139458
If \(x=a t+b t^{2}\), where \(x\) is the distance travelled by the body in kilometer while \(t\) is the time in second, then the unit of \(b\) is
1 \(\mathrm{km} / \mathrm{s}\)
2 \(\mathrm{km}-\mathrm{s}\)
3 \(\mathrm{km} / \mathrm{s}^{2}\)
4 \(\mathrm{km}-\mathrm{s}^{2}\)
Explanation:
C Given that, \(\mathrm{x}=\mathrm{at}+\mathrm{bt}^{2}\) (where \(\mathrm{x}\) is distance) From principle of Homogeneity \([\mathrm{x}]=[\mathrm{at}]=\left[\mathrm{bt}^{2}\right]\) \([\mathrm{L}]=\mathrm{b}\left[\mathrm{T}^{2}\right] \Rightarrow \mathrm{b}=\left[\mathrm{LT}^{-2}\right]\) So, Unit of \(b=\mathrm{km} / \mathrm{s}^{2}\)
AIPMT 1989
Units and Measurements
139459
The unit of permittivity of free space, \(\varepsilon_{0}\) is
1 coulomb/newton-metre
2 newton-metre \({ }^{2} /\) coulomb \(^{2}\)
3 coulomb \({ }^{2} /\) newton-metre \({ }^{2}\)
4 coulomb \(^{2} /\) (newton-metre) \()^{2}\)
Explanation:
C Coulomb law state that. \(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{\mathrm{r}^{2}}\) \(\varepsilon_{0}=\frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{4 \pi \mathrm{F} \cdot \mathrm{r}^{2}}\) When, unit of \(\mathrm{F}=\mathrm{N}\) \(\text { Unit of } \mathrm{r}=\mathrm{m}\) \(\text { Unit of } \mathrm{q}=\text { Coulomb }(\mathrm{c})\) \(\varepsilon_{0}=\frac{\text { Coulomb } \times \text { Coulomb }}{\text { newton }-(\text { metre })^{2}}\) \(\varepsilon_{0}=\frac{\text { C.C }}{\mathrm{N}^{2} \mathrm{~m}^{2}}=\text { Coulomb }^{2} / \text { newton }- \text { metre }^{2}\)
AIPMT 2004
Units and Measurements
139460
The equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=\) constant. The units of a is
1 Dyne \(\times \mathrm{cm}^{5}\)
2 Dyne \(\times \mathrm{cm}^{4}\)
3 Dyne \(/ \mathrm{cm}^{3}\)
4 Dyne \(/ \mathrm{cm}^{2}\)
Explanation:
B Given that, \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{constant}\) In the term \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)\), the units of \(\mathrm{p}\) and \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\) are the same \(\therefore \mathrm{a}=\mathrm{PV}^{2}\) [here \(\mathrm{v}\) is the volume] \(\therefore\) Unit of \(\mathrm{a}=\) unit of \(\mathrm{P} \times\) unit of \(\mathrm{v}^{2}\) \(\therefore\) Unit of \(\mathrm{a}=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times\left(\mathrm{cm}^{3}\right)^{2}\) \(=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times \mathrm{cm}^{6}\) \(=\text { dyne } \times \mathrm{cm}^{4}\) Note:- Here we use CGS system. In SI system unit of a is \(\mathrm{N}-\mathrm{m}^{4}\)
UP CPMT-2014
Units and Measurements
139461
SI unit of intensity of wave is
1 \(\mathrm{J} \mathrm{m}^{-2} \mathrm{~s}^{-1}\)
2 \(\mathrm{J} \mathrm{m}^{-1} \mathrm{~s}^{-2}\)
3 \(\mathrm{W} \mathrm{m}^{-2}\)
4 \(\mathrm{J} \mathrm{m}^{-2}\)
Explanation:
A ,c) : The intensity of waves is defined as the power delivered per Unit area. center of pattern Intensity of wave \(=\frac{\text { energy }}{\text { Area } \times \text { Time }}\) \(=\frac{\mathrm{J}}{\mathrm{m}^{2} \times \mathrm{S}}=\mathrm{Wm}^{-2}\) \(\therefore\) The S.I Unit of intensity of wave is \(\mathrm{Wm}^{-2}\).
