Someone please help me with this.
The speed of ocean waves depends on their wavelength λ (in meters) and the gravitational field strength g (in m/sˆ2) in this way:
V=Kλˆp Gˆq
Where K is a dimentionless constant. Find the value of the exponents p and q.
This is a dimensional analysis problem. If g and λ are the only quantities that can affect the speed (ocean depth is actually another, but we will assume a very deep ocean), then some combination
V * λ^-p * g^-q must be a dimensionless constant, K. The length dimension exponent of the product is 1 -p -q = 0 and the time dimension exponent is -1 +2q = 0.
This leads to -1/2 + q = 0
1/2 - p = 0
p = 1/2 q = 1-p = 1/2
V = K λ^1/2 g^1/2
The crests of some swells or waves in the ocean are 20 m apart. As they pass, a boat at anchor bobs up and down every 5 seconds. The frequency and wavelength of the water wave are
Answer
5 Hz and 20 m
0.2 Hz and 20 m
20 Hz and 5 m
5 Hz and 4 m.
To find the value of the exponents p and q in the equation V = KΛ^p G^q, we can use dimensional analysis.
First, let's analyze the dimensions involved:
- The speed of ocean waves (V) has dimensions of [L]/[T] (length per time).
- The wavelength (Λ) has dimensions of [L] (length).
- The gravitational field strength (G) has dimensions of [L]/[T]^2 (length per time squared).
Now, let's consider the dimensional consistency of the equation:
- The left side of the equation (V) has dimensions of [L]/[T].
- The right side of the equation (KΛ^p G^q) must also have dimensions of [L]/[T].
Since K is a dimensionless constant, we can ignore it for dimensional analysis purposes.
Comparing dimensions, we can set up the following equations:
[L]/[T] = Λ^p * ([L]/[T]^2)^q
Next, let's analyze the dimensions individually:
- On the left side of the equation, we have [L]/[T]. This has a length dimension of 1 and a time dimension of -1.
- On the right side of the equation, we have Λ^p * ([L]/[T]^2)^q. Breaking it down further:
- Λ^p has a length dimension of p and a time dimension of 0.
- ([L]/[T]^2)^q has a length dimension of q and a time dimension of -2q.
Equating the dimensions, we have:
Length dimension: 1 = p + q
Time dimension: -1 = -2q
From the second equation, we can solve for q:
-1 = -2q
q = 1/2
Substituting q = 1/2 into the first equation, we can solve for p:
1 = p + 1/2
p = 1 - 1/2
p = 1/2
Therefore, the values of the exponents p and q are p = 1/2 and q = 1/2.
Substituting these values back into the original equation, we get:
V = KΛ^(1/2) G^(1/2)
So, the speed of ocean waves (V) is equal to a constant (K) multiplied by the square root of the wavelength (Λ) and the square root of the gravitational field strength (G).