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The dimensional formula for angular momentum is [1988]
[Angular momentum ] = [Momentum of inertia] × [Angular velocity] = ML^{2} × T^{–1}
= ML^{2}T^{–1}
OR
In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics.
In SI base units: kg m^{2} s^{−1}
Dimension: M L^{2}T^{−1}
Derivations from other quantities: L = Iω = r × p
If C and R denote capacitance and resistance, the dimensional formula of CR is [1988]
CR = [M ^{1} L^{2} T 4 A ^{2} ][M^{1} L^{2} T ^{3} A^{ 2} ] = [T] = [M^{0} L^{0} T^{1}]
The dimensional formula of torque is [1989]
ζ = [Force × distance] = [MLT^{–2}] [L] = ML^{2}T^{–2}
Dimensional formula of self inductance is [1989]
Induced emf
where L is the self inductance and is the rate of change of current.
∴ Dimensional formula of L =
Of the following quantities, which one has dimension different from the remaining three? [1989]
For angular momentum, the dimensional formula is ML^{2}T^{–1}. For other three, it is ML^{1}T^{–2}.
If x = at + bt^{2}, where x is the distance travelled by the body in kilometers while t is the time in seconds, then the units of b is [1989]
According to Newton, the viscous force acting between liquid layers of area A and velocity gradient ΔV/ΔZ is given by where η is constant called coefficient of viscosity. The dimensional formula of η is [1990]
Substitute the dimension al formula of F, A, ΔV and ΔZ on both sides and find that for η.
The frequency of vibration f of a mass m suspended from a spring of spring constant k is given by a relation of the type f = c m^{x} k^{y}, where c is a dimensionless constant. The values of x and y are
f = c m^{x} k^{y}; Spring constant k = force/length. [M^{0}L^{0}T^{–1}] = [M^{x }(MT^{–2})^{y}] =[ M^{x} + y T^{–2y}]
⇒ x+y = 0, 2y = 1 or
Therefore, x = 
The dimensional formula of pressure is [1990]
[Pressure] = [Force] / [Area]
= ML^{–1}T^{–2}
The dimension al formula for permeability µ is given by [1991]
[n] = L^{–1}, [I] = A
A certain body weighs 22.42 gm and has a measured volume of 4.7 cc. The possible error in the measurement of mass and volume are 0.01 gm and 0.1 cc. Then maximum error in the den sity will be [1991]
D = M/V
=2 %
P represents radiation pressure, crepresents speed of light and S represents radiation energy striking unit area per sec. The non zero integers x, y, z such that P^{x} S^{y} c^{z} is dimensionless are [1992]
Try out the given alter natives.
When x = 1, y = –1, z = 1
= [M^{0}L^{0}T^{0}]
The time dependence of a physical quantity p is given by p = p_{0} exp (– αt^{2}), where α is a constant and t is the time. The constant α [1993]
In p = p_{0} exp (–αt^{2}), αt^{2} dimension less
urpentine oil is flowing through a tube of length l and radius r. The pressure difference between the two ends of the tube is P. The viscosity of oil is given by where v is the velocity of oil at a distance a from the axis of the tube. The dimensions of η are
In a vernier calliper N divisions of vernier scale coincides with (N – 1) divisions of main scale (in which length of one division is 1 mm). The least count of the instrument should be[1994]
Least count = 1MSD – 1 VSD
(∵ N VSD = (N – 1)MSD
In a particular system, the unit of length, mass and time are chosen to be 10 cm, 10 g and 0.1 s respectively. The unit of force in this system will be equivalen t to [1994]
[F] = MLT^{–2} = (10g) (10 cm) (0.1s)^{–2}
= (10^{–2} kg) (10^{–1}m) (10^{–1}s)^{–2} = 10^{–1}N.
Which of the following is a dimensional constant? [1995]
A quantity which has dimensions and a constant value is called dimensional constant. Therefore, gravitational constant (G) is a dimensional constant.
The percentage errors in the measurement of mass and speed are 2% and 3% respectively.The error in kinetic energy obtained by measuring mass and speed will be [1995]
Percentage error in mass
percentage error in speed
Kinetic energy
∴ Error in measurement of kinetic energy
By Binomial Function ,
Reqd. error
∴ %age error = 8%.
