367315
If \(P\) represents radiation pressure, \(c\) represents speed of light and \(Q\) represents radiation striking unit area per second, then non-zero integers \(x\), \(y\) and \(z\) such that \(P^{x} Q^{y} c^{z}\) is dimensionless, are
1 \(x=1, y=1, z=-1\)
2 \(x=1, y=-1, z=1\)
3 \(x=-1, y=1, z=1\)
4 \(x=1, y=1, z=1\)
Explanation:
Dimensions of \(P^{x} Q^{y} c^{z}=\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}\). As it is dimensionless, so \(\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}=\left[M^{0} L^{0} T^{0}\right]\) or \(\left[M^{x+y} L^{-x+z} T^{-2 x-3 y-z}\right]=\left[M^{0} L^{0} T^{0}\right]\) Comparing powers of \(M, L\) and \(T\), we get \(x+y=0,-x+z=0,-2 x-3 y-z=0\) Solving \(x=1, y=-1, z=1\).
PHXI02:UNITS AND MEASUREMENTS
367316
Assertion : Velocity cannot be added to speed. Reason : Both velocity and speed have same dimensions.
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:
Velocity is a vector quantity that includes both magnitude (speed) and direction. Whereas speed has only magnitude. Both cannot be added. Both velocity and speed are measured in units like meters per second \((m{\rm{/}}s)\), indicating that they have the same dimensions. While both the assertion and reason are true, the reason doesn't directly explain why velocity cannot be added to speed. So correct option is (2)
PHXI02:UNITS AND MEASUREMENTS
367317
The quantities \(A\) and \(B\) are related by the relation \(A / B=m\), where \(m\) is the linear density and \(A\) is force, the dimensions of \(B\) will be
1 same as that of pressure
2 same as that of work
3 same as that of momentum
4 same as that of latent heat
Explanation:
\(\dfrac{A}{B}=m, B=\dfrac{A}{m}=\dfrac{\text { force }}{\text { linear density }}=\dfrac{M L T^{-2}}{M L^{-1}}\) \(\therefore B=\left[M^{0} L^{2} T^{-2}\right]\) Latent heat \(=\dfrac{\text { Heat energy }}{\text { mass }}\) \(=\dfrac{M L^{2} T^{-2}}{M}=\left[M^{0} L^{2} T^{-2}\right]\) Thus, \(B\) has same dimensions as that of latent heat.
PHXI02:UNITS AND MEASUREMENTS
367318
A force \(F\) is given by \(F=a t+b t^{2}\), where \(t\) is time. What are the dimensions of \(a\) and \(b\) ?
1 \(\left[M L T^{-1}\right]\) and \(\left[M L T^{0}\right]\)
2 \(\left[M L T^{-3}\right]\) and \(\left[M L^{2} T^{-4}\right]\)
3 \(\left[M L T^{-4}\right]\) and \([M L T]\)
4 \(\left[M L T^{-3}\right]\) and \(\left[M L T^{-4}\right]\)
Explanation:
Given, \(F=a t+b t^{2}\) Only similar quantities can be equated, hence dimensions of the terms on both sides of the equation must be same.\(\left[M L T^{-2}\right]=a[T] \Rightarrow a=\left[M L T^{-3}\right]\) Similarly, \(\left[M L T^{-2}\right]=b\left[T^{2}\right]\) \(b=\left[M L T^{-4}\right]\)
367315
If \(P\) represents radiation pressure, \(c\) represents speed of light and \(Q\) represents radiation striking unit area per second, then non-zero integers \(x\), \(y\) and \(z\) such that \(P^{x} Q^{y} c^{z}\) is dimensionless, are
1 \(x=1, y=1, z=-1\)
2 \(x=1, y=-1, z=1\)
3 \(x=-1, y=1, z=1\)
4 \(x=1, y=1, z=1\)
Explanation:
Dimensions of \(P^{x} Q^{y} c^{z}=\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}\). As it is dimensionless, so \(\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}=\left[M^{0} L^{0} T^{0}\right]\) or \(\left[M^{x+y} L^{-x+z} T^{-2 x-3 y-z}\right]=\left[M^{0} L^{0} T^{0}\right]\) Comparing powers of \(M, L\) and \(T\), we get \(x+y=0,-x+z=0,-2 x-3 y-z=0\) Solving \(x=1, y=-1, z=1\).
PHXI02:UNITS AND MEASUREMENTS
367316
Assertion : Velocity cannot be added to speed. Reason : Both velocity and speed have same dimensions.
