367353
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\) of length \(L\) of time \(T\) and of current \(I\) would be
367354
The magnetic force on a point charge is \(F=q(v \times B)\). Here, \(q=\) electric charge, \(v=\) velocity of point charge \(B=\) magnetic field. The dimensions of \(B\) are
1 \(\left[M L T^{-1} A\right]\)
2 \(\left[M^{2} L T^{-2} A^{-1}\right]\)
3 \(\left[M T^{-2} A^{-1}\right]\)
4 None of these
Explanation:
Magnetic force, \(F=q(v \times B)\) or \(F=q v B \sin \theta\) \(\therefore[B]=\left[\dfrac{F}{q v}\right]=\dfrac{\left[M L T^{-2}\right]}{[A T]\left[L T^{-1}\right]}\) \(=\left[M T^{-2} A^{-1}\right]\)
PHXI02:UNITS AND MEASUREMENTS
367355
Which of the following pairs DOES NOT have the same dimensions?
1 frequency and angular frequency
2 angular velocity and velocity gradient
3 velocity gradient and angular frequency
4 angular frequency and potential energy gradient
Explanation:
Angular frequency \({[\omega]=\left[{T}^{-1}\right]}\) Potential energy gradient \({\left[\dfrac{d V}{d x}\right]=\left[M L T^{-2}\right]}\). So correct option is (4)
PHXI02:UNITS AND MEASUREMENTS
367356
If \(E, M, L\) and \(G\) denote energy, mass, angular momentum and gravitational constant respectively, then the quantity \(\left(E^{2} L^{2} / M^{5} G^{2}\right)\) has the dimensions of
1 angle
2 length
3 mass
4 None of these
Explanation:
Dimensions of \(E=\left[M L^{2} T^{-2}\right]\) Dimensions of \(M=[M]\) Dimensions of \(L=\left[M L^{2} T^{-1}\right]\) Dimensions of \(G=\left[M^{-1} L^{3} T^{-2}\right]\) \(\therefore {\rm{Dimensions}}\,\,{\rm{of}}\) $\begin{aligned}{\left[\dfrac{E^{2} L^{2}}{M^{5} G^{2}}\right] } & =\dfrac{\left[M L^{2} T^{-2}\right]^{2}\left[M L^{2} T^{-1}\right]^{2}}{[M]^{5}\left[M^{-1} L^{3} T^{-2}\right]^{2}} \\& =\left[M L^{2} T^{-2}\right]\end{aligned}$
PHXI02:UNITS AND MEASUREMENTS
367357
Assertion : In \(y=A \sin (\omega t-k x),(\omega t-k x)\) is dimensionless. Reason : Because dimension of \(\omega=\left[M^{0} L^{0} T\right]\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
\(\omega\) (angular velocity) has the dimension of \(\left[T^{-1}\right]\) not \([T]\). \(\omega t\) and hence \((\omega t-k x)\) is dimensionless. So correct option is (3)
367353
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\) of length \(L\) of time \(T\) and of current \(I\) would be
367354
The magnetic force on a point charge is \(F=q(v \times B)\). Here, \(q=\) electric charge, \(v=\) velocity of point charge \(B=\) magnetic field. The dimensions of \(B\) are
1 \(\left[M L T^{-1} A\right]\)
2 \(\left[M^{2} L T^{-2} A^{-1}\right]\)
3 \(\left[M T^{-2} A^{-1}\right]\)
4 None of these
Explanation:
Magnetic force, \(F=q(v \times B)\) or \(F=q v B \sin \theta\) \(\therefore[B]=\left[\dfrac{F}{q v}\right]=\dfrac{\left[M L T^{-2}\right]}{[A T]\left[L T^{-1}\right]}\) \(=\left[M T^{-2} A^{-1}\right]\)
PHXI02:UNITS AND MEASUREMENTS
367355
Which of the following pairs DOES NOT have the same dimensions?
