A student writes an expression of the momentum (p) of a body of mass (m) with total Energy (E) and considers the duration of the time (t) as \(p=\sqrt{2m\frac Et}\). Check its correctness by using dimensional analysis.
1 Answer
Solution:
The given formula is, \(p=\sqrt{\frac{2mE}t}\)
Where,
P is the momentum,
m is the mass,
E is the energy, and
t is the time
The dimension of each terms are as follows:
dimension of momentum, p = [MLT-1],
dimension of mass, m = [M],
dimension of time, t = [T], and
dimension of energy, E = [ML2T2]
Now, substituting the quantities in the above-given relation, we get,
\(\left[MLT^{-1}\right]=\sqrt{\frac{M.ML^2T^{-2}}T}\), because 2 is dimensionless constant
\(=\sqrt{M^2L^2T^{-1}}\)
\(=MLT^{-\frac12}\)
Dimension of LHS ≠ Dimension of RHS
So, the formula is dimensionally incorrect.
- The four applications of dimensional analysis are:
- To check the correctness of physical relation.
- To derive the relation between various physical quantities.
- To convert the value of physical quantities from one system of units into another system of units.
- To find the dimensions of constants in the given equation.
- The correctness of physical relation can be determined by comparing the dimension of the left-hand side (LHS) and the dimension of the right-hand side (RHS)
- We can use the formula, \(N_2=N_1\left[\frac{M_1}{M_2}\right]^a\left[\frac{L_1}{L_2}\right]^b\left[\frac{T_1}{T_2}\right]^c\)\) to convert unit from one to another system using dimension.
- Despite the usefulness of dimensions, there are some limitations. They are:
- The dimensional analysis does not give any information about dimensionless constants.
- If the quantity depends on more than three other physical quantities having dimensions, the formula cannot be derived.
- We cannot derive the formula containing trigonometric functions, logarithmic functions, exponential functions, etc. It is best suited for linear functions only.
- The exact form of a relationship cannot be determined when there is more than one part in any relationship.
- It gives no information about the physical quantity,