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as Sn=u+a/2(2n-1) have no dimensional homogenity then also its widely used..??
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as Sn=u+a/2(2n-1) have no dimensional homogenity then also its widely ...
Here, S=u+(1/2)*a*(2n-1) this equation is used to find distance travelled by body in nth second.
LHS
So, S(nth) = distance travelled in nth second
i.e S(nth) = [L]/[T] = [LT^-1] dimensionally
RHS
Now, U+(1/2)*a*(2n-1) = where,
U= initial velocity = [LT^-1]
a= acceleration = [LT^-2]
2n-1= n=time = [T]
1/2=dimensionless = [MdegreeLdegreeTdegee]
U+(1/2)*a*(2n-1)= [LT^-1]+[LT^-2]*[T]
[LT^-1]+[LT-^1]=[LT^-1] by principle of homogeneity.
As LHS= RHS dimensionally i.e [LT^-1]
Therefore, it is dimensionally correct.
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as Sn=u+a/2(2n-1) have no dimensional homogenity then also its widely ...
Introduction:
The formula Sn = u * (a/2) * (2n - 1) represents the sum of an arithmetic series. Here, Sn represents the sum of the series, u represents the first term, a represents the common difference, and n represents the number of terms.
Explanation of Dimensional Homogeneity:
Dimensional homogeneity refers to the consistency of units in a mathematical equation or formula. For an equation to be dimensionally homogeneous, the units on both sides of the equation should be the same.
In the given formula Sn = u * (a/2) * (2n - 1), the units can be analyzed as follows:
- Sn represents the sum of the series and has units of the first term u.
- u represents the first term of the series and has its own units.
- a represents the common difference between terms and has its own units.
- n represents the number of terms and is dimensionless.
Lack of Dimensional Homogeneity:
Upon analyzing the formula, it becomes evident that the units on both sides of the equation do not match. The left side of the equation, Sn, has units of the first term u, while the right side of the equation has a combination of u and a. Therefore, the formula Sn = u * (a/2) * (2n - 1) lacks dimensional homogeneity.
Widespread Use:
Despite the lack of dimensional homogeneity, the formula Sn = u * (a/2) * (2n - 1) is still widely used in the field of arithmetic series. This is because the formula provides a convenient way to calculate the sum of an arithmetic series, even though the units may not match.
While dimensional homogeneity is an important aspect of mathematical equations in physics, it does not necessarily determine the usability or validity of a formula. In some cases, formulas may lack dimensional homogeneity but still provide accurate results when used appropriately.
It is important to note that when using the formula Sn = u * (a/2) * (2n - 1), one should ensure that the units of the first term u and the common difference a are consistent. This means that both u and a should have the same units to avoid any inconsistencies in the calculations.
In Summary:
- The formula Sn = u * (a/2) * (2n - 1) lacks dimensional homogeneity as the units on both sides of the equation do not match.
- Despite lacking dimensional homogeneity, the formula is widely used in arithmetic series calculations.
- It is important to ensure that the units of the first term u and the common difference a are consistent when using the formula.
The formula Sn = u * (a/2) * (2n - 1) represents the sum of an arithmetic series. Here, Sn represents the sum of the series, u represents the first term, a represents the common difference, and n represents the number of terms.
Explanation of Dimensional Homogeneity:
Dimensional homogeneity refers to the consistency of units in a mathematical equation or formula. For an equation to be dimensionally homogeneous, the units on both sides of the equation should be the same.
In the given formula Sn = u * (a/2) * (2n - 1), the units can be analyzed as follows:
- Sn represents the sum of the series and has units of the first term u.
- u represents the first term of the series and has its own units.
- a represents the common difference between terms and has its own units.
- n represents the number of terms and is dimensionless.
Lack of Dimensional Homogeneity:
Upon analyzing the formula, it becomes evident that the units on both sides of the equation do not match. The left side of the equation, Sn, has units of the first term u, while the right side of the equation has a combination of u and a. Therefore, the formula Sn = u * (a/2) * (2n - 1) lacks dimensional homogeneity.
Widespread Use:
Despite the lack of dimensional homogeneity, the formula Sn = u * (a/2) * (2n - 1) is still widely used in the field of arithmetic series. This is because the formula provides a convenient way to calculate the sum of an arithmetic series, even though the units may not match.
While dimensional homogeneity is an important aspect of mathematical equations in physics, it does not necessarily determine the usability or validity of a formula. In some cases, formulas may lack dimensional homogeneity but still provide accurate results when used appropriately.
It is important to note that when using the formula Sn = u * (a/2) * (2n - 1), one should ensure that the units of the first term u and the common difference a are consistent. This means that both u and a should have the same units to avoid any inconsistencies in the calculations.
In Summary:
- The formula Sn = u * (a/2) * (2n - 1) lacks dimensional homogeneity as the units on both sides of the equation do not match.
- Despite lacking dimensional homogeneity, the formula is widely used in arithmetic series calculations.
- It is important to ensure that the units of the first term u and the common difference a are consistent when using the formula.
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