the potential energy of a particle varies with velocity v as U equals to Av3/v + B where A and B are constant dimensional formula A/B is
(1) ML-1T
(2) M-1LT
(3) L-1T-1
(4) MLT-1
U = A×v^3/v+B
[ML^2T^-2]=A×[LT^-1]^3/[LT^-1]+B
According to the principle of homogeneity, similar quantities can be added or subtracted.
Therefore, Dimensions of B are same as Dimensions of velocity
B=[LT^-1]
Also if we shift the given dimensions then
A= U× (B+v)/ v^3
A= [ML^2T^-2]× [LT^-1] / [LT^-1]^3
A= [ML^3T^-2] / [L^3T^-3]
A=[M]
THUS,
A/B = M/ LT^-1
A/B = [ML^-1T]
U = A×v^3/v+B
[ML^2T^-2]=A×[LT^-1]^3/[LT^-1]+B
According to the principle of homogeneity, similar quantities can be added or subtracted.
Therefore, Dimensions of B are same as Dimensions of velocity
B=[LT^-1]
Also if we shift the given dimensions then
A= U× (B+v)/ v^3
A= [ML^2T^-2]× [LT^-1] / [LT^-1]^3
A= [ML^3T^-2] / [L^3T^-3]
A=[M]
THUS,
A/B = M/ LT^-1
A/B = [ML^-1T]
To determine the dimensional formula for A/B, we need to simplify the given expression for potential energy, U = Av^3/v + B.
First, let's simplify the equation by canceling out v from the numerator and denominator:
U = Av^2 + B
To find the dimensional formula for A/B, we can analyze the terms separately.
The first term, Av^2, implies that A has dimensions equal to the potential energy divided by the square of velocity. Therefore, the dimensions of A can be determined as:
[A] = [U/v^2] = [ML^2T^-2]/[LT^-1]^2 = [ML^2T^-2]/[L^2T^-2] = [MT^0] = [M]
The second term, B, has direct dimensions of potential energy, [U], which is equal to [ML^2T^-2].
Therefore, the dimensional formula for A/B would be:
[M]/[ML^2T^-2] = M/(ML^2T^-2) = M^-1L^2T^2.
So, the correct answer is:
(2) M^-1L^2T^2.
To find the dimensional formula of A/B, we need to analyze the dimensions of each term in the given expression and simplify them.
The given potential energy equation is U = (A v^3)/v + B.
Let's break down each term:
Term 1: (A v^3)/v
The velocity term in the numerator has dimensions of L/T (length/time), raised to the power of 3. The denominator's dimension is simply T (time). Therefore, the overall dimensions of this term are L^3/T.
Term 2: B
Since B is a constant, it does not have any dimensions associated with it.
Combining both terms, the overall dimensions of the potential energy equation are L^3/T.
Now let's consider A/B:
(A/B) = (A v^3)/vB
The dimensions of (A v^3) are L^3/T.
The dimensions of vB are (L/T)B.
To find the dimensions of A/B, we need to simplify these dimensions:
L^3/T / (L/T)^B = L^3/T / L^B / T^B
Simplifying further, we have:
L^3 / L^B * T / T^B = L^(3-B) * T^(1-B)
Comparing this expression with the given answer choices, we can conclude that the correct dimensional formula for A/B is:
(1) ML^(-1)T
Therefore, the answer is option (1).