1 frequency and angular frequency
2 angular velocity and velocity gradient
3 velocity gradient and angular frequency
4 angular frequency and potential energy gradient
Explanation:
Angular frequency \({[\omega]=\left[{T}^{-1}\right]}\) Potential energy gradient \({\left[\dfrac{d V}{d x}\right]=\left[M L T^{-2}\right]}\). So correct option is (4)
PHXI02:UNITS AND MEASUREMENTS
367356
If \(E, M, L\) and \(G\) denote energy, mass, angular momentum and gravitational constant respectively, then the quantity \(\left(E^{2} L^{2} / M^{5} G^{2}\right)\) has the dimensions of
1 angle
2 length
3 mass
4 None of these
Explanation:
Dimensions of \(E=\left[M L^{2} T^{-2}\right]\) Dimensions of \(M=[M]\) Dimensions of \(L=\left[M L^{2} T^{-1}\right]\) Dimensions of \(G=\left[M^{-1} L^{3} T^{-2}\right]\) \(\therefore {\rm{Dimensions}}\,\,{\rm{of}}\) $\begin{aligned}{\left[\dfrac{E^{2} L^{2}}{M^{5} G^{2}}\right] } & =\dfrac{\left[M L^{2} T^{-2}\right]^{2}\left[M L^{2} T^{-1}\right]^{2}}{[M]^{5}\left[M^{-1} L^{3} T^{-2}\right]^{2}} \\& =\left[M L^{2} T^{-2}\right]\end{aligned}$
PHXI02:UNITS AND MEASUREMENTS
367357
Assertion : In \(y=A \sin (\omega t-k x),(\omega t-k x)\) is dimensionless. Reason : Because dimension of \(\omega=\left[M^{0} L^{0} T\right]\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
\(\omega\) (angular velocity) has the dimension of \(\left[T^{-1}\right]\) not \([T]\). \(\omega t\) and hence \((\omega t-k x)\) is dimensionless. So correct option is (3)
367353
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\) of length \(L\) of time \(T\) and of current \(I\) would be
367354
The magnetic force on a point charge is \(F=q(v \times B)\). Here, \(q=\) electric charge, \(v=\) velocity of point charge \(B=\) magnetic field. The dimensions of \(B\) are
1 \(\left[M L T^{-1} A\right]\)
2 \(\left[M^{2} L T^{-2} A^{-1}\right]\)
3 \(\left[M T^{-2} A^{-1}\right]\)
4 None of these
Explanation:
Magnetic force, \(F=q(v \times B)\) or \(F=q v B \sin \theta\) \(\therefore[B]=\left[\dfrac{F}{q v}\right]=\dfrac{\left[M L T^{-2}\right]}{[A T]\left[L T^{-1}\right]}\) \(=\left[M T^{-2} A^{-1}\right]\)
PHXI02:UNITS AND MEASUREMENTS
367355
Which of the following pairs DOES NOT have the same dimensions?
1 frequency and angular frequency
2 angular velocity and velocity gradient
3 velocity gradient and angular frequency
4 angular frequency and potential energy gradient
Explanation:
Angular frequency \({[\omega]=\left[{T}^{-1}\right]}\) Potential energy gradient \({\left[\dfrac{d V}{d x}\right]=\left[M L T^{-2}\right]}\). So correct option is (4)
PHXI02:UNITS AND MEASUREMENTS
367356
If \(E, M, L\) and \(G\) denote energy, mass, angular momentum and gravitational constant respectively, then the quantity \(\left(E^{2} L^{2} / M^{5} G^{2}\right)\) has the dimensions of
1 angle
2 length
3 mass
4 None of these
Explanation:
Dimensions of \(E=\left[M L^{2} T^{-2}\right]\) Dimensions of \(M=[M]\) Dimensions of \(L=\left[M L^{2} T^{-1}\right]\) Dimensions of \(G=\left[M^{-1} L^{3} T^{-2}\right]\) \(\therefore {\rm{Dimensions}}\,\,{\rm{of}}\) $\begin{aligned}{\left[\dfrac{E^{2} L^{2}}{M^{5} G^{2}}\right] } & =\dfrac{\left[M L^{2} T^{-2}\right]^{2}\left[M L^{2} T^{-1}\right]^{2}}{[M]^{5}\left[M^{-1} L^{3} T^{-2}\right]^{2}} \\& =\left[M L^{2} T^{-2}\right]\end{aligned}$
PHXI02:UNITS AND MEASUREMENTS
367357
Assertion : In \(y=A \sin (\omega t-k x),(\omega t-k x)\) is dimensionless. Reason : Because dimension of \(\omega=\left[M^{0} L^{0} T\right]\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
\(\omega\) (angular velocity) has the dimension of \(\left[T^{-1}\right]\) not \([T]\). \(\omega t\) and hence \((\omega t-k x)\) is dimensionless. So correct option is (3)
367353
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\) of length \(L\) of time \(T\) and of current \(I\) would be
367354
The magnetic force on a point charge is \(F=q(v \times B)\). Here, \(q=\) electric charge, \(v=\) velocity of point charge \(B=\) magnetic field. The dimensions of \(B\) are
1 \(\left[M L T^{-1} A\right]\)
2 \(\left[M^{2} L T^{-2} A^{-1}\right]\)
3 \(\left[M T^{-2} A^{-1}\right]\)
4 None of these
Explanation:
Magnetic force, \(F=q(v \times B)\) or \(F=q v B \sin \theta\) \(\therefore[B]=\left[\dfrac{F}{q v}\right]=\dfrac{\left[M L T^{-2}\right]}{[A T]\left[L T^{-1}\right]}\) \(=\left[M T^{-2} A^{-1}\right]\)
PHXI02:UNITS AND MEASUREMENTS
367355
Which of the following pairs DOES NOT have the same dimensions?