UP CPMT-2012
Units and Measurements
139467
If mass is measure in units of \(\alpha \mathrm{kg}\), length in \(\beta\) \(m\) and time in \(\gamma\) sec then calorie would be
1 \(4.2 \alpha \beta^{2} \gamma^{-2}\)
2 \(4.2 \alpha^{-1} \beta^{2} \gamma^{2}\)
3 \(4.2 \alpha^{-1} \beta^{-2} \gamma^{2}\)
4 \(4.2 \alpha^{-2} \beta^{-1} \gamma^{-2}\)
Explanation:
A Given data, Unit of mass \(=\alpha=[\mathrm{M}]\) Unit of length \(=\beta=[\mathrm{L}]\) Unit of time \(=\gamma=[\mathrm{T}]\) Unit of work \(=\) Joule We know that \(\text { Let }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha^{\mathrm{x}} \beta^{\mathrm{y}} \gamma^{\mathrm{z}}\) Both side comparing \(\mathrm{X}=1, \mathrm{y}=2, \mathrm{z}=-2\) Dimension of work \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha \beta^{2} \gamma^{-2}\) As 1 cal \(=4.2\) Joule \(\Rightarrow \quad\) Calorie \(=4.2\) Joule \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\) \(\Rightarrow \quad\) Unit of Calorie \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\)
139458
If \(x=a t+b t^{2}\), where \(x\) is the distance travelled by the body in kilometer while \(t\) is the time in second, then the unit of \(b\) is
1 \(\mathrm{km} / \mathrm{s}\)
2 \(\mathrm{km}-\mathrm{s}\)
3 \(\mathrm{km} / \mathrm{s}^{2}\)
4 \(\mathrm{km}-\mathrm{s}^{2}\)
Explanation:
C Given that, \(\mathrm{x}=\mathrm{at}+\mathrm{bt}^{2}\) (where \(\mathrm{x}\) is distance) From principle of Homogeneity \([\mathrm{x}]=[\mathrm{at}]=\left[\mathrm{bt}^{2}\right]\) \([\mathrm{L}]=\mathrm{b}\left[\mathrm{T}^{2}\right] \Rightarrow \mathrm{b}=\left[\mathrm{LT}^{-2}\right]\) So, Unit of \(b=\mathrm{km} / \mathrm{s}^{2}\)
AIPMT 1989
Units and Measurements
139459
The unit of permittivity of free space, \(\varepsilon_{0}\) is
1 coulomb/newton-metre
2 newton-metre \({ }^{2} /\) coulomb \(^{2}\)
3 coulomb \({ }^{2} /\) newton-metre \({ }^{2}\)
4 coulomb \(^{2} /\) (newton-metre) \()^{2}\)
Explanation:
C Coulomb law state that. \(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{\mathrm{r}^{2}}\) \(\varepsilon_{0}=\frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{4 \pi \mathrm{F} \cdot \mathrm{r}^{2}}\) When, unit of \(\mathrm{F}=\mathrm{N}\) \(\text { Unit of } \mathrm{r}=\mathrm{m}\) \(\text { Unit of } \mathrm{q}=\text { Coulomb }(\mathrm{c})\) \(\varepsilon_{0}=\frac{\text { Coulomb } \times \text { Coulomb }}{\text { newton }-(\text { metre })^{2}}\) \(\varepsilon_{0}=\frac{\text { C.C }}{\mathrm{N}^{2} \mathrm{~m}^{2}}=\text { Coulomb }^{2} / \text { newton }- \text { metre }^{2}\)
AIPMT 2004
Units and Measurements
139460
The equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=\) constant. The units of a is
1 Dyne \(\times \mathrm{cm}^{5}\)
2 Dyne \(\times \mathrm{cm}^{4}\)
3 Dyne \(/ \mathrm{cm}^{3}\)
