write a comparitive account of digestive system of hauman beings,digestive system of ruminents
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Answer:
Dimensions and units
A mechanical system undergoing one-dimensional damped vibrations can be modeled by the equation
mu″+bu′+ku=0,
where m is the mass of the system, b is some damping coefficient, k is a spring constant, and u(t) is the displacement of the system. This is an equation expressing the balance of three physical effects: mu″ (mass times acceleration), bu′ (damping force), and ku (spring force). The different physical quantities, such as m, u(t), b, and k, all have different dimensions, measured in different units, but mu″, bu′, and ku must all have the same dimension, otherwise it would not make sense to add them.
Fundamental concepts
Base units and dimensions
Base units have the important property that all other units derive from them. In the SI system, there are seven such base units and corresponding physical quantities: meter (m) for length, kilogram (kg) for mass, second (s) for time, kelvin (K) for temperature, ampere (A) for electric current, candela (cd) for luminous intensity, and mole (mol) for the amount of substance.
We need some suitable mathematical notation to calculate with dimensions like length, mass, time, and so forth. The dimension of length is written as [L], the dimension of mass as [M], the dimension of time as [T], and the dimension of temperature as [Θ] (the dimensions of the other base units are simply omitted as we do not make much use of them in this text). The dimension of a derived unit like velocity, which is distance (length) divided by time, then becomes [LT−1] in this notation. The dimension of force, another derived unit, is the same as the dimension of mass times acceleration, and hence the dimension of force is [MLT−2].
Let us find the dimensions of the terms in (1). A displacement u(t) has dimension [L]. The derivative u′(t) is change of displacement, which has dimension [L], divided by a time interval, which has dimension [T], implying that the dimension of u′ is [LT−1]. This result coincides with the interpretation of u′ as velocity and the fact that velocity is defined as distance ([L]) per time ([T]).
Looking at (1), and interpreting u(t) as displacement, we realize that the term mu″(mass times acceleration) has dimension [MLT−2]. The term bu′ must have the same dimension, and since u′ has dimension [LT−1], bmust have dimension [MT−1]. Finally, ku must also have dimension [MLT−2], implying that k is a parameter with dimension [MT−2].
The unit of a physical quantity follows from the dimension expression. For example, since velocity has dimension [LT−1]and length is measured in m while time is measured in s, the unit for velocity becomes m/s. Similarly, force has dimension [MLT−2] and unit kg\, m/s2. The k parameter in (1) is measured in kg\,s−2.
Dimension of derivatives
The easiest way to realize the dimension of a derivative, is to express the derivative as a finite difference. For a function u(t) we have
dudt≈u(t+Δt)−u(t)Δt,
where Δt is a small time interval. If u denotes a velocity, its dimension is [LT]−1, and u(t+Δt)−u(t) gets the same dimension. The time interval has dimension [T], and consequently, the finite difference gets the dimension [LT]−2. In general, the dimension of the derivative du/dt is the dimension of u divided by the dimension of t.
Dimensions of common physical quantities
Many derived quantities are measured in derived units that have their own name. Force is one example: Newton (N) is a derived unit for force, equal to kg\, m/s2. Another derived unit is Pascal (Pa) for pressure and stress, i.e., force per area. The unit of Pa then equals N/m2 or kg/ms2. Below are more names for derived quantities, listed with their units.