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:
Velocity is a vector quantity that includes both magnitude (speed) and direction. Whereas speed has only magnitude. Both cannot be added. Both velocity and speed are measured in units like meters per second \((m{\rm{/}}s)\), indicating that they have the same dimensions. While both the assertion and reason are true, the reason doesn't directly explain why velocity cannot be added to speed. So correct option is (2)
PHXI02:UNITS AND MEASUREMENTS
367317
The quantities \(A\) and \(B\) are related by the relation \(A / B=m\), where \(m\) is the linear density and \(A\) is force, the dimensions of \(B\) will be
1 same as that of pressure
2 same as that of work
3 same as that of momentum
4 same as that of latent heat
Explanation:
\(\dfrac{A}{B}=m, B=\dfrac{A}{m}=\dfrac{\text { force }}{\text { linear density }}=\dfrac{M L T^{-2}}{M L^{-1}}\) \(\therefore B=\left[M^{0} L^{2} T^{-2}\right]\) Latent heat \(=\dfrac{\text { Heat energy }}{\text { mass }}\) \(=\dfrac{M L^{2} T^{-2}}{M}=\left[M^{0} L^{2} T^{-2}\right]\) Thus, \(B\) has same dimensions as that of latent heat.
PHXI02:UNITS AND MEASUREMENTS
367318
A force \(F\) is given by \(F=a t+b t^{2}\), where \(t\) is time. What are the dimensions of \(a\) and \(b\) ?
1 \(\left[M L T^{-1}\right]\) and \(\left[M L T^{0}\right]\)
2 \(\left[M L T^{-3}\right]\) and \(\left[M L^{2} T^{-4}\right]\)
3 \(\left[M L T^{-4}\right]\) and \([M L T]\)
4 \(\left[M L T^{-3}\right]\) and \(\left[M L T^{-4}\right]\)
Explanation:
Given, \(F=a t+b t^{2}\) Only similar quantities can be equated, hence dimensions of the terms on both sides of the equation must be same.\(\left[M L T^{-2}\right]=a[T] \Rightarrow a=\left[M L T^{-3}\right]\) Similarly, \(\left[M L T^{-2}\right]=b\left[T^{2}\right]\) \(b=\left[M L T^{-4}\right]\)
367315
If \(P\) represents radiation pressure, \(c\) represents speed of light and \(Q\) represents radiation striking unit area per second, then non-zero integers \(x\), \(y\) and \(z\) such that \(P^{x} Q^{y} c^{z}\) is dimensionless, are
1 \(x=1, y=1, z=-1\)
2 \(x=1, y=-1, z=1\)
3 \(x=-1, y=1, z=1\)
4 \(x=1, y=1, z=1\)
Explanation:
Dimensions of \(P^{x} Q^{y} c^{z}=\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}\). As it is dimensionless, so \(\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}=\left[M^{0} L^{0} T^{0}\right]\) or \(\left[M^{x+y} L^{-x+z} T^{-2 x-3 y-z}\right]=\left[M^{0} L^{0} T^{0}\right]\) Comparing powers of \(M, L\) and \(T\), we get \(x+y=0,-x+z=0,-2 x-3 y-z=0\) Solving \(x=1, y=-1, z=1\).
PHXI02:UNITS AND MEASUREMENTS
367316
Assertion : Velocity cannot be added to speed. Reason : Both velocity and speed have same dimensions.
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:
Velocity is a vector quantity that includes both magnitude (speed) and direction. Whereas speed has only magnitude. Both cannot be added. Both velocity and speed are measured in units like meters per second \((m{\rm{/}}s)\), indicating that they have the same dimensions. While both the assertion and reason are true, the reason doesn't directly explain why velocity cannot be added to speed. So correct option is (2)
PHXI02:UNITS AND MEASUREMENTS
367317
The quantities \(A\) and \(B\) are related by the relation \(A / B=m\), where \(m\) is the linear density and \(A\) is force, the dimensions of \(B\) will be
1 same as that of pressure
2 same as that of work
3 same as that of momentum
4 same as that of latent heat
Explanation:
\(\dfrac{A}{B}=m, B=\dfrac{A}{m}=\dfrac{\text { force }}{\text { linear density }}=\dfrac{M L T^{-2}}{M L^{-1}}\) \(\therefore B=\left[M^{0} L^{2} T^{-2}\right]\) Latent heat \(=\dfrac{\text { Heat energy }}{\text { mass }}\) \(=\dfrac{M L^{2} T^{-2}}{M}=\left[M^{0} L^{2} T^{-2}\right]\) Thus, \(B\) has same dimensions as that of latent heat.
