Desparete Housewives Reruns

EXAMPLES OF GRAPHIC FUNCTIONS

GRAPH FUNCTIONS





Domain and range of a function the domain is the set of all x coordinates of the points on the graph of the function, and the tour is the set of all coordinates in the y-axis The domain values \u200b\u200bare usually associated with the horizontal axis (x axis) and the values \u200b\u200bof travel with the vertical axis (the axis).

Example for discussion:

Determine the domain and path of the function f whose graph is:





Practice Exercise: Determine the domain and path following graph:





functions increasing, decreasing and constant

Definition: Let I be in range in the domain of a function f. Then:
1) f is increasing in the interval I if f (b)> f (a) provided that b> a in I.
2) f is decreasing on the interval I if f (b)
3) f is constant in the interval I if f (b) = f (a) for all b in I.

Examples:

1)

The function f (x) = 2x + 4 is an increasing function on real numbers.

2)

The function g (x) =-x3 is a decreasing in numbers real.

3)
GRAPH FUNCTIONS







Domain and range


The domain of a function is the set of all x coordinates of the points on the graph of the function, and the path is set of all coordinates in the y-axis The domain values \u200b\u200bare usually associated with the horizontal axis (x axis) and the values \u200b\u200bof travel with the vertical axis (the axis).



Example for discussion:



Determines the domain and path of the function f whose graph is:












Practice Exercise: Determine the domain and path following graph:












functions increasing, decreasing and constant


Definition : Let I be in range in the domain of a function f. Then:

1) f is increasing in the interval I if f (b)> f (a) provided that b> a in I.

2) f is decreasing on the interval I if f (b)

3) f is constant in the interval I if f (b) = f (a) for all b in I.



Examples:



1)


The function f (x) = 2x + 4 is an increasing function on real numbers.



2)


The function g (x) =-x3 is a decreasing function in the real numbers.



3)


The function h (x) = 2 is a function it counts in real numbers.



4)


The function f (x) = X2 is a decreasing function in the range from minus infinity to zero and increasing in the range of zero to infinity.





constant function


A constant function is a function of the form f (x) = b. Its graph is a horizontal line, the domain of the set of real numbers and route the set {b}.



Example:






The function f (x) = 2, the domain is the set of real numbers and the route is {2}. The slope (m) is zero.


Identity function





The identity function is the function of the form f (x) = x. The domain and path is the set of real numbers.










linear function




A linear function is a function of the form f (x) = mx + b, where m is different from zero, m and b are real numbers. The restriction m different from zero implies that the graph is a horizontal line. Neither its graph is a vertical line. The domain and path (range) of a function Linear is the set of real numbers.



Remember that if the slope (m) is positive the graph is increasing in real numbers and if the slope is negative the graph is decreasing in real numbers. The intercept is (0, b).







Example:








The function f (x) = 2x + 4, the slope is 2, so the graph is increasing in real numbers. The domain and path is the set of real numbers. The intercept is (0.4).



Exercise: Find the slope, the intercept, the x-intercept, domain and path of f (x) =-3x + 6. Then draw the graph.



Note: A function of the form f (x) = x is also a linear function but the intercept is zero. Its graph is a line that always passes through the origin.










Quadratic function A quadratic function is a function of the form f (x) = ax2 + bx + c, with a non-zero, where a, b and c are real numbers. The graph of a quadratic function is a parabola. If a> 0 then the parabola opens upward and if <0 entonces la parábola abre hacia abajo. El dominio de una función cuadrática es el conjunto de los números reales. El vértice de la parábola se determina por la fórmula:










f (x) = x2 is a quadratic function whose graph is a parabola that opens upward, since a> 0. The vertex is (0.0). The domain is the set of real numbers and the path is zero and positive real. The graph of a function that looks like f (x) = x2 is concave upward.








f (x) =-x2 is a quadratic function whose graph is a parabola that opens to down, because <0. El vértice es (0,0). El dominio es el conjunto de los números reales y el recorrido es el conjunto de los números reales negativos y el cero. La gráfica de una función que luce como f(x) = -x2 es cóncava hacia abajo.






Note: The axis of symmetry is x = h, where h is the abscissa of the vertex of the parabola, parallel to the axis of y.





