
Which of the following quantities are dimensionless? (symbols have their usual meaning)
(A)
(B)
(C)
(D)
[Useful relation , ]
Answer
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Hint:-For finding the dimensionless quantity, we will check option by option putting unit of symbols in it. After getting the unit, we convert it into dimension to check dimensionless quantity.
Complete step-by-step solution:
Dimensionless quantity is defined as physical quantity, which has no dimension. They are also called unit less quantity.
First we check option (A)
Where is moment of inertia, is angular speed, is mass, is velocity and is perpendicular distance.
We know that , therefore unit of moment of inertia is kilogram per meter square ,
Unit of angular speed is radian per second but radian is measurement of angle so, unit is ,
Unit of velocity is meter per second and unit of perpendicular distance is meter .
Putting all the units in the given expression, we have
After simplify, we get
Since, numerator and denominator both are equal. Therefore, a given expression has no unit or says it is dimensionless quantity.
Option (B)
Where is gravitational constant, we derive unit of gravitational constant from the force formula between two masses
Putting unit of force ; kilogram meter per second square , unit of distance ; meter , unit of mass ; kilogram on above formula we get unit of gravitational constant .
After simplify, we get unit of gravitational constant
Now, unit of density is kilogram per meter cube and unit of time is second
Putting units of , and on the expression, we get
After simplify we get,
Since, it has units, therefore it is not a dimensionless quantity.
Now option (C)
We know the unit of density , velocity and distance . So we only derive the unit of viscosity coefficient using the formula . We get after simplification
Since has no unit. So, we put unit of Force , distance and velocity on above equation and we have ,
After simplify, unit of viscosity coefficient
Putting units of on expression and we get
After simplify we get,
Since, the numerator and denominator are the same. It is unit less or say it is dimensionless quantity.
Now, option (D)
We know that the unit of and . is angle and it has no unit. So, we derive the unit of torque using formula .
Put unit of force and distance on above formula and get unit of torque
After simplifying the unit of torque .
Putting units of on the given expression we get,
After simplification we have,
Since it has units. Therefore, it is not a dimensionless quantity.
Hence, option (A) and (C) are correct.
Note:-
Dimensions are represented within the closed brackets and it has no magnitude. But units are represented without any brackets. Dimensions are represented in the form of M, L and T within the closed bracket. For example, the dimension of force is .
Complete step-by-step solution:
Dimensionless quantity is defined as physical quantity, which has no dimension. They are also called unit less quantity.
First we check option (A)
Where
We know that
Unit of angular speed
Unit of velocity
Putting all the units in the given expression, we have
After simplify, we get
Since, numerator and denominator both are equal. Therefore, a given expression has no unit or says it is dimensionless quantity.
Option (B)
Where
Putting unit of force
After simplify, we get unit of gravitational constant
Now, unit of density
Putting units of
After simplify we get,
Since, it has units, therefore it is not a dimensionless quantity.
Now option (C)
We know the unit of density
Since
After simplify, unit of viscosity coefficient
Putting units of
After simplify we get,
Since, the numerator and denominator are the same. It is unit less or say it is dimensionless quantity.
Now, option (D)
We know that the unit of
Put unit of force
After simplifying the unit of torque
Putting units of
After simplification we have,
Since it has units. Therefore, it is not a dimensionless quantity.
Hence, option (A) and (C) are correct.
Note:-
Dimensions are represented within the closed brackets and it has no magnitude. But units are represented without any brackets. Dimensions are represented in the form of M, L and T within the closed bracket. For example, the dimension of force is
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