) \({\dfrac{[L]}{[p]}=\dfrac{\left[{ML}^{2} {~T}^{-1}\right]}{\left[{MLT}^{-1}\right]}=\left[{M}^{0} {~L}^{1} {~T}^{0}\right]}\). So correct option is (2).
PHXI02:UNITS AND MEASUREMENTS
367372
The dimensional formula for impulse is
1 \(ML{T^{ - 1}}\)
2 \(M{L^{ - 1}}\)
3 \({M^{ - 1}}L{T^{ - 1}}\)
4 \(M{L^{ - 1}}{T^{ - 1}}\)
Explanation:
Impulse \(=\) force \(×\) time \( = [ML{T^{ - 2}} \times T] = [ML{T^{ - 1}}]\)
KCET - 2007
PHXI02:UNITS AND MEASUREMENTS
367373
The dimensions of emf in \(M K S\) system of unit is:
367374
If \(G\) be the gravitational constant and \(u\) be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{u G}\).
1 Gravitational potential.
2 Pressure gradient per unit mass.
3 Energy per unit mass.
4 Force per unit mass.
Explanation:
Dimensions of gravitational constant and energy density are given as \([G]=\left[M^{-1} L^{3} T^{-2}\right]\) \([u]=\dfrac{[E]}{[V]}=\dfrac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}=\left[M L^{-1} T^{-2}\right]\) Therefore, dimensions of \(\sqrt{u G}\), \(\begin{aligned}& {[\sqrt{u G}]=\left[\sqrt{\left[M L^{-1} T^{-2}\right] \times\left[M^{-1} L^{3} T^{-2}\right]}\right]} \\& \left.=\left[\sqrt{L^{2} T^{-4}}\right]\right]=\left[L T^{-2}\right]\end{aligned}\) The dimension of \(\sqrt{u G}\) is same as the dimension of Force per unit mass. So, option (4) is correct.
367370
Identify the pair of physical quantities which have different dimensions
1 wave number and Rydberg's constant
2 Stress and Coefficient of elasticity
3 Coercivity and Magnetisation
4 Specific heat capacity and Latent heat
Explanation:
For choice (4) We use \((K)\) for symbol of temperature in dimensional analysis as 'kelvin' is \(SI\) unit of temperture Denote: Specific heat capacity \( = s\) Latent heat \( = {L_{{\rm{heat }}}}\quad \) (Note: \(L\) for length) Their defining formulae are \(Q=m s(\Delta \theta)\) \(Q = m{L_{{\rm{heat}}}}.\) Here \(Q = \) energy (in form of heat) \(m = {\rm{mass}}\) \(\Delta \theta=\) change in temperature \(s=\dfrac{Q}{m(\Delta \theta)}\) \(=\dfrac{M L^{2} T^{-2}}{M(K)}\) \(=\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\). \(L_{\text {heat }}=\dfrac{M L^{2} T^{-2}}{M}\) \( = {L^2}\;{T^{\, - 2}}.\) The pair in choice (4) has different dimensions. For choice (1) \(\bar{v}=\) number of waves per \(cm.\) \(=\left[L^{-1}\right]\) \(R_{H}=\dfrac{1}{\left[\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right] \lambda}\) (where \({n_1},\,\,{n_2}\) are serial numbers of Bohr orbits) Both \(\bar{v}\) and \(R_{H}\) have same dimensional formula \(\operatorname{viz}:\left[L^{-1}\right]\) For choice (2): Stress \(=\dfrac{F}{A}=\dfrac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2}\) Coefficient of elasticity is elastic modulus such as Youngs modulus \((Y)\) \(Y=\dfrac{\text { stress }}{\text { strain }}\) \(=\dfrac{M L^{-1} T^{-2}}{\text { strain }}=M L^{-1} T^{-2}\) \(\left(\because \operatorname{strain}=\dfrac{\Delta l}{l}=\dfrac{[L]}{\left[L^{-1}\right]}=L^{0}\right)\) For choice (3): We know \(B=\mu_{0}(H+M)\) As 'similar quantities' only can be added, the magnetisation \(M\) has same unit as \(H\) (\(i.e.,\) Tesla)
This hystersis graph indicates that 'coercivity' is a particular value of \(H\) for a given magnetic material.