An equation is given as :
where P = Pressure, V = Volume & θ = Absolute temperature. If a and b are constants, then dimensions of a will be [1996]
According to the principle of homogeinity quantity with same dimension can be added or subtracted.
Hence, Dimension of P = Dimension of
⇒ Dimension of = Dimension of
a = [M L^{5} T^{–2}]
The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and length are 4% and 3% respectively, the maximum error in the measurement of den sity will be [1996]
Density =
% error in density = % error in Mass + 3 (% error in length]
= 4 + 3(3) = 13%
Which of the following will have the dimensions of time [1996]
The force F on a sphere of radius a moving in a medium with velocity v is given by F = 6πηav.The dimensions of η are [1997]
F= 6πη av
= ML^{–1}T^{–1}
The dimensional formula for magnetic flux is [1999]
Dimension of magnetic flux = Dimension of voltage × Dimension of time = [ML^{2}T^{–3}A^{–1}] [T] = [ML^{2}T^{–2}A^{–1}]
∵ Voltage =
Which one of the following groups have quantities that do not have the same dimensions? [2000]
Force has dimension [MLT^{–2}] while impulse has dimension [MLT^{–1}], both have different dimensions.
The dimensions of Planck’s constant are same as [2001]
We know that E = hv
Angular momentum = Iω
= [ML^{2}][T^{–1}] = [ML^{2}T^{–1}]
The unit of the StefanBoltzmann's constant is [2002]
E = σAT^{4}
E is energy dissipated per second.
The unit of permittivity of free space, ε_{o} is [2004]
⇒ unit of ε_{o} is (coulomb)^{2}/ newton metre^{2}
The dimensions of universal gravitational constant are [1992, 2004]
∴ dimension of G is
= M^{–1}L^{3}T^{–2}
The ratio of the dimension of Planck’s constant and that of the moment of inertia has the dimension of [2005]
The velocity v of a particle at time t is given by where a, b and c are constant.The dimensions of a, b and c are respectively [2006]
Dimension of a. t = dimension of velocity a . t = LT^{1} ⇒ a = LT^{2 }
Dimension of c = dimension of t (two physical quantity of same dimension can only be added)
So, dimension of c = T
Dimension of = Dimension of v
So, answer is LT^{–2}, L & T
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 [2007]
Dimensions of Resistance,
If the error in the measurement of radius of a sphere is 2%, then the error in the determination of volume of the sphere will be: [2008]
Error in the measurement of radius of a sphere = 2%
Volume of the sphere =
∴ Error in the volume =
= 3 × 2% = 6%
Which two of the following five physical parameters have the same dimensions?
(A) Energy density [2008]
(B) Refractive index
(C) Dielectric constant
(D) Young’s modulus
(E) Magnetic field
[Energy density] =
[Young’s Modulus] =
If the dimensions of a physical quantity are given by M^{a} L^{b} T^{c}, then the physical quantity will be:
The dimension of where ε_{0} is permittivity of free space and E is electric field, is:[2010]
represents energy density i.e., energy per unit volume.
The dimensions of are [2011]
, so, dimensions are [LTT^{–1}].
The density of material in CGS system of units is 4g/cm^{3}. In a system of units in which unit of length is 10 cm and unit of mass is 100 g, the value of density of material will be [2011M]
In CGS system,
The unit of mass is 100g and unit of length is 10 cm, so
density =
= 40 unit
The dimensions of (μ_{0}ε_{0})^{–1/2 }are : [2012M]
: speed of light
where ε_{0} = permittivity of free space
μ_{0} = permeability of free space
So dimension LT^{–1}
In an experiment four quantities a, b, c and d are measured with percentage error 1%, 2%, 3% and 4% respectively. Quantity P is calculated as follows error in P is [NEET 2013]
100%.
= 3 × 1% + 2 × 2% + 3% + 4% = 14%
The pair of quantities having same dimensions is [NEET Kar. 2013]
Work = Force × displacement Torque = Force × force arm
= mass × acceleration × length
= [M] × [LT^{–2}] × [L] = [M L^{2}T^{–2}]
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