Examples for discussion: Find the vertex, x intercepts, intercept, domain, path and axis of symmetry. Interval indicates that the function is increasing and decreasing. Draw the graph for each of the following functions:

1) f (x) = x2 - 2x - 3

2) g (x) =-x2 - 2x + 3



Practice Exercise: Let f (x) =-x2 + 4x - 4. Halla the vertex, x intercepts, intercept, domain and range. Interval indicates that the function is increasing and decreasing. Draw the graph.





absolute value function


The function is the absolute value of x. The domain is the set of real numbers and the path is zero and positive real numbers. Its graph is:










domain function


party domain match functions are functions that are formed by different equations for different parts of the domain. For example:







The graph of this function is:





































The domain is the set of real numbers except zero, expressed as a range is (- ¥, 0) È (0, ¥). The journey is the set of real numbers except -1 and 1 and real numbers between -1 and 1, ie (- ¥, -1) E (1, ¥). The open points (0, -1) and (0.1) indicates that the points do not belong to the graph of f. Due to the separation of the graph at x = 0, we say that f is discontinuous at x = 0.








radical Function The function is the square root function. Its graph is as follows:






Its domain is [0, ¥) and travel is [0, ¥).




The function h (x) = 2 is a function it counts in real numbers.

4)

The function f (x) = x2 is a decreasing function in the range from minus infinity to zero and increasing in the range of zero to infinity.


constant function

A constant function is a function of the form f (x) = b. Its graph is a horizontal line, the domain of the set of real numbers and route the set {b}.

Example:


The function f (x) = 2, the domain is the set of real numbers and travel is {2}. The slope (m) is zero. Identity function




The identity function is the function of the form f (x) = x. The domain and path is the set of real numbers.




linear function


A linear function is a function of the form f (x) = mx + b, where m is different from zero, m and b are real numbers. The restriction m different from zero implies that the graph is a horizontal line. Neither its graph is a vertical line. The domain and path (range) of a linear function is the set of real numbers.

Remember that if the slope (m) is positive the graph is increasing in real numbers and if the slope is negative the graph is decreasing in real numbers. The intercept is (0, b).



Example:



The function f (x) = 2x + 4, the slope is 2, so the graph is increasing in real numbers. The domain and path is the set of real numbers. The intercept is (0.4).

Exercise: Find the slope, the intercept, the x-intercept, domain and path of f (x) =-3x + 6. Then draw the graph.

Note: A function of the form f (x) = x is also a linear function but the intercept is zero. Its graph is a line that always passes through the origin.





Quadratic function A quadratic function is a function of the form f (x) = ax2 + bx + c, with a non-zero, where a, b and c are real numbers. The graph of a quadratic function is a parabola. If a> 0 then the parabola opens upward and if <0 entonces la parábola abre hacia abajo. El dominio de una función cuadrática es el conjunto de los números reales. El vértice de la parábola se determina por la fórmula:





f (x) = x2 is a quadratic function whose graph is a parabola that opens upward, since a> 0. The vertex is (0.0). The domain is the set of real numbers and the path is zero and positive real. The graph of a function that looks like f (x) = x2 is concave upward.



f (x) =-x2 is a quadratic function whose graph is a parabola that opens downward, because <0. El vértice es (0,0). El dominio es el conjunto de los números reales y el recorrido es el conjunto de los números reales negativos y el cero. La gráfica de una función que luce como f(x) = -x2 es cóncava hacia abajo.



Note: The axis of symmetry is x = h, where h is the abscissa of the vertex parable, parallel to the axis of y.


Examples for discussion: Find the vertex, x intercepts, intercept, domain, path and axis of symmetry. Interval indicates that the function is increasing and decreasing. Draw the graph for each of the following functions:
1) f (x) = x2 - 2x - 3
2) g (x) =-x2 - 2x + 3

Practice Exercise: Let f (x) =-x2 + 4x - 4. Find the vertex, x intercepts, intercept, domain and range. Interval indicates that the function is increasing and decreasing. Draw the graph. Function




The absolute value function is the absolute value of x. The domain is the set of real numbers and the path is zero and positive real numbers. Its graph is:




domain function

party functions party domain are functions that are composed of different equations for different parts of the domain. For example:



The graph of this function is:

















The domain is the set of real numbers except zero, expressed as interval is (- ¥, 0) E (0, ¥). The journey is the set of real numbers except -1 and 1 and real numbers between -1 and 1, ie (- ¥, -1) E (1, ¥). The open points (0, -1) and (0.1) indicates that the points do not belong to the graph of f. Due to the separation of the graph at x = 0, we say that f is discontinuous at x = 0. Radical function




The function is the square root function. Its graph is as follows:


Its domain is [0, ¥) and travel is [0, ¥).