JEE - 2022
PHXI02:UNITS AND MEASUREMENTS
367371
The dimension of the ratio of angular momentum to linear momentum is
1 \({{L}^{0}}\)
2 \({{L}^{1}}\)
3 \({{L}^{2}}\)
4 \(MLT\)
Explanation:
Angular momentum \({L=m v r}\) Linen momentum \({=p=m v}\) \({\dfrac{[L]}{[p]}=\dfrac{\left[{ML}^{2} {~T}^{-1}\right]}{\left[{MLT}^{-1}\right]}=\left[{M}^{0} {~L}^{1} {~T}^{0}\right]}\). So correct option is (2).
PHXI02:UNITS AND MEASUREMENTS
367372
The dimensional formula for impulse is
1 \(ML{T^{ - 1}}\)
2 \(M{L^{ - 1}}\)
3 \({M^{ - 1}}L{T^{ - 1}}\)
4 \(M{L^{ - 1}}{T^{ - 1}}\)
Explanation:
Impulse \(=\) force \(×\) time \( = [ML{T^{ - 2}} \times T] = [ML{T^{ - 1}}]\)
KCET - 2007
PHXI02:UNITS AND MEASUREMENTS
367373
The dimensions of emf in \(M K S\) system of unit is:
367374
If \(G\) be the gravitational constant and \(u\) be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{u G}\).
1 Gravitational potential.
2 Pressure gradient per unit mass.
3 Energy per unit mass.
4 Force per unit mass.
Explanation:
Dimensions of gravitational constant and energy density are given as \([G]=\left[M^{-1} L^{3} T^{-2}\right]\) \([u]=\dfrac{[E]}{[V]}=\dfrac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}=\left[M L^{-1} T^{-2}\right]\) Therefore, dimensions of \(\sqrt{u G}\), \(\begin{aligned}& {[\sqrt{u G}]=\left[\sqrt{\left[M L^{-1} T^{-2}\right] \times\left[M^{-1} L^{3} T^{-2}\right]}\right]} \\& \left.=\left[\sqrt{L^{2} T^{-4}}\right]\right]=\left[L T^{-2}\right]\end{aligned}\) The dimension of \(\sqrt{u G}\) is same as the dimension of Force per unit mass. So, option (4) is correct.
367370
Identify the pair of physical quantities which have different dimensions
1 wave number and Rydberg's constant
2 Stress and Coefficient of elasticity
3 Coercivity and Magnetisation
4 Specific heat capacity and Latent heat
Explanation:
For choice (4) We use \((K)\) for symbol of temperature in dimensional analysis as 'kelvin' is \(SI\) unit of temperture Denote: Specific heat capacity \( = s\) Latent heat \( = {L_{{\rm{heat }}}}\quad \) (Note: \(L\) for length) Their defining formulae are \(Q=m s(\Delta \theta)\) \(Q = m{L_{{\rm{heat}}}}.\) Here \(Q = \) energy (in form of heat) \(m = {\rm{mass}}\) \(\Delta \theta=\) change in temperature \(s=\dfrac{Q}{m(\Delta \theta)}\) \(=\dfrac{M L^{2} T^{-2}}{M(K)}\) \(=\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\). \(L_{\text {heat }}=\dfrac{M L^{2} T^{-2}}{M}\) \( = {L^2}\;{T^{\, - 2}}.\) The pair in choice (4) has different dimensions. For choice (1) \(\bar{v}=\) number of waves per \(cm.\) \(=\left[L^{-1}\right]\) \(R_{H}=\dfrac{1}{\left[\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right] \lambda}\) (where \({n_1},\,\,{n_2}\) are serial numbers of Bohr orbits) Both \(\bar{v}\) and \(R_{H}\) have same dimensional formula \(\operatorname{viz}:\left[L^{-1}\right]\) For choice (2): Stress \(=\dfrac{F}{A}=\dfrac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2}\) Coefficient of elasticity is elastic modulus such as Youngs modulus \((Y)\) \(Y=\dfrac{\text { stress }}{\text { strain }}\) \(=\dfrac{M L^{-1} T^{-2}}{\text { strain }}=M L^{-1} T^{-2}\) \(\left(\because \operatorname{strain}=\dfrac{\Delta l}{l}=\dfrac{[L]}{\left[L^{-1}\right]}=L^{0}\right)\) For choice (3): We know \(B=\mu_{0}(H+M)\) As 'similar quantities' only can be added, the magnetisation \(M\) has same unit as \(H\) (\(i.e.,\) Tesla)
This hystersis graph indicates that 'coercivity' is a particular value of \(H\) for a given magnetic material.