4 Dyne \(/ \mathrm{cm}^{2}\)
Explanation:
B Given that, \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{constant}\) In the term \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)\), the units of \(\mathrm{p}\) and \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\) are the same \(\therefore \mathrm{a}=\mathrm{PV}^{2}\) [here \(\mathrm{v}\) is the volume] \(\therefore\) Unit of \(\mathrm{a}=\) unit of \(\mathrm{P} \times\) unit of \(\mathrm{v}^{2}\) \(\therefore\) Unit of \(\mathrm{a}=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times\left(\mathrm{cm}^{3}\right)^{2}\) \(=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times \mathrm{cm}^{6}\) \(=\text { dyne } \times \mathrm{cm}^{4}\) Note:- Here we use CGS system. In SI system unit of a is \(\mathrm{N}-\mathrm{m}^{4}\)
UP CPMT-2014
Units and Measurements
139461
SI unit of intensity of wave is
1 \(\mathrm{J} \mathrm{m}^{-2} \mathrm{~s}^{-1}\)
2 \(\mathrm{J} \mathrm{m}^{-1} \mathrm{~s}^{-2}\)
3 \(\mathrm{W} \mathrm{m}^{-2}\)
4 \(\mathrm{J} \mathrm{m}^{-2}\)
Explanation:
A ,c) : The intensity of waves is defined as the power delivered per Unit area. center of pattern Intensity of wave \(=\frac{\text { energy }}{\text { Area } \times \text { Time }}\) \(=\frac{\mathrm{J}}{\mathrm{m}^{2} \times \mathrm{S}}=\mathrm{Wm}^{-2}\) \(\therefore\) The S.I Unit of intensity of wave is \(\mathrm{Wm}^{-2}\).
UP CPMT-2012
Units and Measurements
139467
If mass is measure in units of \(\alpha \mathrm{kg}\), length in \(\beta\) \(m\) and time in \(\gamma\) sec then calorie would be
1 \(4.2 \alpha \beta^{2} \gamma^{-2}\)
2 \(4.2 \alpha^{-1} \beta^{2} \gamma^{2}\)
3 \(4.2 \alpha^{-1} \beta^{-2} \gamma^{2}\)
4 \(4.2 \alpha^{-2} \beta^{-1} \gamma^{-2}\)
Explanation:
A Given data, Unit of mass \(=\alpha=[\mathrm{M}]\) Unit of length \(=\beta=[\mathrm{L}]\) Unit of time \(=\gamma=[\mathrm{T}]\) Unit of work \(=\) Joule We know that \(\text { Let }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha^{\mathrm{x}} \beta^{\mathrm{y}} \gamma^{\mathrm{z}}\) Both side comparing \(\mathrm{X}=1, \mathrm{y}=2, \mathrm{z}=-2\) Dimension of work \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha \beta^{2} \gamma^{-2}\) As 1 cal \(=4.2\) Joule \(\Rightarrow \quad\) Calorie \(=4.2\) Joule \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\) \(\Rightarrow \quad\) Unit of Calorie \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\)
139458
If \(x=a t+b t^{2}\), where \(x\) is the distance travelled by the body in kilometer while \(t\) is the time in second, then the unit of \(b\) is
1 \(\mathrm{km} / \mathrm{s}\)
2 \(\mathrm{km}-\mathrm{s}\)
3 \(\mathrm{km} / \mathrm{s}^{2}\)
4 \(\mathrm{km}-\mathrm{s}^{2}\)
Explanation:
C Given that, \(\mathrm{x}=\mathrm{at}+\mathrm{bt}^{2}\) (where \(\mathrm{x}\) is distance) From principle of Homogeneity \([\mathrm{x}]=[\mathrm{at}]=\left[\mathrm{bt}^{2}\right]\) \([\mathrm{L}]=\mathrm{b}\left[\mathrm{T}^{2}\right] \Rightarrow \mathrm{b}=\left[\mathrm{LT}^{-2}\right]\) So, Unit of \(b=\mathrm{km} / \mathrm{s}^{2}\)
AIPMT 1989
Units and Measurements
139459
The unit of permittivity of free space, \(\varepsilon_{0}\) is
1 coulomb/newton-metre
2 newton-metre \({ }^{2} /\) coulomb \(^{2}\)
3 coulomb \({ }^{2} /\) newton-metre \({ }^{2}\)
4 coulomb \(^{2} /\) (newton-metre) \()^{2}\)
Explanation:
C Coulomb law state that. \(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{\mathrm{r}^{2}}\) \(\varepsilon_{0}=\frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{4 \pi \mathrm{F} \cdot \mathrm{r}^{2}}\) When, unit of \(\mathrm{F}=\mathrm{N}\) \(\text { Unit of } \mathrm{r}=\mathrm{m}\) \(\text { Unit of } \mathrm{q}=\text { Coulomb }(\mathrm{c})\) \(\varepsilon_{0}=\frac{\text { Coulomb } \times \text { Coulomb }}{\text { newton }-(\text { metre })^{2}}\) \(\varepsilon_{0}=\frac{\text { C.C }}{\mathrm{N}^{2} \mathrm{~m}^{2}}=\text { Coulomb }^{2} / \text { newton }- \text { metre }^{2}\)
AIPMT 2004
Units and Measurements
139460
The equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=\) constant. The units of a is
1 Dyne \(\times \mathrm{cm}^{5}\)
2 Dyne \(\times \mathrm{cm}^{4}\)
3 Dyne \(/ \mathrm{cm}^{3}\)
4 Dyne \(/ \mathrm{cm}^{2}\)
Explanation:
B Given that, \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{constant}\) In the term \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)\), the units of \(\mathrm{p}\) and \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\) are the same \(\therefore \mathrm{a}=\mathrm{PV}^{2}\) [here \(\mathrm{v}\) is the volume] \(\therefore\) Unit of \(\mathrm{a}=\) unit of \(\mathrm{P} \times\) unit of \(\mathrm{v}^{2}\) \(\therefore\) Unit of \(\mathrm{a}=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times\left(\mathrm{cm}^{3}\right)^{2}\) \(=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times \mathrm{cm}^{6}\) \(=\text { dyne } \times \mathrm{cm}^{4}\) Note:- Here we use CGS system. In SI system unit of a is \(\mathrm{N}-\mathrm{m}^{4}\)
UP CPMT-2014
Units and Measurements
139461
SI unit of intensity of wave is
1 \(\mathrm{J} \mathrm{m}^{-2} \mathrm{~s}^{-1}\)
2 \(\mathrm{J} \mathrm{m}^{-1} \mathrm{~s}^{-2}\)
3 \(\mathrm{W} \mathrm{m}^{-2}\)
4 \(\mathrm{J} \mathrm{m}^{-2}\)
Explanation:
A ,c) : The intensity of waves is defined as the power delivered per Unit area. center of pattern Intensity of wave \(=\frac{\text { energy }}{\text { Area } \times \text { Time }}\) \(=\frac{\mathrm{J}}{\mathrm{m}^{2} \times \mathrm{S}}=\mathrm{Wm}^{-2}\) \(\therefore\) The S.I Unit of intensity of wave is \(\mathrm{Wm}^{-2}\).
