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|“||It is natural to begin this discussion by considering the various possible types of motion in themselves, leaving out of account for a time the causes to which the initiation of motion may be ascribed; this preliminary enquiry constitutes the science of Kinematics. — ET Whittaker||”|
Kinematics is not to be confused with another branch of classical mechanics: analytical dynamics (the study of the relationship between the motion of objects and its causes), sometimes subdivided into kinetics (the study of the relation between external forces and motion) and statics (the study of the relations in a system at equilibrium). Kinematics also differs from dynamics as used in modern-day physics to describe time-evolution of a system.
The term kinematics is less common today than in the past, but still has a role in physics. (See analytical dynamics for more detail on usage). The term "kinematics" also finds use in biomechanics and animal locomotion.
The simplest application of kinematics is for particle motion, translational or rotational. The next level of complexity is introduced by the introduction of rigid bodies, which are collections of particles having time invariant distances amongst themselves. Rigid bodies might undergo translation and rotation or a combination of both. A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints.
Some kinds of computational fluid dynamics use a (complicated) particle-based kinematic model of the fluid, but fluid flow is generally thought of using a continuum mechanics model rather than kinematics.
Linear or translational kinematics is the description of the motion in space of a point along a line, also known as trajectory or path.[note 1] This path can be either straight (rectilinear) or curved (curvilinear).
Particle kinematics is the study of the kinematics of a single particle. The results obtained in particle kinematics are used to study the kinematics of collection of particles, dynamics and in many other branches of mechanics.
The position of a point in space is the most fundamental idea in particle kinematics. To specify the position of a point, one must specify three things: the reference point (often called the origin), distance from the reference point and the direction in space in which the straight line from the reference point to the particle makes. Exclusion of any of these three parameters renders the description of position inconsistent. Consider for example a tower 50 m south from your home. The reference point is home, the distance 50 m and the direction south. If one only says that the tower is 50 m south, the natural question that arises is " from where?". If one says that the tower is southwards from your home, the question that arises is "how far?". If one says the tower is 50 m from your home, the question that arises is "in which direction?". Hence, all these three parameters are crucial to defining uniquely the position of a point in space.
Position is usually described by mathematical quantities that have all these three attributes: the most common are vectors and complex numbers. Usually, only vectors are used. For measurement of distances and directions, usually three dimensional coordinate systems are used the origin coinciding with the reference point. A three dimensional coordinate system (whose origin coincides with the reference point) with some provision for time measurement is called a reference frame or frame of reference or simply frame. All observations in physics are incomplete without the reference frame being specified.
The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three-dimensions, the position of point A can be expressed as
where xA, yA, and zA are the Cartesian coordinates of the point. The magnitude of the position vector |r| gives the distance between the point A and the origin.
The direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the position vector of a particle isn't unique. Relative to different frames of reference, the position vector of the same particle can be different.
Once the notion of position is firmly established, the ideas of rest and motion naturally follow. If the position vector of the particle (relative to a given reference frame) changes with time, then the particle is said to be in motion with respect to the chosen reference frame. However, if the position vector of the particle (relative to a given reference frame) remains same with time, then the particle is said to be at rest with respect to the chosen frame. Note that rest and motion are relative to the reference frame chosen. It is quite possible that a particle at rest relative to a particular reference frame is in motion relative to the other. Hence, rest and motion aren't absolute terms, rather they are dependent on the reference frame.
The path of locus of endpoints of the reference frame dependent. The path of a particle may be rectilinear (straight line) in one frame, and curved in another.
Displacement is a vector describing the difference in position between two points, i.e. it is the change in position the particle undergoes during the time interval. If point A has position rA = (xA,yA,zA) and point B has position rB = (xB,yB,zB), the displacement rAB of B from A is given by
Geometrically, displacement is the shortest distance between the points A and B. Displacement, distinct from position vector, is independent of the reference frame. This can be understood as follows: the positions of points is frame dependent, however, the shortest distance between any pair of points is invariant on translation from one frame to another (barring relativistic cases).
When considering the motion of a particle over time, distance is the length of the particle's path as opposed to displacement which is the change from its initial position to its final position. For example, a race car traversing a 10 km closed loop from start to finish travels a distance of 10 km; its displacement, however, is zero because it arrives back at its initial position.
If the position of the particle is known as a function of time (r = r(t)), the distance s it travels from time t1 to time t2 can be found by
The formula utilizes the fact that over an infinitesimal time interval, the magnitude of the displacement equals the distance covered in that interval. This is analogous to the geometric fact that infinitesimal arcs on a curved line coincide with the chord drawn between the ends of the arc itself.
Average velocity is defined as
where Δr is the change in displacement and Δt is the interval of time over which displacement changes.The direction ofis same as the direction of the displacement Δr as Δt>0.
Velocity is the measure of the rate of change in displacement with respect to time; that is, how the distance of a point changes with each instant of time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) can be defined as the limiting value of average velocity as the time interval Δt becomes smaller and smaller. Both Δr and Δt approach zero but the ratio approaches a non-zero limit v. That is,
where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.[note 2] As per its definition in the derivative form, velocity can be said to be the time rate of change of displacement. Further, as dr is tangential to the actual path, so is the velocity.
As position vector itself is frame dependent, velocity is also dependent on the reference frame.
