D Dimension of pressure \([\mathrm{P}]=\frac{[\mathrm{F}]}{[\mathrm{A}]}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{L}^{2}\right]}\) \({[\mathrm{P}]=\left[\mathrm{M}^{+1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]}\) Dimension of Distance \((\mathrm{x})=[\mathrm{L}]\) Dimension of Time \((t)=\left[\mathrm{T}^{1}\right]\) \(\mathrm{P}=\frac{\mathrm{a}-\mathrm{t}^{2}}{\mathrm{bx}}=\frac{\mathrm{a}}{\mathrm{bx}}-\frac{\mathrm{t}^{2}}{\mathrm{bx}}\) Dimension of \(\mathrm{P}=\) Dimension of \(\frac{\mathrm{a}}{\mathrm{bx}}\) \({[\mathrm{P}]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}\) \({\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\frac{\mathrm{a}}{\mathrm{b}}\right] \frac{1}{[\mathrm{~L}]}}\) The dimension of \(\left[\frac{\mathrm{a}}{\mathrm{b}}\right]=\left[\mathrm{MT}^{-2}\right]\)
JCECE-2011
Units and Measurements
139492
\(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\) stand for dimension of
1 work
2 torque
3 linear momentum
4 coefficient of viscosity
Explanation:
D (i) Dimension of work \(\mathrm{W} =\text { f.d }\) \(\mathrm{W} =\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (ii) Dimension of torque \(\mathrm{T} =\mathrm{f} \times \mathrm{r}\) \(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(\mathrm{T} =\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (iii) Linear momentum \(\mathrm{P} =\mathrm{m} . \mathrm{v} .\) \(=[\mathrm{M}]\left[\mathrm{LT}^{-1}\right]\) \(=\left[\mathrm{MLT}{ }^{-1}\right]\) (iv) Dimension of coefficient of viscosity \(\mathrm{F} =6 \pi \eta \mathrm{rv}\) \(\eta =\frac{\mathrm{F}}{6 \pi \mathrm{r} \cdot \mathrm{v}}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{L}]\left[\mathrm{LT}^{-1}\right]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)
UPSEE - 2013
Units and Measurements
139493
If \(p\) represents radiation pressure, \(c\) represents speed of light and \(S\) represents radiation energy striking unit area per sec. The non-zero integers \(x, y, z\) such that \(p^{x} \quad S^{y} \quad c^{z}\) is dimensionless are
1 \(\mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1\)
2 \(x=-1, y=1, z=1\)
3 \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\)
4 \(x=1, y=1, z=-1\)
Explanation:
C Given, \(\mathrm{P}=\text { radiation pressure }\) \(\mathrm{C}=\text { speed of light }\) \(\mathrm{S}=\text { radiation energy }\) \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are non zero intezers. \(\left[\mathrm{P}^{\mathrm{x}} \mathrm{S}^{\mathrm{y}} \mathrm{C}^{\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) The dimension of \(\mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) The dimension of \(\mathrm{S}=\left[\mathrm{MT}^{-3}\right]\) The dimension of \(\mathrm{C}=\left[\mathrm{LT}^{-1}\right]\) Putting the dimension in equation (i) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{MT}^{-3}\right]^{\mathrm{y}} \cdot\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\left[\mathrm{M}^{\mathrm{x}+\mathrm{y}} \cdot \mathrm{L}^{-\mathrm{x}+\mathrm{z}} \cdot \mathrm{T}^{-2 \mathrm{x}-3 \mathrm{y}-\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\mathrm{x}+\mathrm{y}=0\) \(z-x=0\) \(-2 x-3 y-z=0\) From equation (iii) \(\mathrm{x}=\mathrm{z}\) From equation (ii) \(\therefore \quad \mathrm{z}=-\mathrm{y}\) By solving, we get, \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\) Hence, option (c) is correct.