UP CPMT-2012
Units and Measurements
139467
If mass is measure in units of \(\alpha \mathrm{kg}\), length in \(\beta\) \(m\) and time in \(\gamma\) sec then calorie would be
1 \(4.2 \alpha \beta^{2} \gamma^{-2}\)
2 \(4.2 \alpha^{-1} \beta^{2} \gamma^{2}\)
3 \(4.2 \alpha^{-1} \beta^{-2} \gamma^{2}\)
4 \(4.2 \alpha^{-2} \beta^{-1} \gamma^{-2}\)
Explanation:
A Given data, Unit of mass \(=\alpha=[\mathrm{M}]\) Unit of length \(=\beta=[\mathrm{L}]\) Unit of time \(=\gamma=[\mathrm{T}]\) Unit of work \(=\) Joule We know that \(\text { Let }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha^{\mathrm{x}} \beta^{\mathrm{y}} \gamma^{\mathrm{z}}\) Both side comparing \(\mathrm{X}=1, \mathrm{y}=2, \mathrm{z}=-2\) Dimension of work \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha \beta^{2} \gamma^{-2}\) As 1 cal \(=4.2\) Joule \(\Rightarrow \quad\) Calorie \(=4.2\) Joule \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\) \(\Rightarrow \quad\) Unit of Calorie \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\)
139458
If \(x=a t+b t^{2}\), where \(x\) is the distance travelled by the body in kilometer while \(t\) is the time in second, then the unit of \(b\) is
1 \(\mathrm{km} / \mathrm{s}\)
2 \(\mathrm{km}-\mathrm{s}\)
3 \(\mathrm{km} / \mathrm{s}^{2}\)
4 \(\mathrm{km}-\mathrm{s}^{2}\)
Explanation:
C Given that, \(\mathrm{x}=\mathrm{at}+\mathrm{bt}^{2}\) (where \(\mathrm{x}\) is distance) From principle of Homogeneity \([\mathrm{x}]=[\mathrm{at}]=\left[\mathrm{bt}^{2}\right]\) \([\mathrm{L}]=\mathrm{b}\left[\mathrm{T}^{2}\right] \Rightarrow \mathrm{b}=\left[\mathrm{LT}^{-2}\right]\) So, Unit of \(b=\mathrm{km} / \mathrm{s}^{2}\)
AIPMT 1989
Units and Measurements
139459
The unit of permittivity of free space, \(\varepsilon_{0}\) is
1 coulomb/newton-metre
2 newton-metre \({ }^{2} /\) coulomb \(^{2}\)
3 coulomb \({ }^{2} /\) newton-metre \({ }^{2}\)
4 coulomb \(^{2} /\) (newton-metre) \()^{2}\)
Explanation:
C Coulomb law state that. \(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{\mathrm{r}^{2}}\) \(\varepsilon_{0}=\frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{4 \pi \mathrm{F} \cdot \mathrm{r}^{2}}\) When, unit of \(\mathrm{F}=\mathrm{N}\) \(\text { Unit of } \mathrm{r}=\mathrm{m}\) \(\text { Unit of } \mathrm{q}=\text { Coulomb }(\mathrm{c})\) \(\varepsilon_{0}=\frac{\text { Coulomb } \times \text { Coulomb }}{\text { newton }-(\text { metre })^{2}}\) \(\varepsilon_{0}=\frac{\text { C.C }}{\mathrm{N}^{2} \mathrm{~m}^{2}}=\text { Coulomb }^{2} / \text { newton }- \text { metre }^{2}\)
AIPMT 2004
Units and Measurements
139460
The equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=\) constant. The units of a is
1 Dyne \(\times \mathrm{cm}^{5}\)
2 Dyne \(\times \mathrm{cm}^{4}\)
3 Dyne \(/ \mathrm{cm}^{3}\)
4 Dyne \(/ \mathrm{cm}^{2}\)
Explanation:
B Given that, \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{constant}\) In the term \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)\), the units of \(\mathrm{p}\) and \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\) are the same \(\therefore \mathrm{a}=\mathrm{PV}^{2}\) [here \(\mathrm{v}\) is the volume] \(\therefore\) Unit of \(\mathrm{a}=\) unit of \(\mathrm{P} \times\) unit of \(\mathrm{v}^{2}\) \(\therefore\) Unit of \(\mathrm{a}=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times\left(\mathrm{cm}^{3}\right)^{2}\) \(=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times \mathrm{cm}^{6}\) \(=\text { dyne } \times \mathrm{cm}^{4}\) Note:- Here we use CGS system. In SI system unit of a is \(\mathrm{N}-\mathrm{m}^{4}\)
UP CPMT-2014
Units and Measurements
139461
SI unit of intensity of wave is
1 \(\mathrm{J} \mathrm{m}^{-2} \mathrm{~s}^{-1}\)
2 \(\mathrm{J} \mathrm{m}^{-1} \mathrm{~s}^{-2}\)
3 \(\mathrm{W} \mathrm{m}^{-2}\)
4 \(\mathrm{J} \mathrm{m}^{-2}\)
Explanation:
A ,c) : The intensity of waves is defined as the power delivered per Unit area. center of pattern Intensity of wave \(=\frac{\text { energy }}{\text { Area } \times \text { Time }}\) \(=\frac{\mathrm{J}}{\mathrm{m}^{2} \times \mathrm{S}}=\mathrm{Wm}^{-2}\) \(\therefore\) The S.I Unit of intensity of wave is \(\mathrm{Wm}^{-2}\).
