Units & Dimensions Test - 9

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 newton-second, 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 one-newton force applied for one second.

F = ma
[F] = kg m/s²

Unit of velocity = m/s
1m = 10⁶  Micro meter
And 1 sec = 10⁶  Micro sec
Thus 1 m/s = 10⁶ 10⁶  micro meter /micro second

Dimension of KE is ML²T^-2
That of pressure is ML^-2T^-2
That of momentum is ML/T 
That of power is ML²T^-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.

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.

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] = ML²T^-2 / T^-1
= ML²T^-1
[P] = MLT^-1
[F] = MLT^-2
[Angular momentum] = [P x r] = ML²T^-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.

Moment of inertia =1/2​×mass×(Radius of gyration)²=[M¹L²T⁰]
Moment of force = torque =N.m=[M¹L²T⁻²]
Both are different from each other.

Volume=4πR³/3
ΔV/V×100=3∆R/R×100
=3×1/100​×100
=3
∴ Percentage error in volume is 3%

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

Time period of oscillation of pendulam 

where L is length of pendulam and g is acceleration due to gravity.
hence we have  ...........................(1)
hence 

We know that any quantity in power as a dimension of 1, thus [at²] = 1
Thus we get [a] = T^-2

Dimension of [A]=[MLT^-2]
Dimension of [B]=[MLT^-2]
 
 [A]/[B]= [ M⁰L⁰T⁰]
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=[M⁰L⁰T⁰]
Ct=∠⇒C=Angle/Time=1/T=T^-1
Dx=∠⇒D=Angle/Distance=1/L=L^-1
∴C/D=T^-1L^-1 =[M0 LT^-1 ]

The answer is PV²
Solution,
As we know the Vander Waals equation is {P+(a/V²)} (V−b) =RT
Then,
The dimension of a is,
(a/V²) = (P)
Or, a=PV²

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]

Both are different from each other.
Energy × Time=(M¹L²T⁻²)×(T¹)=M¹L²T⁻¹
Force × Velocity=(M¹L¹T⁻²)×(L¹T⁻¹)=M¹L²T⁻³
Impulse × distance=(M¹L¹T⁻¹)×(L¹)=M¹L²T⁻¹
Power=M¹L²T⁻³
Angular Energy=M¹L¹T⁻²

Argument of sin is dimensionless and using principle of dimensional homogeneity, we get
∣bt∣=∣cx∣=[M₀L₀T₀]
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.