What Is Ic Butal-apap-325-caff Tab

1.1.3 Examples of functions and their graphs
The graph of a function
The graph of a function is the set of points in the plane of the form (x, y) where x is in the domain of function and also y = f (x).
We discuss some important types of functions and observe their graphs. Pay attention to the manner in which the graphs of these functions. All examples are algebraic functions, we will discuss other types of functions such as trigonometric functions later. For now, observe the following functions and their graphs.
constant function: f (x) = k, where k is some constant

What all have in common the graphics? How do they differ?
linear function: f (x) = ax + b

What have in common the graphics? How do they differ?
quadratic function: f (x) = ax2 + bx + c = a (x - x0) 2 + y0
The red dot is called the vertex of the parabola.
What are your coordinates?
How are the coordinates of the vertices with the numbers in the form f (x) = a (x-x0) 2 + y0?
f (x) = x2 + 2 x + 1 = (x + 1) 2

The red dot is called the vertex of the parabola.
What are your coordinates?
How are the coordinates of the vertices with the numbers in the form f (x) = a (x-x0) 2 + y0?
f (x) = 2 x2 + x = (x + 1) 2 to 1

The red dot is called the vertex of the parabola.
What are your coordinates?
How are the coordinates of the vertices with the numbers in the form f (x) = a (x-x0) 2 + y0?
f (x) = 2 x - x2 = 1 - (x - 1) 2

The red dot is called the vertex of the parabola.
What are your coordinates?
How are the coordinates of the vertices with the numbers in the form f (x) = a (x-x0) 2 + y0?
What significance have the numbers, x0, y0 to the graph of the function f (x) = a (x-x0) 2 + y0?

f (x)
= 10 + 2 x - 2 x2 1
21




= - 2 [- (
) + x2] 2



2


polynomial function P (x) = x3 - 3x2 + 2x - 7
racionalUna function rational function is a quotient of two polynomials, f (x) = P (x) / Q (x) x + 4

f (x) = x2

- 16 What happens in x values \u200b\u200bwhere the denominator equals zero?
Power function: f (x) = k xnEn where k is any real constant and n is a real number.
For now we will restrict ourselves to rational exponents. Functions as xpi will discussed later. The domain of a power function depends on the exponent n.
f (x) = x-1


f (x) = x1 / 3


f (x) = x1 / 2


f (x) = x2 / 3



defined function by sections
is not necessary that a function is defined by a single formula. The matching rule may depend on what part of the domain from the independent variable.
In the following two graphs look at two examples of functions defined in sections.



f (x) = {
x2,
4 x, if 0
<= x <= 5
f (x) = {
-x2,
if x < 0
3,
if 0 <= x < 1
2 x - 1,
if x> = 1

Long Time Effect Syphilis

Quadratic Equations

Quadratic Equations

1 .- What is an equation?
2 .- What is a quadratic equation?
3 .- Solutions of a quadratic equation: Formula resolvent
4 .- Types of solutions: Real and imaginary
5 .- Examples. Verification solutions
6 .- Exercises quadratic equations are solved
1 .- What is an equation?
is an algebraic expression consisting of two separate members by an equal sign. One or both sides of the equation must have at least one variable or letter, call unknown. The equations become identities only for certain values \u200b\u200bof (s) unknown (s). These particular values \u200b\u200bare called solutions of the equation. Example:
equation: 3x - 8 = 10 only holds for X = 6, since if we substitute this value will be the identity equation: 10 = 10. So we say that X = 6 is the solution of the given equation. In fact, the only solution. If we used, for example, X = 2, be 2 = 10 (nonsense)
solve an equation is finding the values \u200b\u200bof X that satisfy through various mathematical techniques. If the equation is in first grade, a punt is the general procedure. If the degree of the equation is greater than one, should use other methods.
2 .- What is a quadratic equation?
is a particular type of equation where the unknown variable or is squared, that is, second degree. An example would be: 2x2 - 3x = 9. In this type of equation can not easily clear the X, therefore requires a general procedure to find solutions.
3 .- Solutions of a quadratic equation: Formula resolvent
The procedure is to perform changes in the algebraic equation of the quadratic equation: ax2 + bx + c = 0 until the X is clear. This procedure is not covered in this document. Solving a quadratic equation is called the resolvent formula:
The formula generates two responses: one with the + and one with the - sign before the root. Solving a quadratic equation is then limited, to identify the letters a, b and c and substitute their values \u200b\u200bin the formula solver.
It should be noted that, using the formula solver is a procedure that requires careful and requires extracting the square root of a number, either by calculator or manual process.
These difficulties make the mistake inexperienced student constantly on the solution. There are special procedures, applicable only to certain cases in which the roots can be found more easily and quickly. They are about factoring techniques.
4 .- Types of solutions: Real and imaginary
A quadratic equation can generate three types of solutions, also called roots, namely two distinct real roots

A real root (or two equal roots)
Two imaginary roots different
The approach distinguishes between these cases is the sign of the discriminant. Discriminant D is defined as:
D = b2 - 4.ac
If the discriminant is positive, then the square root is a real number and generates two distinct real roots
If the discriminant is zero, the result is zero, and both roots are the same number.
If the discriminant is negative, the square root is imaginary, producing two imaginary or complex roots. 5 .- Examples
. Verification solutions continación be resolved
A few examples will show all possible cases mentioned above. 5.1 .- Solving
- 5x2 + 13x + 6 = 0
points are identified, taking care of that equation is ordered respect to x, of degree greater than minor. With this condition we have: a = - 5, b = 13, c = 6. The following formula applies:

=

=


Since square roots are not usually stored, should be removed with a calculator, by trial or by the manual procedure. The desired root is 17, since the square of 17 is precisely 289. It is then that:
There are two different roots, one using the + sign and another sign -. Whether you call them X1 and X2 to the two solutions, which are:

Both values \u200b\u200bof x satisfy the equation, ie to replace it, produce an identity. Replacement procedure to test whether the values \u200b\u200bfound satisfy the equation is called the check is made.
Testing with x = 3. Result: -5. (3) 2 + 13. (3) + 6 = -45 + 39 + 6 = 0, as expected in the second member. Testing
X = -2 / 5, there is
Note that the fraction 20/25 was simplified to 4 / 5 before adding it to the other. As both responses produced identities, it is now certain that 2 and -2 / 5 are the roots of - 5x2 + 13x + 6 = 0
5.2 .- Solve: 6x - 2x = 9
letters can not be identified directly, since equation is messy and there is a zero on the right side of equality, therefore, make the necessary changes so that the equation is the desired shape. Transposing and changing place is: - x2 +6 x - 9 = 0. Now identify letters: a = -1, b = 6, c = -9, and applies the formula solver:
Note that the discriminant is zero, so there are two roots equal to 3, ie x1 = x2 = 3. Substituting the values \u200b\u200bin the original equation, it holds that: 6.3 - 32 = 18 - 9 = 9 thus has found the answer.
5.3 .- Solving:-6x + 13 = - x2
Again there to sort and transpose to obtain: x2-6x + 13 = 0; Identifying letters: a = 1, b = -6, c = 13. Applying the resolvent we have:
Oops! The discriminant is negative and no calculator will evaluate the square root of a negative number because this is a result which belongs to the complex numbers. Without going into details beyond the scope of this document, the root of -16 is 4i, where i is the basis of complex numbers or imagiarios, ie. The roots are then:
Separating the two answers, the solutions are: X1 = -3 + 2.i; X2 = -3 - 2.i. Require verification operations with complex numbers in rectangular form. It leaves the reader interested, investigate and verify.
6 .- Exercises quadratic equations are solved
The following exercises are approaches that generate a quadratic equation. Must first consider the logic of the problem, calling xa of the variables that the problem sets, but then must write the relations between the variables, according to approach and finally, solve the equation.
There is no general procedure for handling the logic of such problems, only experience will give the expertise to raise them. The interested reader can consult the book "Algebra" by Aurelio Baldor, considered by many as the bible of algebra.
6.1 .- The sum of two numbers is 10 and the sum of their squares is 58. Halle
both numbers are first assigned the variable x to one of the unknowns of the problem. There are two unknowns that are both numbers, as the problem does not distinguish between a and another, can be assigned either xa, eg
x = first number / / as the sum of both is 10, then necessarily the other will be:
10 - x = Second number
worth better explain this: If between your friend and you have Bs 1000, Bs and his friend is 400, how are you?, obviously, by subtracting the total minus 400, ie 1000 to 400 = B 600. If they have B x, the bill does not change, just do not know the value but in terms of x, ie you have 1000 - x
The final condition of the problem states that the sum of the squares of two numbers is 58 , then:
x2 + (10 - x) 2 = 58 This is the equation to solve
To solve it, apply some techniques of elementary algebra and then rearranged to apply the resolvent. The operation described in parentheses is the square of a binomial. It is a common misconception among students (very difficult to eradicate, by the way) to write: (a - b) 2 = a2 - b2, which is incorrect. The correct expression is: (a - b) 2 = a2 - b2 +
2.ab Developing the equation is: x2 + 102 - 2.10.x + x 2 = 58 => x2 + 100 - 20.x + x 2 = 58
Sorting and grouping: 2x2 - 20.x + 42 = 0, dividing by 2 whole equation: x2 - 10x + 21 = 0
Applying the resolvent is x1 = 3 and x2 = 7. The problem generated (apparently) two solutions, so you have to try both. Suppose we take the first (x = 3). Reviewing the initial approach, it appears that: First issue: x = 3, second number = 10 - 3 = 7.
If you take the second answer (x = 7), is: First issue: x = 7, second number = 10 - 7 = 3. In both cases, since there is no differentiation between the two numbers, the only answer is sought are numbers 3 and 7.
6.2 .- The length of a rectangular room is 3 meters longer than the width. If the width increases by 3 m and 2 m length increases, the area is doubled. Find the original floor area.
In this case, if no differentiation between long and wide, so be careful with the allocation and especially the interpretation of the variable x. This problem can easily be placed on the x either unknown, long or wide. Suppose that:
x = width of the room / / The length is 3 meters longer than the width, so:
x + 3 = length of the room. / / The area of \u200b\u200ba rectangle is the multiplication of both:
x. (X + 3) = area of \u200b\u200bthe room. Note that these are the initial data. The conditions of the problem
explain that the width increases by 3 meters and 2 meters long increases, so, after the increase are:
x + 3 = new width of the room
x + 5 = across the room again
(x + 3). (x + 5) = new living area
The new area is twice the first, so we propose the equation:
(x + 3). (x + 5) = 2. x. (X + 3)
multiplications are performed: x2 + 5x + 3x + 15 = 2x2 + 6x
It happens all the first member: x2 + 5x + 3x + 15 - 2x2 - 6x = 0
is simplified: - x2 + 2x + 15 = 0 This is the equation to solve. Solver is applied and is: x1 = 5 and x2 = - 3. The solution x = -3 is discarded, since x is the width of the room and can not be negative. Is taken as the only response that the original width was 5 meters. Looking at the initial conditions, it follows that the length is: x + 3 = 8 meters. So the original area was 8m.5m = 40 m2.
6.3 .- Find the area and perimeter of tríángulorectángulo shown. Dimensions are in meters