AIPMT 1992
Units and Measurements
139494
The dimensional formula for permeability of free space, \(\mu_{0}\) is
A The dimensional formula for permeability of free space, \(\mu_{0}=\frac{2 \pi \times \text { force } \times \text { distance }}{\text { current } \times \text { current } \times \text { length }}\) Dimension of force \(=\left[\mathrm{MLT}^{-2}\right]\) Dimension of current \(=[\mathrm{A}]\) Dimension of length and distance \(=[\mathrm{L}]\) \(\therefore \mu_{0}=\frac{\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]}{[\mathrm{A}][\mathrm{A}][\mathrm{L}]}\) \(\mu_{0}=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\)
D Dimension of pressure \([\mathrm{P}]=\frac{[\mathrm{F}]}{[\mathrm{A}]}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{L}^{2}\right]}\) \({[\mathrm{P}]=\left[\mathrm{M}^{+1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]}\) Dimension of Distance \((\mathrm{x})=[\mathrm{L}]\) Dimension of Time \((t)=\left[\mathrm{T}^{1}\right]\) \(\mathrm{P}=\frac{\mathrm{a}-\mathrm{t}^{2}}{\mathrm{bx}}=\frac{\mathrm{a}}{\mathrm{bx}}-\frac{\mathrm{t}^{2}}{\mathrm{bx}}\) Dimension of \(\mathrm{P}=\) Dimension of \(\frac{\mathrm{a}}{\mathrm{bx}}\) \({[\mathrm{P}]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}\) \({\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\frac{\mathrm{a}}{\mathrm{b}}\right] \frac{1}{[\mathrm{~L}]}}\) The dimension of \(\left[\frac{\mathrm{a}}{\mathrm{b}}\right]=\left[\mathrm{MT}^{-2}\right]\)
JCECE-2011
Units and Measurements
139492
\(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\) stand for dimension of
1 work
2 torque
3 linear momentum
4 coefficient of viscosity
Explanation:
D (i) Dimension of work \(\mathrm{W} =\text { f.d }\) \(\mathrm{W} =\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (ii) Dimension of torque \(\mathrm{T} =\mathrm{f} \times \mathrm{r}\) \(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(\mathrm{T} =\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (iii) Linear momentum \(\mathrm{P} =\mathrm{m} . \mathrm{v} .\) \(=[\mathrm{M}]\left[\mathrm{LT}^{-1}\right]\) \(=\left[\mathrm{MLT}{ }^{-1}\right]\) (iv) Dimension of coefficient of viscosity \(\mathrm{F} =6 \pi \eta \mathrm{rv}\) \(\eta =\frac{\mathrm{F}}{6 \pi \mathrm{r} \cdot \mathrm{v}}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{L}]\left[\mathrm{LT}^{-1}\right]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)
UPSEE - 2013
Units and Measurements
139493
If \(p\) represents radiation pressure, \(c\) represents speed of light and \(S\) represents radiation energy striking unit area per sec. The non-zero integers \(x, y, z\) such that \(p^{x} \quad S^{y} \quad c^{z}\) is dimensionless are
1 \(\mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1\)
2 \(x=-1, y=1, z=1\)
3 \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\)
4 \(x=1, y=1, z=-1\)
Explanation:
C Given, \(\mathrm{P}=\text { radiation pressure }\) \(\mathrm{C}=\text { speed of light }\) \(\mathrm{S}=\text { radiation energy }\) \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are non zero intezers. \(\left[\mathrm{P}^{\mathrm{x}} \mathrm{S}^{\mathrm{y}} \mathrm{C}^{\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) The dimension of \(\mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) The dimension of \(\mathrm{S}=\left[\mathrm{MT}^{-3}\right]\) The dimension of \(\mathrm{C}=\left[\mathrm{LT}^{-1}\right]\) Putting the dimension in equation (i) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{MT}^{-3}\right]^{\mathrm{y}} \cdot\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\left[\mathrm{M}^{\mathrm{x}+\mathrm{y}} \cdot \mathrm{L}^{-\mathrm{x}+\mathrm{z}} \cdot \mathrm{T}^{-2 \mathrm{x}-3 \mathrm{y}-\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\mathrm{x}+\mathrm{y}=0\) \(z-x=0\) \(-2 x-3 y-z=0\) From equation (iii) \(\mathrm{x}=\mathrm{z}\) From equation (ii) \(\therefore \quad \mathrm{z}=-\mathrm{y}\) By solving, we get, \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\) Hence, option (c) is correct.
