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
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