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Which of the following sets cannot enter into the list of fundamental quantities in any system of units ?
We define length and time separately as it is not possible to define velocity without using these quantities. This means that one fundamental quantity depends on the other. So, these quantities cannot be listed as fundamental quantities in any system of units.
Parallactic second is the unit of distance because parallactic second is an abbreviation of parsec. Parsec = Parsec is the Unit for larger distances.It is the distance at which a star would make parallax of one Second of arc.
The newton second (also newtonsecond, symbol N s or N.s) is the derived SI unit of impulse. It is dimensionally equivalent to the momentum unit kilogram metre per second (kg.m/s). One newton second corresponds to a onenewton force applied for one second.
F = ma
[F] = kg m/s^{2}
If the unit of length is micrometer and the unit of time is microsecond, the unit of velocity will be :
Unit of velocity = m/s
1m = 10^{6} Micro meter
And 1 sec = 10^{6} Micro sec
Thus 1 m/s = 10^{6} 10^{6} micro meter /micro second
What is the physical quantity whose dimensions are M L^{2} T^{2} ?
Dimension of KE is ML^{2}T^{2}
That of pressure is ML^{2}T^{2}
That of momentum is ML/T
That of power is ML^{2}T^{3}
A dimensionless quantity can have unit. for example angle (radian). But oppisite is not true. A unitless quantity can never have dimensions.it is the unit that give dimensions.
If a and b are two physical quantities having different dimensions then which of the following can denote a new physical quantity
Quantities with different dimensions cant be added or subtracted. Also the dimension of the product of the quantities in power is always one. But when two quantities are multiplied the dimension of the product is the product of the dimensions of the initial quantities.
Two physical quantities whose dimensions are not same, cannot be :
The two physical quantities which do not have the same dimensions can’t be added or subtracted in the same expression because doing so would lead to equating two quantities of different dimensions, which is non possible. Ex [F] [P].
Correct Answer : d
Explanation : a) A topological sort of a directed acyclic graph (DAG) is any ordering m1, m2, …, mn of the nodes
of the graph, such that if mimj is an edge then mi appears before mj . Any topological sort of a
the dependency graph gives a valid evaluation order for the semantic rules.
b) The parse tree can be annotated with synthesized or inherited attributes. The parse tree can also be indicated with an arrow mark to indicate the manner in which the value gets propagated between the nodes of the parse tree. This graph is called as dependency graph as it indicates the dependency between nodes for deriving the values. This graph is an acyclic graph which doesn’t have a cycle. The presence of a cycle indicates that the graph is incorrect as the dependence of nodes for deriving values cannot be predicted. Edges in the dependence graph show the evaluation order for attribute values and thus the graph is a directed one.
We know that E = hv , where E is energy and v is frequency.
Thus we get h = E/v
And [h] = [E/v] = ML^{2}T^{2} / T^{1}
= ML^{2}T^{1}
[P] = MLT^{1}
[F] = MLT^{2}
[Angular momentum] = [P x r] = ML^{2}T^{1}
The vander Waals gas equation is
where P is the pressure, V is molar volume and T is the temperature of the given sample of gas. R is called mola gas constant, a and b are called vander Waals constants.
Which pair of following quantities has dimensions different from each other
Moment of inertia =1/2×mass×(Radius of gyration)^{2}=[M^{1}L^{2}T^{0}]
Moment of force = torque =N.m=[M^{1}L^{2}T^{−2}]
Both are different from each other.
If the error in measurement of radius of sphere is 1% , what will be the error in measurement of volume
Volume=4πR^{3}/3
ΔV/V×100=3∆R/R×100
=3×1/100×100
=3
∴ Percentage error in volume is 3%
The velocity 'v' (in cm/s) of a particle is given in terms of time 't' (in s) by the equation
v = at +
The dimensions of a, b and c are
As v = at + b / (t + c)
We get that v, at and b / (t+c) have the same dimensions as they are equated and added.
Similarly c and t have also same dimensions, [c] = T and [a] = L/T
And [b] = L
In case of measurement of ‘g’, if error in measurement of length of pendulum is 2%, the percentage error in time period is1 %. The maximum error in measurement of g is
Time period of oscillation of pendulam
where L is length of pendulam and g is acceleration due to gravity.
hence we have ...........................(1)
hence
The time dependence of a physical quantity ?
P = P_{0}exp(_at^{2})
where a is a constant and t is time. The constant a
We know that any quantity in power as a dimension of 1, thus [at^{2}] = 1
Thus we get [a] = T^{2}
Force F is given in terms of time t and distance x by
F = A sin C t + B cos D x
Then the dimensions of and are given by
Dimension of [A]=[MLT^{2}]
Dimension of [B]=[MLT^{2}]
[A]/[B]= [ M^{0}L^{0}T^{0}]
CT=1
Dimension of [C]=1/[T]=[T^{1}]
DL=1
Dimension of [D]=1/[L]=[L^{1}]
[C][D] =[T^{1}]/[L^{1}]
=LT^{1}
A/B=Force/Force=[M^{0}L^{0}T^{0}]
Ct=∠⇒C=Angle/Time=1/T=T^{1}
Dx=∠⇒D=Angle/Distance=1/L=L^{1}
∴C/D=T^{1}L^{1} =[M0 LT^{1} ]
The Van der Waal equation for 1 mole of a real gas is
where P is the pressure, V is the volume, T is the absolute temperature, R is the molar gas constant and a, b are Van dar Waal constants. The dimensions of a are the same as those of
The answer is PV^{2}
Solution,
As we know the Vander Waals equation is {P+(a/V^{2})} (V−b) =RT
Then,
The dimension of a is,
(a/V^{2}) = (P)
Or, a=PV^{2}
The dimensional formula of coefficient of viscosity is
If force (F) is given by F = Pt^{_1 }+ Qt, where t is time. The unit of P is same as that of
According to the principle of homogeneity of dimension, an equation is dimensionally correct when each term has same dimension on both sides of the equation.
Since left side has dimension of force so the term Pt^{−1} will have also dimension of force.
Thus, [P][T^{−1}]=[F]=[MLT^{−2}]
or [P]=[MLT^{−1}]
We know that the dimension of momentum is [p]=[MLT^{−1}]
The product of energy and time is called action. The dimensional formula for action is same as that for
Both are different from each other.
Energy × Time=(M^{1}L^{2}T^{−2})×(T^{1})=M^{1}L^{2}T^{−1}
Force × Velocity=(M^{1}L^{1}T^{−2})×(L^{1}T^{−1})=M^{1}L^{2}T^{−3}
Impulse × distance=(M^{1}L^{1}T^{−1})×(L^{1})=M^{1}L^{2}T^{−1}
Power=M^{1}L^{2}T^{−3}
Angular Energy=M^{1}L^{1}T^{−2}
When a wave traverses a medium, the displacement of a particle located at x at time t is given by y = a sin (bt  cx) where a, b and c are constants of the wave. The dimensions of b/c are the same as those of
Argument of sin is dimensionless and using principle of dimensional homogeneity, we get
∣bt∣=∣cx∣=[M_{0}L_{0}T_{0}]
Thus, ∣b∣=T−1 and ∣c∣=[L^{−1}]
Thus, dimensions of b/c are [LT^{−1}], which is the same as the velocity of wave.
In the above question dimensions of b/c are the same as those of
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