The surface r of a liquid depends on the density p acceleration due to gravity g nd height h use the method of dimension to derive the actual relationship between this quantities.
An object is tie to the end of a string move in a circle the force exerted by the string depend on the mass of an subject it speed and the radius of the circle use method of dimension to see how it variables can be combined correctly.
To derive the relationship between the surface area (r) of a liquid, density (p), acceleration due to gravity (g), and height (h), we will use the method of dimensions.
Step 1: Identify the dimensions of each quantity involved.
- Surface area (r) has dimensions of length squared (L^2).
- Density (p) has dimensions of mass divided by volume (M/L^3).
- Acceleration due to gravity (g) has dimensions of length divided by time squared (L/T^2).
- Height (h) has dimensions of length (L).
Step 2: Express each quantity in terms of its fundamental units.
- Surface area (r) = [L^2]
- Density (p) = [M][L^-3]
- Acceleration due to gravity (g) = [L][T^-2]
- Height (h) = [L]
Step 3: Form an equation using the Principle of Homogeneity, ensuring both sides have the same dimensions.
Since the surface area depends on density, acceleration due to gravity, and height, we can express this relationship as:
r = k * p^a * g^b * h^c
Here, k is the constant of proportionality, while a, b, and c are the exponents that will determine the relationship between the variables.
Step 4: Determine the exponents 'a', 'b', and 'c' using dimensional analysis.
Comparing dimensions on both sides of the equation:
[L^2] = [M]^a * [L]^(3a) * [L/T^2]^b * [L]^c
Equating exponents of similar dimensions:
- For length (L): 2 = 3a + b + c
- For mass (M): 0 = a
- For time (T): 0 = -2b
Solving the three equations simultaneously:
a = 0, b = 0, c = 2/3
Step 5: Substitute the exponents into the equation to get the relationship:
r = k * p^0 * g^0 * h^(2/3)
r = k * h^(2/3)
Therefore, using the method of dimensions, we have derived the relationship between surface area (r), density (p), acceleration due to gravity (g), and height (h) as r = k * h^(2/3).
Now, let's move on to the second question.
To determine how the variables of mass (m), speed (v), and radius (r) are combined correctly in the force exerted by the string, we will again use the method of dimensions.
Step 1: Identify the dimensions of each quantity involved.
- Mass (m) has dimensions of mass (M).
- Speed (v) has dimensions of length divided by time (L/T).
- Radius (r) has dimensions of length (L).
- Force (F) has dimensions of mass times length divided by time squared (ML/T^2).
Step 2: Express each quantity in terms of its fundamental units.
- Mass (m) = [M]
- Speed (v) = [L/T]
- Radius (r) = [L]
- Force (F) = [ML/T^2]
Step 3: Form an equation using the Principle of Homogeneity, ensuring both sides have the same dimensions.
Since the force exerted by the string depends on mass, speed, and radius, we can express this relationship as:
F = k * m^a * v^b * r^c
Here, k is the constant of proportionality, while a, b, and c are the exponents that will determine the relationship between the variables.
Step 4: Determine the exponents 'a', 'b', and 'c' using dimensional analysis.
Comparing dimensions on both sides of the equation:
[ML/T^2] = [M]^a * [L/T]^b * [L]^c
Equating exponents of similar dimensions:
- For mass (M): 1 = a
- For length (L): 1 = c
- For time (T): -2 = -b
Solving the three equations simultaneously:
a = 1, b = 2, c = 1
Step 5: Substitute the exponents into the equation to get the relationship:
F = k * m^1 * v^2 * r^1
F = k * m * v^2 * r
Therefore, using the method of dimensions, we have derived the relationship between the force (F) exerted by the string, mass (m), speed (v), and radius (r) as F = k * m * v^2 * r.