If the triangle is a right triangle, then it fulfills the Pythagorean Theorem: The square of the hypotenuse equals the sum of the squares of the legs. " The hypotenuse is the longest side (2x-5) and the other two are legs, it raises the equation:
(x + 3) 2 + (x - 4) 2 = (2x - 5) 2 each binomial Developing
squared, we have:
x2 + x2 + 32 + 2.3.x - 2.4.x + 42 = (2x) 2 - 2. (2x) 5 + 52 = x2 + 6x + 9 + x2 - 8x + 16 = 4x2 - 20x + 25
Regrouping: x2 + 6x + 9 + x2 - 8x + 16 - 4x2 + 20x - 25 = 0
Finally: -2 x2 + 18x = 0 This is the equation to solve
The roots of the equation are x1 = 0 and x2 = 9. The solution x = 0 is discarded, and then a leg would be -4 m, which is not possible. The solution is then x = 9. Thus, the triangle with equal sides is 12 meters and 5 meters and 13 meters hypotenuse. The area of \u200b\u200ba triangle is base times height over 2, the base and height are the two legs that are 90 °, so the area is A = 12. 5 / 2 = 30 m2. The perimeter is the sum of the sides, ie, P = 12 m + 5 m + 13 m = 30 m. Exercises
developed resolved and reviewed by: Carlos E. Utrera, Rev: June 2006

Wording Candyland Invitation

DYNAMICS IS ONE THAT IS THE RELATIONSHIP wedges DEVICES IN THREE DIMENSIONS ARE YOUR SPEED GROUND FORCE ACCELERATION AND ALSO OTHER WEIGHT

Work and Energy

Ikusu Otome Valkyrie 1

COMMENT WORK AND DYNAMIC CRITICAL ENERGY PHYSICS




Labour Dynamics and energy energíaTrabajo The simple pendulum The spring strength (I) The elastic spring (II) The elastic spring (III) particle attached to a rubber
Work and Energy
(the loop) The conical pendulum
Balance and stability (I)
Balance and stability (II)
Balance and stability (III)
Balance and Stability (IV)
Movement on a cycloid (I) Movement on hemispherical dome
Movement on
sup. Career semicircular two skiers

Movement on a cycloid (II)
Movement on a parable

Working Concept Concept kinetic energy
conservative force. Potential energy
Principle of conservation of energy
nonconservative forces

Energy Balance Concept work

work is called infinitesimal, the vector dot product of force by the displacement vector.
Where Ft is the component of force along the displacement, ds is the magnitude of the displacement vector dr, q the angle between the force vector with the vector displacement.
The total work along the path between points A and B is the sum of all infinitesimal
His work is the geometric mean area under the graph of the function which relates the tangential component of force Ft, and the shift s.
Example: Calculate the work required to stretch a spring 5 cm, if the spring constant is 1000 N / m.
The force required to deform a spring is F = 1000 ° x N, where x is the strain. The work of this force is calculated by integral
Triangle area the figure is (0.05 • 50) / 2 = 1.25 J
When the force is constant, the work is obtained by multiplying the force component along displacement by displacement.
W = ft • s
Example: Calculate the work
a constant force of 12 N, whose application point moves 7 m, if the angle between the directions of force and displacement are 0 º, 60 º, 90 º, 135 º, 180 º.
If the force and displacement have the same sense, the work is positive
If the force and displacement are in opposite directions, the work is negative
If the force is perpendicular to the displacement, the work is zero.