PHXI02:UNITS AND MEASUREMENTS
367318
A force \(F\) is given by \(F=a t+b t^{2}\), where \(t\) is time. What are the dimensions of \(a\) and \(b\) ?
1 \(\left[M L T^{-1}\right]\) and \(\left[M L T^{0}\right]\)
2 \(\left[M L T^{-3}\right]\) and \(\left[M L^{2} T^{-4}\right]\)
3 \(\left[M L T^{-4}\right]\) and \([M L T]\)
4 \(\left[M L T^{-3}\right]\) and \(\left[M L T^{-4}\right]\)
Explanation:
Given, \(F=a t+b t^{2}\) Only similar quantities can be equated, hence dimensions of the terms on both sides of the equation must be same.\(\left[M L T^{-2}\right]=a[T] \Rightarrow a=\left[M L T^{-3}\right]\) Similarly, \(\left[M L T^{-2}\right]=b\left[T^{2}\right]\) \(b=\left[M L T^{-4}\right]\)
367315
If \(P\) represents radiation pressure, \(c\) represents speed of light and \(Q\) represents radiation striking unit area per second, then non-zero integers \(x\), \(y\) and \(z\) such that \(P^{x} Q^{y} c^{z}\) is dimensionless, are
1 \(x=1, y=1, z=-1\)
2 \(x=1, y=-1, z=1\)
3 \(x=-1, y=1, z=1\)
4 \(x=1, y=1, z=1\)
Explanation:
Dimensions of \(P^{x} Q^{y} c^{z}=\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}\). As it is dimensionless, so \(\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}=\left[M^{0} L^{0} T^{0}\right]\) or \(\left[M^{x+y} L^{-x+z} T^{-2 x-3 y-z}\right]=\left[M^{0} L^{0} T^{0}\right]\) Comparing powers of \(M, L\) and \(T\), we get \(x+y=0,-x+z=0,-2 x-3 y-z=0\) Solving \(x=1, y=-1, z=1\).
PHXI02:UNITS AND MEASUREMENTS
367316
Assertion : Velocity cannot be added to speed. Reason : Both velocity and speed have same dimensions.
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:
Velocity is a vector quantity that includes both magnitude (speed) and direction. Whereas speed has only magnitude. Both cannot be added. Both velocity and speed are measured in units like meters per second \((m{\rm{/}}s)\), indicating that they have the same dimensions. While both the assertion and reason are true, the reason doesn't directly explain why velocity cannot be added to speed. So correct option is (2)
PHXI02:UNITS AND MEASUREMENTS
367317
The quantities \(A\) and \(B\) are related by the relation \(A / B=m\), where \(m\) is the linear density and \(A\) is force, the dimensions of \(B\) will be
1 same as that of pressure
2 same as that of work
3 same as that of momentum
4 same as that of latent heat
Explanation:
\(\dfrac{A}{B}=m, B=\dfrac{A}{m}=\dfrac{\text { force }}{\text { linear density }}=\dfrac{M L T^{-2}}{M L^{-1}}\) \(\therefore B=\left[M^{0} L^{2} T^{-2}\right]\) Latent heat \(=\dfrac{\text { Heat energy }}{\text { mass }}\) \(=\dfrac{M L^{2} T^{-2}}{M}=\left[M^{0} L^{2} T^{-2}\right]\) Thus, \(B\) has same dimensions as that of latent heat.
PHXI02:UNITS AND MEASUREMENTS
367318
A force \(F\) is given by \(F=a t+b t^{2}\), where \(t\) is time. What are the dimensions of \(a\) and \(b\) ?
1 \(\left[M L T^{-1}\right]\) and \(\left[M L T^{0}\right]\)
2 \(\left[M L T^{-3}\right]\) and \(\left[M L^{2} T^{-4}\right]\)
3 \(\left[M L T^{-4}\right]\) and \([M L T]\)
4 \(\left[M L T^{-3}\right]\) and \(\left[M L T^{-4}\right]\)
Explanation:
Given, \(F=a t+b t^{2}\) Only similar quantities can be equated, hence dimensions of the terms on both sides of the equation must be same.\(\left[M L T^{-2}\right]=a[T] \Rightarrow a=\left[M L T^{-3}\right]\) Similarly, \(\left[M L T^{-2}\right]=b\left[T^{2}\right]\) \(b=\left[M L T^{-4}\right]\)
