Check the correctness of the following formulas: 1) T = 2π√(l/g), 2) Work = P/T, 3) E = mc², 4) F = Gm₁m₂/r². Find the dimension of 9/6 in the equation F = a√(r) + b*t² when F is f... Check the correctness of the following formulas: 1) T = 2π√(l/g), 2) Work = P/T, 3) E = mc², 4) F = Gm₁m₂/r². Find the dimension of 9/6 in the equation F = a√(r) + b*t² when F is force (N), r is distance, and t is time. The Van der Waals equation for a gas is (P + a)(v - b) = RT. Determine the dimension of a and b, hence find the SI units of a and b.
Understand the Problem
The question is asking to verify the correctness of a series of physical formulas and to find the dimensions of certain variables in given equations. This includes checking specific formulas related to mechanics and thermodynamics.
Answer
Correct formulas: 1) Yes, 2) No, 3) Yes, 4) Yes. Dimensions of \( a \): \( [M][L^{-1}][T^{-2}] \), \( b \): \( [L^3] \). Units: \( a \) is \( \text{Pa} \), \( b \) is \( \text{m}^3 \).
Answer for screen readers
- The first formula is correct.
- The second formula is incorrect.
- The third formula is correct.
- The fourth formula is correct.
- Dimension of ( a ): ( [M][L^{-1}][T^{-2}] ), SI unit: ( \text{Pa} )
Dimension of ( b ): ( [L^3] ), SI unit: ( \text{m}^3 )
Steps to Solve
-
Check the formulas
Let's verify each formula for its correctness:- (1) ( T = 2\pi \sqrt{\frac{l}{g}} ) is correct for the period of a simple pendulum.
- (2) Work = ( \frac{P}{T} ) is incorrect; Work = Power (\times) Time, typically represented as ( W = P \times t ).
- (3) ( E = mc^2 ) is correct as it represents the mass-energy equivalence.
- (4) ( F = \frac{Gm_1m_2}{r^2} ) is correct; it represents the gravitational force between two masses.
-
Find the dimension of ( \frac{9}{6} )
In the equation ( F = a\sqrt{r} + b t^2 ):- Dimensional formula of Force ( F ) is ( [M][L][T^{-2}] ).
- Dimensional formula of distance ( r ) is ( [L] ).
- Dimensional formula of time ( t ) is ( [T] ).
For ( a\sqrt{r} ), we have:
[ a\sqrt{[L]} = [M][L][T^{-2}] \implies a = [M][L^{\frac{3}{2}}][T^{-2}] ]For ( b t^2 ), we have:
[ bt^2 = [M][L][T^{-2}] \implies b = [M][L][T^{-4}] ] -
Find dimensions of ( a ) and ( b ) in the Van der Waals equation
The Van der Waals equation is ((P + a)(v - b) = RT).- Here, Pressure ( P ) has dimensions of ( [M][L^{-1}][T^{-2}] ).
- Volume ( v ) has dimensions of ( [L^3] ).
- Gas constant ( R ) is ( \frac{[P][v]}{[T]} ), hence ( R = \frac{[M][L^{-1}][T^{-2}][L^3]}{[T]} = [M][L^2][T^{-3}] ).
Analyzing ( a ) and ( b ):
From ( (P + a) ):- Thus, ( a ) must have the same dimensions as ( P ), so:
( a = [M][L^{-1}][T^{-2}] ).
From ( (v - b) ): - Thus, ( b ) has dimensions of volume ( [L^3] ).
-
SI Units of ( a ) and ( b )
- For ( a ) (Pressure): SI unit is ( \text{Pa} ) (Pascals) or ( \text{N/m}^2 ).
- For ( b ) (Volume): SI unit is ( \text{m}^3 ).
- The first formula is correct.
- The second formula is incorrect.
- The third formula is correct.
- The fourth formula is correct.
- Dimension of ( a ): ( [M][L^{-1}][T^{-2}] ), SI unit: ( \text{Pa} )
Dimension of ( b ): ( [L^3] ), SI unit: ( \text{m}^3 )
More Information
- The formulas highlighted are mainly from classical mechanics and thermodynamics. Understanding their dimensions ensures consistency in physical equations.
- SI units offer a universal language for measurements in science and engineering.
Tips
- Mixing up the relationships in equations, especially between work, power, and time. Work should always be equated as ( W = P \times t ).
- Failing to convert the units correctly when deriving dimensions can lead to inaccurate conclusions.
