Math, asked by darkknightforces, 1 month ago

The formula v2=3mgl (1-cos theta)/ M(1+Sin2 theta) is obtained as the solution of a problem. Use the method of dimensions to find whether this is a reasonable solution

Answers

Answered by Afreenakbar

Answer:

We examine the dimensions of each term in the formula in order to assess the reasonableness of the presented formula using the method of dimensions.

Let's assign the following dimensions:

v = velocity ( $\frac{L}{T}$ )

m = mass (M)

g = acceleration due to gravity ( $\frac{ L}{T^2}$ )

l = length (L)

θ = angle (dimensionless)

M = a constant (dimensionless)

We can now determine the size of each term in the formula:

1. $3mgl: (M)(L)( \frac{L}{T^2})(L) = M \times \frac{L^2}{T^2}$

2. (1 - cos θ): (dimensionless)

3. M(1 + sin²θ): (M)(dimensionless) = M

According to the equations,

$v^2 = \frac{3mgl(1 - cos \theta)}{ M(1 + sin^2\theta)}$

The dimensions of both sides of the equation are examined:

Left-hand side (LHS):

$v^2 = ( \frac{ L}{T})^2 = \frac{L^2}{T^2}$

Right-hand side (RHS):

$\frac{3mgl(1 - cos \theta) }{M(1 + sin^2 \theta)} = ( \frac{M \times L^2}{T^2}) \times \frac{(dimensionless) }{ (M) }= \frac{L^2}{T^2}$

From the standpoint of dimensional analysis, the formula makes sense because the dimensions of both sides of the equation match. It is crucial to remember that dimensional analysis by itself cannot ensure the precision or validity of the formula. It only evaluates how consistently the units in question operate.