concept of kinetic energy
Suppose that F is the resultant of forces acting on a particle of mass m. The work of this force is equal to the difference between the actual value and the initial value of the kinetic energy of the particle.
In the first line we have applied Newton's second law, the tangential component of force equals mass times acceleration tangential.
In the second line at the tangential acceleration is equal to the derivative of the magnitude of the velocity, and the ratio between the displacement ds and dt the time it takes to move is equal to the velocity v of the mobile.
kinetic energy is defined as the expression
The work-energy theorem states that the work of the resultant of forces acting on a particle changes its kinetic energy.
Example: Find the speed with which a bullet exits after passing through a table of 7 cm thick and constant resistance opposed to F = 1800 N. The initial velocity of the bullet is 450 m / s and mass is 15 g.
The work force is F =- 0.07 126 -1800 · J
The final velocity v is

conservative force. potential energy
A force is conservative when the work of this force is equal to the difference between initial and final values \u200b\u200bof a function that only depends on the coordinates. In this function is called potential energy.
The work of a conservative force does not depend on the path taken to get from point A to point B.
The work of a conservative force along a closed path is zero. Example

acting on a particle force F = N
2xyi + x2j Calculate the work done by force along the closed path ABCA.
The curve AB is the segment of a parabola y = x2 / 3.
BC is the segment of the line through the points (0.1) and (3.3) and
CA is the portion of the Y axis that runs from the origin to the point (0.1)
infinitesimal work dW is the vector dot product of force by the displacement vector
dW = F × dr = (Fxi + f and j) · (dxi + D AND J) = FXDX + Fydy
The variables x and y are related through the equation of the trajectory y = f (x) and infinitesimal displacements dx and dy are related through the geometric interpretation of the derivative dy = f '(x) · dx. Where f '(x) means, derivative of the function f (x) with respect to x. We
to calculate the work in each of the sections and the total work on the closed road. Career Segment AB

y = x2 / 3, d = (2 / 3) x · dx. Section BC

The trajectory is the line through the points (0.1) and (3.3). This is a line of slope 2 / 3 and whose intercept is 1.
y = (2 / 3) x +1, dy = (2 / 3) · dx
Section CD
The trajectory is the line x = 0, dx = 0, the force F = 0 and therefore, work WCA = 0
The total work
WABC = WAB + WBC + WCA = 27 + (-27) +0 = 0
Weight is a conservative force
calculate the work force F =- mg weight j when the body moves from position A which is already ordered to position B whose ordinate is yB.
The potential energy Ep for the conservative force is a functional weight
where c is an additive constant that allows us to set the zero of potential energy.
The force of a spring is conservative
As shown in the figure when a spring is deformed x, exerts a force on the particle proportional to strain xy opposite to it.
For x> 0, F =- kx x
<0, F=kx
For work of this force is, when the particle moves from position to position xA xB is

Ep potential energy function for the conservative force F goes
The zero level potential energy is set as follows: when the strain is zero x = 0, the value of the potential energy is taken zero, Ep = 0, so that the true additive constant c = 0.

Principle of conservation of energy
If only a conservative force F acting on a particle, the work of this force is equal to the difference between the initial and final potential energy
As seen in the previous section The work of the resultant of forces acting on the particle is equal to the difference between the final and initial kinetic energy. Equating
both jobs, we get the expression of the principle energy conservation
EKA EKB + + AfL = EPB
The mechanical energy of the particle (sum of the kinetic potential energy) is constant at all points of his career.
Checking the principle of conservation of energy
A body of 2 kg is dropped from a height of 3 m.
calculating speed of the body when to 1 m high and when it reaches the ground, according to the formula of uniformly accelerated rectilinear motion
The total potential and kinetic energy in these positions
Take g = 10 m/s2
initial position x = 3 m, v = 0.
Ep = 2.10.3 = 60 J, Ek = 0, EA = Ek + Ep = 60 J
When x = 1 m
Ep = 2.10.1 = 20 J, Ek = 40, EB = Ek + Ep = 60 J
When x = 0 m
Ep = 2.10.0 = 0 J, Ek = 60, EC = Ek + Ep = 60 J
total energy of the body is constant. The potential energy decreases and kinetic energy increases.

non-conservative forces
To realize the significance of a nonconservative force, we will compare it with the conservative force weight.
Weight is a conservative force.
calculate the weight work force when the particle moves from A to B, then when moving from B to A.
WAB = WBA
mg x =- mg x
The total work along the closed road ABA, WABA is zero.
The force of friction is a nonconservative force
When the particle moves from A to B or from B to the frictional force is opposite to the motion, the work is negative because the force is opposite to
displacement WAB WBA
=- =- Fr x Fr x
The total work along the closed path ABA, is nonzero WABA WABA
2FR x =-

