If a is acceleration, v is velocity, x is position, and t is time, then
check the validity (wrong or correct) of the following equations using
dimensional analysis:
a) t2 = 2x/a
b) t = x/v
c) a = v/x
d) v = a/t
ALSO ,
The term 1/2 PV^2
rv2 occurs in Bernoulli’s equation in Chapter 15,
with P being the density of a fluid and v its speed. Find the dimensions of
1/2 PV^2
Thank you in advance .
on the first, I will be happy to critique your thinking.
on the second
Pressure*velocity^2=force/area * distance^2/time^2=
force/time^2
in the first i chose last one
but in second p mean density so the formula should be p=m/v m mass , v volume is not it ?
Q2)
(M/L^3) x (L^2/T^2) = M/LT = ML^-1T^-2
To check the validity of the equations using dimensional analysis, we need to compare the dimensions on both sides of the equation. The dimensions of each variable are as follows:
Acceleration (a): [L/T^2]
Velocity (v): [L/T]
Position (x): [L]
Time (t): [T]
a) t^2 = (2x/a)
On the left-hand side (LHS), we have [T^2]. On the right-hand side (RHS), we have [(2x) / (L/T^2)], which simplifies to [2LT^2 / L]. After canceling out the L, the RHS becomes [2T^2]. Since [T^2] and [2T^2] have different dimensions, this equation is INCORRECT.
b) t = (x/v)
On the LHS, we have [T]. On the RHS, we have [(L) / (L/T)], which simplifies to [LT / L]. After canceling out the L, the RHS becomes [T]. The equation is CORRECT.
c) a = (v/x)
On the LHS, we have [L/T^2]. On the RHS, we have [(L/T) / L], which simplifies to [LT / L^2]. After canceling out the L, the RHS becomes [T/L]. Since [L/T^2] and [T/L] have different dimensions, this equation is INCORRECT.
d) v = (a/t)
On the LHS, we have [L/T]. On the RHS, we have [(L/T^2) / T], which simplifies to [L / T^3]. Since [L/T] and [L/T^3] have different dimensions, this equation is INCORRECT.
Therefore, equations b) and c) are incorrect.
Moving on to the second part of your question regarding the dimensions of 1/2 PV^2 in Bernoulli's equation:
The term 1/2 PV^2 has the dimensions of energy. To find the dimensions, we can decompose the individual terms:
1/2: It is a dimensionless constant.
P: It represents pressure, which has the dimensions of [M/LT^2] (mass per unit area).
V: It represents velocity, which has the dimensions of [L/T].
Multiplying these dimensions together, we get:
1/2 * [M/LT^2] * [L/T] = [M(L/T)^2] = [ML^2/T^2]
Therefore, the dimensions of 1/2 PV^2 in Bernoulli's equation are [ML^2/T^2], which represent the dimensions of energy.
I hope this helps! Let me know if you have any further questions.