JEE - 2022
PHXI02:UNITS AND MEASUREMENTS
367371
The dimension of the ratio of angular momentum to linear momentum is
1 \({{L}^{0}}\)
2 \({{L}^{1}}\)
3 \({{L}^{2}}\)
4 \(MLT\)
Explanation:
Angular momentum \({L=m v r}\) Linen momentum \({=p=m v}\) \({\dfrac{[L]}{[p]}=\dfrac{\left[{ML}^{2} {~T}^{-1}\right]}{\left[{MLT}^{-1}\right]}=\left[{M}^{0} {~L}^{1} {~T}^{0}\right]}\). So correct option is (2).
PHXI02:UNITS AND MEASUREMENTS
367372
The dimensional formula for impulse is
1 \(ML{T^{ - 1}}\)
2 \(M{L^{ - 1}}\)
3 \({M^{ - 1}}L{T^{ - 1}}\)
4 \(M{L^{ - 1}}{T^{ - 1}}\)
Explanation:
Impulse \(=\) force \(×\) time \( = [ML{T^{ - 2}} \times T] = [ML{T^{ - 1}}]\)
KCET - 2007
PHXI02:UNITS AND MEASUREMENTS
367373
The dimensions of emf in \(M K S\) system of unit is:
367374
If \(G\) be the gravitational constant and \(u\) be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{u G}\).
1 Gravitational potential.
2 Pressure gradient per unit mass.
3 Energy per unit mass.
4 Force per unit mass.
Explanation:
Dimensions of gravitational constant and energy density are given as \([G]=\left[M^{-1} L^{3} T^{-2}\right]\) \([u]=\dfrac{[E]}{[V]}=\dfrac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}=\left[M L^{-1} T^{-2}\right]\) Therefore, dimensions of \(\sqrt{u G}\), \(\begin{aligned}& {[\sqrt{u G}]=\left[\sqrt{\left[M L^{-1} T^{-2}\right] \times\left[M^{-1} L^{3} T^{-2}\right]}\right]} \\& \left.=\left[\sqrt{L^{2} T^{-4}}\right]\right]=\left[L T^{-2}\right]\end{aligned}\) The dimension of \(\sqrt{u G}\) is same as the dimension of Force per unit mass. So, option (4) is correct.