Energy balance
In general, a particle forces are conservative and nonconservative Fnc Fc. The work of the resultant of forces acting on the particle is equal to the difference between the final kinetic energy less than the initial.
The work of conservative forces is equal to the potential energy difference between initial and final
Applying the distributive property of the scalar product we obtain
The work of a nonconservative force changes the mechanical energy (kinetic plus potential) of the particle.
Example 1:
A block of mass 0.2 kg starts its upward movement in a plane 30 ° tilt, with an initial speed of 12 m / s. If the coefficient of friction between the block and the plane is 0.16. Determine the length x
that crosses the block along the plane until it comes to speed v
that will block the return to the base plane
When the body
ascends the inclined plane of the body
energy at A is EA = ½ 122 = 14.4 0.2 J
The energy of the body B is EB = 0.2 ° 9.8 ° h = 1.96 ⋅ h = 0.98 · x J
work friction force when the body moves from A to B is W =-
Fr · x =- μ · mg · cosθ · x =- 0.16 · 0.2 ° 9.8 ° cos30 · x =- 0.272 · x J
De the energy balance equation W = EB-EA, solve for x = 11.5 m, h = x · sen30 º = 5.75 m
When the body falls
The energy of the body B is EB = 0.2 ° 9.8 ° h = 1.96 • h = 0.98 · x = 0.98 5.11 = 28.11 J ·
The body's energy level at the base of 0.2 ° ½ EA == v2
work friction force when the body moves from B to A is
W =- Fr · x =- μ · mg · cosθ · x =- 0.16 · 0.2 ° 9.8 ° 11.5 cos30 =- 3.12 J ·
From equation energy balance W = EA-EB, we solve v = 3.9 m / s.
Example 2:
A particle of mass m slides on a surface in a quarter-circle of radius R, as shown in Fig.
The forces acting on the particle are: Weight mg

The reaction of the surface N, whose address is radial
The frictional force Fr, whose direction is tangential and whose meaning is opposite to the velocity of the particle. Breaking
mg weight along the normal and tangential direction, we write the equation of motion of the particle in the tangential direction
mat = mg · cosθ-Fr
Where t = dv / dt is the tangential component of acceleration. Write an equation of motion differential equation calculated
Wr work done by the force of friction. The frictional force is opposite to the displacement

Given that the deslazamiento is a small circular arc dl = R · dθ and
The work done by nonconservative force Fr
If the phone goes from rest v = 0, θ = 0 position. When it comes to the position θ
Energy Kinetic has increased mv2 / 2.
The potential energy has decreased mgRsenθ.
The work of the frictional force is equal to the energy difference between final and initial energy or the sum of the change in kinetic energy plus the change of potential energy.
The total work force of friction when the particle describes the quarter circle is

For an explicit calculation of the work of the frictional force see " Movement on a hemispherical dome with friction"

De

Dressy Outfits With Leggings And Boots

Dynamics Wikipedia, the free encyclopedia
(Redirected from Dynamic (Physical) )
Jump to: navigation , search
For musical sense, see Dynamics (music) .
The dynamic is part of physics that describes the evolution over time of a physical system in relation to the causes of changes physical and / or state of motion. The aim is to describe the dynamics of the factors that may cause alterations of a physical system, quantify and propose equations of motion or evolution equations for this system.
The study of the dynamics is prominent in the mechanical systems (classical, relativistic or quantum), but also thermodynamics and electrodynamics . In this paper we develop the main aspects of the dynamics of mechanical systems, leaving for other items to study the dynamics in non-mechanical. Content
[ hide] 1 History

2 Calculus 2.1
dynamic conservation laws
2.2 Equations of Motion
3 Dynamics of mechanical systems
3.1 Particle Dynamics
3.2 Dynamics Rigid body
4 Concepts related to the dynamic inertia

4.1 4.2 Work and Energy
5 See also
/ /

History [edit ]
The first major contribution is Galileo Galilei because . His experiments led to uniformly accelerated bodies Isaac Newton to formulate its fundamental laws of motion, which he presented in his major work Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy ") in 1687 .
Scientists believe current laws that made Newton give the correct answers to most of the problems of moving bodies, but there are exceptions. In particular, the equations to describe the motion are not adequate when a body travels at high speeds with regard to the speed of light or when objects are extremely small size comparable to molecular sizes.
Understanding the laws of classical dynamics has allowed man to determine the value and direction of the force to be applied to produce a particular movement or change in the body. For example, to make a rocket away from Earth, apply a certain force to overcome the force of gravity that attracts them in the same way, for a given load transport mechanism is necessary to apply adequate force in the right place.

