Answer :
### 1. Difference between Accurate and Precise Measurements:
- Accuracy refers to how close a measured value is to the true or accepted value. An accurate measurement is one that is very close to the actual value.
- Precision refers to the consistency or repeatability of measurements. A precise measurement is one that yields very similar results under unchanged conditions, even if these results are not close to the true value.
For example, if you measure the length of a table multiple times and get values like 2.01 m, 2.00 m, and 2.02 m, these measurements are precise as they are close to each other. If the actual length of the table is 2.00 m, those measurements are also accurate.
### 2. Dimensional Correctness and Physical Relation:
A dimensionally correct equation means that the dimensions on both sides of the equation match. This is a necessary condition but not a sufficient one for a physically correct relation. A dimensionally correct equation ensures that the units are compatible and balanced. However, just because an equation is dimensionally correct does not mean it accurately represents the physical phenomena involved. For an equation to be a correct physical relation, it must also be derived from physical principles and verified by experimental data.
### 3. Check the Correctness of the Formula [tex]\( t = 2\pi \sqrt{m/k} \)[/tex]:
#### Understanding the Variables and their Dimensions:
- [tex]\( t \)[/tex]: Time period (dimension: [tex]\([T]\)[/tex])
- [tex]\( m \)[/tex]: Mass (dimension: [tex]\([M]\)[/tex])
- [tex]\( k \)[/tex]: Force per unit displacement (dimension [tex]\([F] / [L]\)[/tex])
Since the force [tex]\( F \)[/tex] is given by [tex]\( F = ma \)[/tex] (where [tex]\( a \)[/tex] is the acceleration with dimension [tex]\([L T^{-2}]\)[/tex]), the dimension of force [tex]\( F \)[/tex] is:
[tex]\[ [F] = [M] [L][T^{-2}] \][/tex]
Thus, the dimension of [tex]\( k \)[/tex] (force per unit displacement) is:
[tex]\[ [k] = \frac{[F]}{[L]} = \frac{[M] [L] [T^{-2}]}{[L]} = [M] [T^{-2}] \][/tex]
#### Dimensional Analysis of the Formula:
Given the formula:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
We must analyze the dimensions on the right-hand side.
The dimension of [tex]\( \frac{m}{k} \)[/tex]:
[tex]\[ \frac{m}{k} = \frac{[M]}{[M][T^{-2}]} = \frac{[M]}{[M][T^{-2}]} = [T^{2}] \][/tex]
So, the dimension of [tex]\( \sqrt{\frac{m}{k}} \)[/tex] would be:
[tex]\[ \sqrt{\frac{m}{k}} = \sqrt{[T^{2}]} = [T] \][/tex]
Multiplying by the dimensionless constant [tex]\( 2\pi \)[/tex]:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
Thus, the dimensional analysis of [tex]\( t \)[/tex] gives:
[tex]\[ [t] = [T] \][/tex]
#### Conclusion:
The given formula [tex]\( t = 2\pi \sqrt{\frac{m}{k}} \)[/tex] is dimensionally correct because both sides of the equation have the dimension [tex]\([T]\)[/tex].
However, as discussed earlier, while the formula is dimensionally correct, it must also be verified physically to be considered a correct physical relation. In this case, the formula represents the time period of a simple harmonic oscillator, which is a physically correct relationship verified by physical principles and experimental data.
- Accuracy refers to how close a measured value is to the true or accepted value. An accurate measurement is one that is very close to the actual value.
- Precision refers to the consistency or repeatability of measurements. A precise measurement is one that yields very similar results under unchanged conditions, even if these results are not close to the true value.
For example, if you measure the length of a table multiple times and get values like 2.01 m, 2.00 m, and 2.02 m, these measurements are precise as they are close to each other. If the actual length of the table is 2.00 m, those measurements are also accurate.
### 2. Dimensional Correctness and Physical Relation:
A dimensionally correct equation means that the dimensions on both sides of the equation match. This is a necessary condition but not a sufficient one for a physically correct relation. A dimensionally correct equation ensures that the units are compatible and balanced. However, just because an equation is dimensionally correct does not mean it accurately represents the physical phenomena involved. For an equation to be a correct physical relation, it must also be derived from physical principles and verified by experimental data.
### 3. Check the Correctness of the Formula [tex]\( t = 2\pi \sqrt{m/k} \)[/tex]:
#### Understanding the Variables and their Dimensions:
- [tex]\( t \)[/tex]: Time period (dimension: [tex]\([T]\)[/tex])
- [tex]\( m \)[/tex]: Mass (dimension: [tex]\([M]\)[/tex])
- [tex]\( k \)[/tex]: Force per unit displacement (dimension [tex]\([F] / [L]\)[/tex])
Since the force [tex]\( F \)[/tex] is given by [tex]\( F = ma \)[/tex] (where [tex]\( a \)[/tex] is the acceleration with dimension [tex]\([L T^{-2}]\)[/tex]), the dimension of force [tex]\( F \)[/tex] is:
[tex]\[ [F] = [M] [L][T^{-2}] \][/tex]
Thus, the dimension of [tex]\( k \)[/tex] (force per unit displacement) is:
[tex]\[ [k] = \frac{[F]}{[L]} = \frac{[M] [L] [T^{-2}]}{[L]} = [M] [T^{-2}] \][/tex]
#### Dimensional Analysis of the Formula:
Given the formula:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
We must analyze the dimensions on the right-hand side.
The dimension of [tex]\( \frac{m}{k} \)[/tex]:
[tex]\[ \frac{m}{k} = \frac{[M]}{[M][T^{-2}]} = \frac{[M]}{[M][T^{-2}]} = [T^{2}] \][/tex]
So, the dimension of [tex]\( \sqrt{\frac{m}{k}} \)[/tex] would be:
[tex]\[ \sqrt{\frac{m}{k}} = \sqrt{[T^{2}]} = [T] \][/tex]
Multiplying by the dimensionless constant [tex]\( 2\pi \)[/tex]:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
Thus, the dimensional analysis of [tex]\( t \)[/tex] gives:
[tex]\[ [t] = [T] \][/tex]
#### Conclusion:
The given formula [tex]\( t = 2\pi \sqrt{\frac{m}{k}} \)[/tex] is dimensionally correct because both sides of the equation have the dimension [tex]\([T]\)[/tex].
However, as discussed earlier, while the formula is dimensionally correct, it must also be verified physically to be considered a correct physical relation. In this case, the formula represents the time period of a simple harmonic oscillator, which is a physically correct relationship verified by physical principles and experimental data.