1 frequency and angular frequency
2 angular velocity and velocity gradient
3 velocity gradient and angular frequency
4 angular frequency and potential energy gradient
Explanation:
Angular frequency \({[\omega]=\left[{T}^{-1}\right]}\) Potential energy gradient \({\left[\dfrac{d V}{d x}\right]=\left[M L T^{-2}\right]}\). So correct option is (4)
PHXI02:UNITS AND MEASUREMENTS
367356
If \(E, M, L\) and \(G\) denote energy, mass, angular momentum and gravitational constant respectively, then the quantity \(\left(E^{2} L^{2} / M^{5} G^{2}\right)\) has the dimensions of
1 angle
2 length
3 mass
4 None of these
Explanation:
Dimensions of \(E=\left[M L^{2} T^{-2}\right]\) Dimensions of \(M=[M]\) Dimensions of \(L=\left[M L^{2} T^{-1}\right]\) Dimensions of \(G=\left[M^{-1} L^{3} T^{-2}\right]\) \(\therefore {\rm{Dimensions}}\,\,{\rm{of}}\) $\begin{aligned}{\left[\dfrac{E^{2} L^{2}}{M^{5} G^{2}}\right] } & =\dfrac{\left[M L^{2} T^{-2}\right]^{2}\left[M L^{2} T^{-1}\right]^{2}}{[M]^{5}\left[M^{-1} L^{3} T^{-2}\right]^{2}} \\& =\left[M L^{2} T^{-2}\right]\end{aligned}$
PHXI02:UNITS AND MEASUREMENTS
367357
Assertion : In \(y=A \sin (\omega t-k x),(\omega t-k x)\) is dimensionless. Reason : Because dimension of \(\omega=\left[M^{0} L^{0} T\right]\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
\(\omega\) (angular velocity) has the dimension of \(\left[T^{-1}\right]\) not \([T]\). \(\omega t\) and hence \((\omega t-k x)\) is dimensionless. So correct option is (3)
367353
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\) of length \(L\) of time \(T\) and of current \(I\) would be
367354
The magnetic force on a point charge is \(F=q(v \times B)\). Here, \(q=\) electric charge, \(v=\) velocity of point charge \(B=\) magnetic field. The dimensions of \(B\) are
1 \(\left[M L T^{-1} A\right]\)
2 \(\left[M^{2} L T^{-2} A^{-1}\right]\)
3 \(\left[M T^{-2} A^{-1}\right]\)
4 None of these
Explanation:
Magnetic force, \(F=q(v \times B)\) or \(F=q v B \sin \theta\) \(\therefore[B]=\left[\dfrac{F}{q v}\right]=\dfrac{\left[M L T^{-2}\right]}{[A T]\left[L T^{-1}\right]}\) \(=\left[M T^{-2} A^{-1}\right]\)
PHXI02:UNITS AND MEASUREMENTS
367355
Which of the following pairs DOES NOT have the same dimensions?
1 frequency and angular frequency
2 angular velocity and velocity gradient
3 velocity gradient and angular frequency
4 angular frequency and potential energy gradient
Explanation:
Angular frequency \({[\omega]=\left[{T}^{-1}\right]}\) Potential energy gradient \({\left[\dfrac{d V}{d x}\right]=\left[M L T^{-2}\right]}\). So correct option is (4)
PHXI02:UNITS AND MEASUREMENTS
367356
If \(E, M, L\) and \(G\) denote energy, mass, angular momentum and gravitational constant respectively, then the quantity \(\left(E^{2} L^{2} / M^{5} G^{2}\right)\) has the dimensions of
1 angle
2 length
3 mass
4 None of these
Explanation:
Dimensions of \(E=\left[M L^{2} T^{-2}\right]\) Dimensions of \(M=[M]\) Dimensions of \(L=\left[M L^{2} T^{-1}\right]\) Dimensions of \(G=\left[M^{-1} L^{3} T^{-2}\right]\) \(\therefore {\rm{Dimensions}}\,\,{\rm{of}}\) $\begin{aligned}{\left[\dfrac{E^{2} L^{2}}{M^{5} G^{2}}\right] } & =\dfrac{\left[M L^{2} T^{-2}\right]^{2}\left[M L^{2} T^{-1}\right]^{2}}{[M]^{5}\left[M^{-1} L^{3} T^{-2}\right]^{2}} \\& =\left[M L^{2} T^{-2}\right]\end{aligned}$
PHXI02:UNITS AND MEASUREMENTS
367357
Assertion : In \(y=A \sin (\omega t-k x),(\omega t-k x)\) is dimensionless. Reason : Because dimension of \(\omega=\left[M^{0} L^{0} T\right]\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
\(\omega\) (angular velocity) has the dimension of \(\left[T^{-1}\right]\) not \([T]\). \(\omega t\) and hence \((\omega t-k x)\) is dimensionless. So correct option is (3)