dynamic Calculus [edit ]
Through the concepts of displacement , speed and acceleration is possible to describe the movements of a body or object without considering how they were made, discipline known for cinematic . By contrast, the dynamic is part of the mechanical that deals with the study the movement of bodies under the action of forces .
The dynamic analysis approach is based on the equations of motion of and integration. For extremely simple problems using equations of Newtonian mechanics directly aided the conservation laws .

conservation laws [edit ]
Main article:
conservation law Conservation laws can be formulated in terms of theorems that establish specific conditions under which a particular magnitude "is preserved" (Ie, remains constant in value over time as the system moves or changes over time). Besides the law of conservation of energy other important conservation laws take the form of theorems vector. These theorems are:
theorem of momentum, which for a system of point particles requires particle forces only depend on the distance between them and are directed along the line that connects them. In continuum mechanics and rigid body mechanics vector can be formulated theorems of momentum conservation.
The angular momentum theorem states that under conditions similar to the previous theorem vector sum of torques about an axis is equal to the temporal variation of angular momentum .

Equations of Motion [edit ]
Main article:
equation of motion there are several ways to raise equations of motion to predict the evolution in time of a mechanical system based on the initial conditions and acting forces. In classical mechanics there are several possible formulations to propose equations:
The that uses Newtonian mechanics to write directly ordinary differential equations of second order in terms of forces and in Cartesian coordinates. This system leads to difficult equations integrable by elementary means and is only used in very simple problems, usually using inertial reference systems. The Lagrangian mechanics
, this method also uses ordinary differential equations of second order, while allowing the use of fully general coordinates, called generalized coordinates , better adapted to the geometry of the problem. In addition the equations are valid in any reference frame it is inertial or not. Besides getting easier to integrate the systems Noether theorem and the coordinate transformations can find comprehensive movement, also called conservation laws, more easily than the Newtonian approach. The Hamiltonian mechanics
is similar to above but in the equations of motion are ordinary differential equations are first order. Moreover, the range of admissible coordinate transformations is much wider than in Lagrangian mechanics, which makes it even easier to find integrals of motion and conserved quantities.
The Hamilton-Jacobi method is a method based on the resolution of a partial differential equation using the method separation of variables, which is the easiest way when you know the right set of integrals of motion.

Dynamics of mechanical systems [ edit]
In physics there are two major types of physical systems finite systems of particles and fields . The evolution over time of the first can be described by a finite set of ordinary differential equations, why is said to have a finite number of degrees of freedom . In contrast, the evolution over time of the fields requires a set of complex equations. Partial, and in some informal sense behave like a particle system with an infinite number of degrees of freedom.
Most mechanical systems are the first type, although there are mechanical systems which are described more simply as fields, as with fluid or deformable solid . It also happens that some mechanical systems ideally formed by an infinite number of particles, such as rigid bodies can be described by a finite number of degrees of freedom.

particle dynamics [edit ]
Main article: Dynamics of material point
The dynamics of the particle is a part of Newtonian mechanics in which the systems are analyzed as systems of point particles and distance forces are exerted snapshots

rigid body dynamics [edit ]
Main article: rigid body mechanics
The mechanics of a rigid body is that which looks for movement and balance of solid material deformation ignoring their . It is, therefore, a mathematical model useful to study some of the mechanics of solids, since all real solids are deformable. Rigid means a set of points in space that move in such a way that does not alter the distances between them, whatever the force acting (mathematically, the motion of a rigid body is given by a group of isometries uniparametrical) .

Concepts related to the dynamics [edit ]

Inertia [edit ]
Main articles: inertia and inertial mass
Inertia is the difficulty or resistance opposing a physical system or a social system to change.
In physics we say that a system has more inertia when it is more difficult to change in the physical state of the same. The two most common uses in physics are the mechanical inertia and thermal inertia. The first one appears in mechanics and is a measure of difficulty to change the state of motion or rest of a body. The mechanical inertia depends on the amount of mass and inertia tensor. Thermal inertia measures the difficulty with which a body changes its temperature by contact with other bodies or to be heated. The thermal inertia depends on the amount of mass and heat capacity.
Calls inertial forces are fictitious forces and apparent that an observer in a non-inertial reference ....
Inertial mass is a measure of mass resistance to the change in speed relative to an inertial reference system. In classical physics the inertial mass of point particles is defined by the following equation, where a particle is taken as the unit (m1 = 1):
where I is the inertial mass of the particle i, and the acceleration ai1 i initial particle in the direction of the particle i to particle 1 in a volume occupied only by particles i 1, where two particles are initially at rest at a distance unit. No external forces but exert force particles into each other.

Work and Energy [edit ]
The work and energy displayed on the mechanics through energy theorems. The primary, and from which they derive the other theorems, the theorem energy. This theorem can be stated in differential version or full version. Henceforth we will refer to the Theorem of kinetic energy as TEC. Thanks to the TEC
can establish a relationship between mechanics and other