UP CPMT-2012
Units and Measurements
139467
If mass is measure in units of \(\alpha \mathrm{kg}\), length in \(\beta\) \(m\) and time in \(\gamma\) sec then calorie would be
1 \(4.2 \alpha \beta^{2} \gamma^{-2}\)
2 \(4.2 \alpha^{-1} \beta^{2} \gamma^{2}\)
3 \(4.2 \alpha^{-1} \beta^{-2} \gamma^{2}\)
4 \(4.2 \alpha^{-2} \beta^{-1} \gamma^{-2}\)
Explanation:
A Given data, Unit of mass \(=\alpha=[\mathrm{M}]\) Unit of length \(=\beta=[\mathrm{L}]\) Unit of time \(=\gamma=[\mathrm{T}]\) Unit of work \(=\) Joule We know that \(\text { Let }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha^{\mathrm{x}} \beta^{\mathrm{y}} \gamma^{\mathrm{z}}\) Both side comparing \(\mathrm{X}=1, \mathrm{y}=2, \mathrm{z}=-2\) Dimension of work \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha \beta^{2} \gamma^{-2}\) As 1 cal \(=4.2\) Joule \(\Rightarrow \quad\) Calorie \(=4.2\) Joule \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\) \(\Rightarrow \quad\) Unit of Calorie \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\)
139458
If \(x=a t+b t^{2}\), where \(x\) is the distance travelled by the body in kilometer while \(t\) is the time in second, then the unit of \(b\) is
1 \(\mathrm{km} / \mathrm{s}\)
2 \(\mathrm{km}-\mathrm{s}\)
3 \(\mathrm{km} / \mathrm{s}^{2}\)
4 \(\mathrm{km}-\mathrm{s}^{2}\)
Explanation:
C Given that, \(\mathrm{x}=\mathrm{at}+\mathrm{bt}^{2}\) (where \(\mathrm{x}\) is distance) From principle of Homogeneity \([\mathrm{x}]=[\mathrm{at}]=\left[\mathrm{bt}^{2}\right]\) \([\mathrm{L}]=\mathrm{b}\left[\mathrm{T}^{2}\right] \Rightarrow \mathrm{b}=\left[\mathrm{LT}^{-2}\right]\) So, Unit of \(b=\mathrm{km} / \mathrm{s}^{2}\)
AIPMT 1989
Units and Measurements
139459
The unit of permittivity of free space, \(\varepsilon_{0}\) is
1 coulomb/newton-metre
2 newton-metre \({ }^{2} /\) coulomb \(^{2}\)
3 coulomb \({ }^{2} /\) newton-metre \({ }^{2}\)
4 coulomb \(^{2} /\) (newton-metre) \()^{2}\)
Explanation:
C Coulomb law state that. \(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{\mathrm{r}^{2}}\) \(\varepsilon_{0}=\frac{\mathrm{q}_{1} \cdot \mathrm{q}_{2}}{4 \pi \mathrm{F} \cdot \mathrm{r}^{2}}\) When, unit of \(\mathrm{F}=\mathrm{N}\) \(\text { Unit of } \mathrm{r}=\mathrm{m}\) \(\text { Unit of } \mathrm{q}=\text { Coulomb }(\mathrm{c})\) \(\varepsilon_{0}=\frac{\text { Coulomb } \times \text { Coulomb }}{\text { newton }-(\text { metre })^{2}}\) \(\varepsilon_{0}=\frac{\text { C.C }}{\mathrm{N}^{2} \mathrm{~m}^{2}}=\text { Coulomb }^{2} / \text { newton }- \text { metre }^{2}\)
AIPMT 2004
Units and Measurements
139460
The equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=\) constant. The units of a is
1 Dyne \(\times \mathrm{cm}^{5}\)
2 Dyne \(\times \mathrm{cm}^{4}\)
3 Dyne \(/ \mathrm{cm}^{3}\)
4 Dyne \(/ \mathrm{cm}^{2}\)
Explanation:
B Given that, \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{constant}\) In the term \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)\), the units of \(\mathrm{p}\) and \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\) are the same \(\therefore \mathrm{a}=\mathrm{PV}^{2}\) [here \(\mathrm{v}\) is the volume] \(\therefore\) Unit of \(\mathrm{a}=\) unit of \(\mathrm{P} \times\) unit of \(\mathrm{v}^{2}\) \(\therefore\) Unit of \(\mathrm{a}=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times\left(\mathrm{cm}^{3}\right)^{2}\) \(=\frac{\text { dyne }}{\mathrm{cm}^{2}} \times \mathrm{cm}^{6}\) \(=\text { dyne } \times \mathrm{cm}^{4}\) Note:- Here we use CGS system. In SI system unit of a is \(\mathrm{N}-\mathrm{m}^{4}\)
UP CPMT-2014
Units and Measurements
139461
SI unit of intensity of wave is
1 \(\mathrm{J} \mathrm{m}^{-2} \mathrm{~s}^{-1}\)
2 \(\mathrm{J} \mathrm{m}^{-1} \mathrm{~s}^{-2}\)
3 \(\mathrm{W} \mathrm{m}^{-2}\)
4 \(\mathrm{J} \mathrm{m}^{-2}\)
Explanation:
A ,c) : The intensity of waves is defined as the power delivered per Unit area. center of pattern Intensity of wave \(=\frac{\text { energy }}{\text { Area } \times \text { Time }}\) \(=\frac{\mathrm{J}}{\mathrm{m}^{2} \times \mathrm{S}}=\mathrm{Wm}^{-2}\) \(\therefore\) The S.I Unit of intensity of wave is \(\mathrm{Wm}^{-2}\).
UP CPMT-2012
Units and Measurements
139467
If mass is measure in units of \(\alpha \mathrm{kg}\), length in \(\beta\) \(m\) and time in \(\gamma\) sec then calorie would be
1 \(4.2 \alpha \beta^{2} \gamma^{-2}\)
2 \(4.2 \alpha^{-1} \beta^{2} \gamma^{2}\)
3 \(4.2 \alpha^{-1} \beta^{-2} \gamma^{2}\)
4 \(4.2 \alpha^{-2} \beta^{-1} \gamma^{-2}\)
Explanation:
A Given data, Unit of mass \(=\alpha=[\mathrm{M}]\) Unit of length \(=\beta=[\mathrm{L}]\) Unit of time \(=\gamma=[\mathrm{T}]\) Unit of work \(=\) Joule We know that \(\text { Let }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha^{\mathrm{x}} \beta^{\mathrm{y}} \gamma^{\mathrm{z}}\) Both side comparing \(\mathrm{X}=1, \mathrm{y}=2, \mathrm{z}=-2\) Dimension of work \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\alpha \beta^{2} \gamma^{-2}\) As 1 cal \(=4.2\) Joule \(\Rightarrow \quad\) Calorie \(=4.2\) Joule \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\) \(\Rightarrow \quad\) Unit of Calorie \(=4.2\left(\alpha \beta^{2} \gamma^{-2}\right)\)