AIPMT 1992
Units and Measurements
139494
The dimensional formula for permeability of free space, \(\mu_{0}\) is
A The dimensional formula for permeability of free space, \(\mu_{0}=\frac{2 \pi \times \text { force } \times \text { distance }}{\text { current } \times \text { current } \times \text { length }}\) Dimension of force \(=\left[\mathrm{MLT}^{-2}\right]\) Dimension of current \(=[\mathrm{A}]\) Dimension of length and distance \(=[\mathrm{L}]\) \(\therefore \mu_{0}=\frac{\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]}{[\mathrm{A}][\mathrm{A}][\mathrm{L}]}\) \(\mu_{0}=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\)
D Dimension of pressure \([\mathrm{P}]=\frac{[\mathrm{F}]}{[\mathrm{A}]}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{L}^{2}\right]}\) \({[\mathrm{P}]=\left[\mathrm{M}^{+1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]}\) Dimension of Distance \((\mathrm{x})=[\mathrm{L}]\) Dimension of Time \((t)=\left[\mathrm{T}^{1}\right]\) \(\mathrm{P}=\frac{\mathrm{a}-\mathrm{t}^{2}}{\mathrm{bx}}=\frac{\mathrm{a}}{\mathrm{bx}}-\frac{\mathrm{t}^{2}}{\mathrm{bx}}\) Dimension of \(\mathrm{P}=\) Dimension of \(\frac{\mathrm{a}}{\mathrm{bx}}\) \({[\mathrm{P}]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}\) \({\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\frac{\mathrm{a}}{\mathrm{b}}\right] \frac{1}{[\mathrm{~L}]}}\) The dimension of \(\left[\frac{\mathrm{a}}{\mathrm{b}}\right]=\left[\mathrm{MT}^{-2}\right]\)
JCECE-2011
Units and Measurements
139492
\(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\) stand for dimension of
1 work
2 torque
3 linear momentum
4 coefficient of viscosity
Explanation:
D (i) Dimension of work \(\mathrm{W} =\text { f.d }\) \(\mathrm{W} =\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (ii) Dimension of torque \(\mathrm{T} =\mathrm{f} \times \mathrm{r}\) \(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(\mathrm{T} =\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (iii) Linear momentum \(\mathrm{P} =\mathrm{m} . \mathrm{v} .\) \(=[\mathrm{M}]\left[\mathrm{LT}^{-1}\right]\) \(=\left[\mathrm{MLT}{ }^{-1}\right]\) (iv) Dimension of coefficient of viscosity \(\mathrm{F} =6 \pi \eta \mathrm{rv}\) \(\eta =\frac{\mathrm{F}}{6 \pi \mathrm{r} \cdot \mathrm{v}}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{L}]\left[\mathrm{LT}^{-1}\right]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)
UPSEE - 2013
Units and Measurements
139493
If \(p\) represents radiation pressure, \(c\) represents speed of light and \(S\) represents radiation energy striking unit area per sec. The non-zero integers \(x, y, z\) such that \(p^{x} \quad S^{y} \quad c^{z}\) is dimensionless are
1 \(\mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1\)
2 \(x=-1, y=1, z=1\)
3 \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\)
4 \(x=1, y=1, z=-1\)
Explanation:
C Given, \(\mathrm{P}=\text { radiation pressure }\) \(\mathrm{C}=\text { speed of light }\) \(\mathrm{S}=\text { radiation energy }\) \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are non zero intezers. \(\left[\mathrm{P}^{\mathrm{x}} \mathrm{S}^{\mathrm{y}} \mathrm{C}^{\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) The dimension of \(\mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) The dimension of \(\mathrm{S}=\left[\mathrm{MT}^{-3}\right]\) The dimension of \(\mathrm{C}=\left[\mathrm{LT}^{-1}\right]\) Putting the dimension in equation (i) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{MT}^{-3}\right]^{\mathrm{y}} \cdot\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\left[\mathrm{M}^{\mathrm{x}+\mathrm{y}} \cdot \mathrm{L}^{-\mathrm{x}+\mathrm{z}} \cdot \mathrm{T}^{-2 \mathrm{x}-3 \mathrm{y}-\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\mathrm{x}+\mathrm{y}=0\) \(z-x=0\) \(-2 x-3 y-z=0\) From equation (iii) \(\mathrm{x}=\mathrm{z}\) From equation (ii) \(\therefore \quad \mathrm{z}=-\mathrm{y}\) By solving, we get, \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\) Hence, option (c) is correct.