367370
Identify the pair of physical quantities which have different dimensions
1 wave number and Rydberg's constant
2 Stress and Coefficient of elasticity
3 Coercivity and Magnetisation
4 Specific heat capacity and Latent heat
Explanation:
For choice (4) We use \((K)\) for symbol of temperature in dimensional analysis as 'kelvin' is \(SI\) unit of temperture Denote: Specific heat capacity \( = s\) Latent heat \( = {L_{{\rm{heat }}}}\quad \) (Note: \(L\) for length) Their defining formulae are \(Q=m s(\Delta \theta)\) \(Q = m{L_{{\rm{heat}}}}.\) Here \(Q = \) energy (in form of heat) \(m = {\rm{mass}}\) \(\Delta \theta=\) change in temperature \(s=\dfrac{Q}{m(\Delta \theta)}\) \(=\dfrac{M L^{2} T^{-2}}{M(K)}\) \(=\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\). \(L_{\text {heat }}=\dfrac{M L^{2} T^{-2}}{M}\) \( = {L^2}\;{T^{\, - 2}}.\) The pair in choice (4) has different dimensions. For choice (1) \(\bar{v}=\) number of waves per \(cm.\) \(=\left[L^{-1}\right]\) \(R_{H}=\dfrac{1}{\left[\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right] \lambda}\) (where \({n_1},\,\,{n_2}\) are serial numbers of Bohr orbits) Both \(\bar{v}\) and \(R_{H}\) have same dimensional formula \(\operatorname{viz}:\left[L^{-1}\right]\) For choice (2): Stress \(=\dfrac{F}{A}=\dfrac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2}\) Coefficient of elasticity is elastic modulus such as Youngs modulus \((Y)\) \(Y=\dfrac{\text { stress }}{\text { strain }}\) \(=\dfrac{M L^{-1} T^{-2}}{\text { strain }}=M L^{-1} T^{-2}\) \(\left(\because \operatorname{strain}=\dfrac{\Delta l}{l}=\dfrac{[L]}{\left[L^{-1}\right]}=L^{0}\right)\) For choice (3): We know \(B=\mu_{0}(H+M)\) As 'similar quantities' only can be added, the magnetisation \(M\) has same unit as \(H\) (\(i.e.,\) Tesla)
This hystersis graph indicates that 'coercivity' is a particular value of \(H\) for a given magnetic material.
JEE - 2022
PHXI02:UNITS AND MEASUREMENTS
367371
The dimension of the ratio of angular momentum to linear momentum is
1 \({{L}^{0}}\)
2 \({{L}^{1}}\)
3 \({{L}^{2}}\)
4 \(MLT\)
Explanation:
Angular momentum \({L=m v r}\) Linen momentum \({=p=m v}\) \({\dfrac{[L]}{[p]}=\dfrac{\left[{ML}^{2} {~T}^{-1}\right]}{\left[{MLT}^{-1}\right]}=\left[{M}^{0} {~L}^{1} {~T}^{0}\right]}\). So correct option is (2).
PHXI02:UNITS AND MEASUREMENTS
367372
The dimensional formula for impulse is
1 \(ML{T^{ - 1}}\)
2 \(M{L^{ - 1}}\)
3 \({M^{ - 1}}L{T^{ - 1}}\)
4 \(M{L^{ - 1}}{T^{ - 1}}\)
Explanation:
Impulse \(=\) force \(×\) time \( = [ML{T^{ - 2}} \times T] = [ML{T^{ - 1}}]\)
KCET - 2007
PHXI02:UNITS AND MEASUREMENTS
367373
The dimensions of emf in \(M K S\) system of unit is:
367374
If \(G\) be the gravitational constant and \(u\) be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{u G}\).
1 Gravitational potential.
2 Pressure gradient per unit mass.
3 Energy per unit mass.
4 Force per unit mass.
Explanation:
Dimensions of gravitational constant and energy density are given as \([G]=\left[M^{-1} L^{3} T^{-2}\right]\) \([u]=\dfrac{[E]}{[V]}=\dfrac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}=\left[M L^{-1} T^{-2}\right]\) Therefore, dimensions of \(\sqrt{u G}\), \(\begin{aligned}& {[\sqrt{u G}]=\left[\sqrt{\left[M L^{-1} T^{-2}\right] \times\left[M^{-1} L^{3} T^{-2}\right]}\right]} \\& \left.=\left[\sqrt{L^{2} T^{-4}}\right]\right]=\left[L T^{-2}\right]\end{aligned}\) The dimension of \(\sqrt{u G}\) is same as the dimension of Force per unit mass. So, option (4) is correct.
