A student writes an expression of the force causing a body of mass (m) to move in a circular motion with a velocity (v) as F = mv2 uses the dimensional method to check its correctness.
1 Answer
Solution:
The given relation F = mv2, where F is the force, m is the mass and v is the speed of the particle.
The dimension of force F = [MLT-2]
The dimension of velocity, v = [LT-1]
The dimension of mass m = [M]
The dimension of left hand side, F = [MLT-2]
Dimension of the right hand side mv2 = [M][LT-1]2 = [ML2T-2]
Since the dimensions on the left-hand side are not equal to the dimension on the right-hand side of the equation. So, the formula is dimensionally wrong.
Things to remember from Physical Quantities
- 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,
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