Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as
\(v_2=\frac{n}{m^2}v_1\) and \(a_2=\frac{a_1}{mn}\)
respectively. Here m and n are constants. The relations for distance and time in two systems respectively are :
Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as
\(v_2=\frac{n}{m^2}v_1\) and \(a_2=\frac{a_1}{mn}\)
respectively. Here m and n are constants. The relations for distance and time in two systems respectively are :
\(\frac{n^3}{m_3}L_1=L_2\) and\(\frac{n^2}{m}T_1=T_2\)
\(L_1=\frac{n^4}{m^2}L_2\) and \(T_1=\frac{n^2}{m}T_2\)
\(L_1=\frac{n^2}{m}L_2\) and \(T_1=\frac{n^4}{m^2}T_2\)
\(\frac{n^2}{m}L_1=L_2 \) and \(\frac{n^4}{m^2}T_1=T_2\)
The Correct Option is A
Solution and Explanation
To find the relation for distance \(L\) and time \(T\) between two different systems of units, we will use the given relationships for velocity \(v\) and acceleration \(a\) between the two systems.
The relationship for velocity between the two systems is given by:
\(v_2=\frac{n}{m^2}v_1\)
The relationship for acceleration between the two systems is given by:
\(a_2=\frac{a_1}{mn}\)
We know that velocity \(v\) is the rate of change of distance \(L\) with respect to time \(T\). Thus, we can write:
\(v = \frac{L}{T}\)
Similarly, acceleration \(a\) is the rate of change of velocity \(v\) with respect to time \(T\). So:
\(a = \frac{v}{T} = \frac{L}{T^2}\)
For the system of units 1 and 2, the formulas for length \(L\) and time \(T\) can be derived by connecting the given equations for velocity and acceleration.
Equate the expressions of acceleration from both systems using the relation \(a = \frac{L}{T^2}\):
\(\frac{L_2}{T_2^2} = \frac{1}{mn} \cdot \frac{L_1}{T_1^2}\)
Similarly, equate the expressions of velocity:
\(\frac{L_2}{T_2} = \frac{n}{m^2} \cdot \frac{L_1}{T_1}\)
Solving these two equations, we get:
\(\frac{L_1}{L_2} = \frac{n^3}{m^3}\) and \(\frac{T_1^2}{T_2^2} = \frac{n^2}{m^2}\)
Which simplifies to:
\(\frac{n^3}{m^3}L_1=L_2\) and \(\frac{n^2}{m}T_1=T_2\)
Thus, the correct relation for distance and time in the two systems is:
\(\frac{n^3}{m^3}L_1=L_2\) and \(\frac{n^2}{m}T_1=T_2\)
This matches the given correct option.
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Concepts Used:
Dimensional Analysis
Dimensional Analysis is a process which helps verify any formula by the using the principle of homogeneity. Basically dimensions of each term of a dimensional equation on both sides should be the same.
Limitation of Dimensional Analysis: Dimensional analysis does not check for the correctness of value of constants in an equation.
Using Dimensional Analysis to check the correctness of the equation
Let us understand this with an example:
Suppose we don’t know the correct formula relation between speed, distance and time,
We don’t know whether
(i) Speed = Distance/Time is correct or
(ii) Speed =Time/Distance.
Now, we can use dimensional analysis to check whether this equation is correct or not.
By reducing both sides of the equation in its fundamental units form, we get
(i) [L][T]-¹ = [L] / [T] (Right)
(ii) [L][T]-¹ = [T] / [L] (Wrong)
From the above example it is evident that the dimensional formula establishes the correctness of an equation.