AIPMT 1992
Units and Measurements
139494
The dimensional formula for permeability of free space, \(\mu_{0}\) is
A The dimensional formula for permeability of free space, \(\mu_{0}=\frac{2 \pi \times \text { force } \times \text { distance }}{\text { current } \times \text { current } \times \text { length }}\) Dimension of force \(=\left[\mathrm{MLT}^{-2}\right]\) Dimension of current \(=[\mathrm{A}]\) Dimension of length and distance \(=[\mathrm{L}]\) \(\therefore \mu_{0}=\frac{\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]}{[\mathrm{A}][\mathrm{A}][\mathrm{L}]}\) \(\mu_{0}=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\)
D Dimension of pressure \([\mathrm{P}]=\frac{[\mathrm{F}]}{[\mathrm{A}]}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{L}^{2}\right]}\) \({[\mathrm{P}]=\left[\mathrm{M}^{+1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]}\) Dimension of Distance \((\mathrm{x})=[\mathrm{L}]\) Dimension of Time \((t)=\left[\mathrm{T}^{1}\right]\) \(\mathrm{P}=\frac{\mathrm{a}-\mathrm{t}^{2}}{\mathrm{bx}}=\frac{\mathrm{a}}{\mathrm{bx}}-\frac{\mathrm{t}^{2}}{\mathrm{bx}}\) Dimension of \(\mathrm{P}=\) Dimension of \(\frac{\mathrm{a}}{\mathrm{bx}}\) \({[\mathrm{P}]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}\) \({\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\frac{\mathrm{a}}{\mathrm{b}}\right] \frac{1}{[\mathrm{~L}]}}\) The dimension of \(\left[\frac{\mathrm{a}}{\mathrm{b}}\right]=\left[\mathrm{MT}^{-2}\right]\)
JCECE-2011
Units and Measurements
139492
\(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\) stand for dimension of
1 work
2 torque
3 linear momentum
4 coefficient of viscosity
Explanation:
D (i) Dimension of work \(\mathrm{W} =\text { f.d }\) \(\mathrm{W} =\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (ii) Dimension of torque \(\mathrm{T} =\mathrm{f} \times \mathrm{r}\) \(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(\mathrm{T} =\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (iii) Linear momentum \(\mathrm{P} =\mathrm{m} . \mathrm{v} .\) \(=[\mathrm{M}]\left[\mathrm{LT}^{-1}\right]\) \(=\left[\mathrm{MLT}{ }^{-1}\right]\) (iv) Dimension of coefficient of viscosity \(\mathrm{F} =6 \pi \eta \mathrm{rv}\) \(\eta =\frac{\mathrm{F}}{6 \pi \mathrm{r} \cdot \mathrm{v}}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{L}]\left[\mathrm{LT}^{-1}\right]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)
UPSEE - 2013
Units and Measurements
139493
If \(p\) represents radiation pressure, \(c\) represents speed of light and \(S\) represents radiation energy striking unit area per sec. The non-zero integers \(x, y, z\) such that \(p^{x} \quad S^{y} \quad c^{z}\) is dimensionless are
1 \(\mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1\)
2 \(x=-1, y=1, z=1\)
3 \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\)
4 \(x=1, y=1, z=-1\)
Explanation:
C Given, \(\mathrm{P}=\text { radiation pressure }\) \(\mathrm{C}=\text { speed of light }\) \(\mathrm{S}=\text { radiation energy }\) \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are non zero intezers. \(\left[\mathrm{P}^{\mathrm{x}} \mathrm{S}^{\mathrm{y}} \mathrm{C}^{\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) The dimension of \(\mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) The dimension of \(\mathrm{S}=\left[\mathrm{MT}^{-3}\right]\) The dimension of \(\mathrm{C}=\left[\mathrm{LT}^{-1}\right]\) Putting the dimension in equation (i) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{MT}^{-3}\right]^{\mathrm{y}} \cdot\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\left[\mathrm{M}^{\mathrm{x}+\mathrm{y}} \cdot \mathrm{L}^{-\mathrm{x}+\mathrm{z}} \cdot \mathrm{T}^{-2 \mathrm{x}-3 \mathrm{y}-\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\mathrm{x}+\mathrm{y}=0\) \(z-x=0\) \(-2 x-3 y-z=0\) From equation (iii) \(\mathrm{x}=\mathrm{z}\) From equation (ii) \(\therefore \quad \mathrm{z}=-\mathrm{y}\) By solving, we get, \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\) Hence, option (c) is correct.
AIPMT 1992
Units and Measurements
139494
The dimensional formula for permeability of free space, \(\mu_{0}\) is
A The dimensional formula for permeability of free space, \(\mu_{0}=\frac{2 \pi \times \text { force } \times \text { distance }}{\text { current } \times \text { current } \times \text { length }}\) Dimension of force \(=\left[\mathrm{MLT}^{-2}\right]\) Dimension of current \(=[\mathrm{A}]\) Dimension of length and distance \(=[\mathrm{L}]\) \(\therefore \mu_{0}=\frac{\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]}{[\mathrm{A}][\mathrm{A}][\mathrm{L}]}\) \(\mu_{0}=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\)
D Dimension of pressure \([\mathrm{P}]=\frac{[\mathrm{F}]}{[\mathrm{A}]}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{L}^{2}\right]}\) \({[\mathrm{P}]=\left[\mathrm{M}^{+1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]}\) Dimension of Distance \((\mathrm{x})=[\mathrm{L}]\) Dimension of Time \((t)=\left[\mathrm{T}^{1}\right]\) \(\mathrm{P}=\frac{\mathrm{a}-\mathrm{t}^{2}}{\mathrm{bx}}=\frac{\mathrm{a}}{\mathrm{bx}}-\frac{\mathrm{t}^{2}}{\mathrm{bx}}\) Dimension of \(\mathrm{P}=\) Dimension of \(\frac{\mathrm{a}}{\mathrm{bx}}\) \({[\mathrm{P}]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}\) \({\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\frac{\mathrm{a}}{\mathrm{b}}\right] \frac{1}{[\mathrm{~L}]}}\) The dimension of \(\left[\frac{\mathrm{a}}{\mathrm{b}}\right]=\left[\mathrm{MT}^{-2}\right]\)
JCECE-2011
Units and Measurements
139492
\(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\) stand for dimension of
1 work
2 torque
3 linear momentum
4 coefficient of viscosity
Explanation:
D (i) Dimension of work \(\mathrm{W} =\text { f.d }\) \(\mathrm{W} =\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (ii) Dimension of torque \(\mathrm{T} =\mathrm{f} \times \mathrm{r}\) \(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]\) \(\mathrm{T} =\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (iii) Linear momentum \(\mathrm{P} =\mathrm{m} . \mathrm{v} .\) \(=[\mathrm{M}]\left[\mathrm{LT}^{-1}\right]\) \(=\left[\mathrm{MLT}{ }^{-1}\right]\) (iv) Dimension of coefficient of viscosity \(\mathrm{F} =6 \pi \eta \mathrm{rv}\) \(\eta =\frac{\mathrm{F}}{6 \pi \mathrm{r} \cdot \mathrm{v}}\) \(=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{L}]\left[\mathrm{LT}^{-1}\right]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)
UPSEE - 2013
Units and Measurements
139493
If \(p\) represents radiation pressure, \(c\) represents speed of light and \(S\) represents radiation energy striking unit area per sec. The non-zero integers \(x, y, z\) such that \(p^{x} \quad S^{y} \quad c^{z}\) is dimensionless are
1 \(\mathrm{x}=1, \mathrm{y}=1, \mathrm{z}=1\)
2 \(x=-1, y=1, z=1\)
3 \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\)
4 \(x=1, y=1, z=-1\)
Explanation:
C Given, \(\mathrm{P}=\text { radiation pressure }\) \(\mathrm{C}=\text { speed of light }\) \(\mathrm{S}=\text { radiation energy }\) \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are non zero intezers. \(\left[\mathrm{P}^{\mathrm{x}} \mathrm{S}^{\mathrm{y}} \mathrm{C}^{\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) The dimension of \(\mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) The dimension of \(\mathrm{S}=\left[\mathrm{MT}^{-3}\right]\) The dimension of \(\mathrm{C}=\left[\mathrm{LT}^{-1}\right]\) Putting the dimension in equation (i) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{MT}^{-3}\right]^{\mathrm{y}} \cdot\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\left[\mathrm{M}^{\mathrm{x}+\mathrm{y}} \cdot \mathrm{L}^{-\mathrm{x}+\mathrm{z}} \cdot \mathrm{T}^{-2 \mathrm{x}-3 \mathrm{y}-\mathrm{z}}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) \(\mathrm{x}+\mathrm{y}=0\) \(z-x=0\) \(-2 x-3 y-z=0\) From equation (iii) \(\mathrm{x}=\mathrm{z}\) From equation (ii) \(\therefore \quad \mathrm{z}=-\mathrm{y}\) By solving, we get, \(\mathrm{x}=1, \mathrm{y}=-1, \mathrm{z}=1\) Hence, option (c) is correct.
AIPMT 1992
Units and Measurements
139494
The dimensional formula for permeability of free space, \(\mu_{0}\) is
A The dimensional formula for permeability of free space, \(\mu_{0}=\frac{2 \pi \times \text { force } \times \text { distance }}{\text { current } \times \text { current } \times \text { length }}\) Dimension of force \(=\left[\mathrm{MLT}^{-2}\right]\) Dimension of current \(=[\mathrm{A}]\) Dimension of length and distance \(=[\mathrm{L}]\) \(\therefore \mu_{0}=\frac{\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]}{[\mathrm{A}][\mathrm{A}][\mathrm{L}]}\) \(\mu_{0}=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\)