The speed of an object is the magnitude |v| of its velocity. It is a scalar quantity:
The distance traveled by a particle over time is a non-decreasing quantity. Hence, ds/dt is non-negative, which implies that speed is also non-negative.
Average acceleration (acceleration over a length of time) is defined as:
where Δv is the change in velocity and Δt is the interval of time over which velocity changes.
Acceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as the limiting value of average acceleration as Δt becomes smaller and smaller. Under such a limit, → a.
where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time.
If the acceleration of a particle is zero, then the velocity of the particle is constant over time and the motion is said to be uniform. Otherwise, the motion is non-uniform.
If the acceleration is non-zero but constant, the motion is said to be motion with constant acceleration. On the other hand, if the acceleration is variable, the motion is called motion with variable acceleration. In motion with variable acceleration, the rate of change of acceleration is called the jerk
The above definitions can be inverted by mathematical integration to find:
Many physical situations can be modeled as constant-acceleration processes, such as projectile motion.
Integrating acceleration a with respect to time t gives the change in velocity. When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the integral relations can be simplified:
Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v0)/t,
By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.
A relationship without explicit time dependence may also be derived for one-dimensional motion. By setting r0 = 0 and noting that at = v − v0,
where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:
In the case of straight-line motion, r is parallel to a, and r has magnitude equal to the path length s = r − r0 at time t. Then
This relation is useful when time is not known explicitly.
To describe the motion of object A with respect to object B, when we know how each is moving with respect to a reference object O, we can use vector algebra. Choose an origin for reference, and let the positions of objects A, B, and O be denoted by rA, rB, and rO. Then the position of A relative to the reference object O is
Consequently, the position of A relative to B is
The above relative equation states that the motion of A relative to B is equal to the motion of A relative to O minus the motion of B relative to O. It may be easier to visualize this result if the terms are re-arranged:
or, in words, the motion of A relative to the reference is that of B plus the relative motion of A with respect to B. These relations between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations.
For example, let Ann move with velocity relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity ), the equation above gives:
To find we simply rearrange this equation to obtain:
|Example: Rectilinear (1D) motion|
Consider an object that is fired directly upwards and falls back to the ground so that its trajectory is contained in a straight line. If we adopt the convention that the upward direction is the positive direction, the object experiences a constant acceleration of approximately −9.81 m s−2. Therefore, its motion can be modeled with the equations governing uniformly accelerated motion.
For the sake of example, assume the object has an initial velocity of +50 m s−1. There are several interesting kinematic questions we can ask about the particle's motion:
To answer this question, we apply the formula
Since the question asks for the length of time between the object leaving the ground and hitting the ground on its fall, the displacement is zero.
There are two solutions: the first, t = 0, is trivial. The solution of interest is
In this case, we use the fact that the object has a velocity of zero at the apex of its trajectory. Therefore, the applicable equation is:
If the origin of our coordinate system is at the ground, then xi is zero. Then we solve for xf and substitute known values:
To answer this question, we use the fact that the object has an initial velocity of zero at the apex before it begins its descent. We can use the same equation we used for the last question, using the value of 127.55 m for xi.
Assuming this experiment were performed in a vacuum (negating drag effects), we find that the final and initial speeds are equal, a result which agrees with conservation of energy.
|Example: Projectile (2D) motion|
Suppose that an object is not fired vertically but is fired at an angle θ from the ground. The object will then follow a parabolic trajectory, and its horizontal motion can be modeled independently of its vertical motion. Assume that the object is fired at an initial velocity of 50 m s−1 and 30° from the horizontal.
The object experiences an acceleration of −9.81 m s−2 in the vertical direction and no acceleration in the horizontal direction. Therefore, the horizontal displacement is
Solving the equation requires finding t. This can be done by analyzing the motion in the vertical direction. If we impose that the vertical displacement is zero, we can use the same procedure we did for rectilinear motion to find t.
We now solve for t and substitute this expression into the original expression for horizontal displacement.
Note the use of the trigonometric identity 2sinθ cosθ = sin 2θ.
Rotational or angular kinematics is the description of the rotation of an object. The description of rotation requires some method for describing orientation, for example, the Euler angles. In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The z-axis has been chosen for convenience.
Description of rotation then involves these three quantities:
The angular velocity is represented in Figure 1 by a vector Ω pointing along the axis of rotation with magnitude ω and sense determined by the direction of rotation as given by the right-hand rule.
The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:
Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf are, respectively, the initial and final angular velocities, and α is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.
This example deals with a "point" object, by which is meant that complications due to rotation of the body itself about its own center of mass are ignored.
Displacement. An object in circular motion is located at a position r(t) given by:
where uR is a unit vector pointing outward from the axis of rotation toward the periphery of the circle of motion, located at a radius R from the axis.
Linear velocity. The velocity of the object is then
The magnitude of the unit vector uR (by definition) is fixed, so its time dependence is entirely due to its rotation with the radius to the object, that is,
where uθ is a unit vector perpendicular to uR pointing in the direction of rotation, ω(t) is the (possibly time varying) angular rate of rotation, and the symbol × denotes the vector cross product. The velocity is then:
The velocity therefore is tangential to the circular orbit of the object, pointing in the direction of rotation, and increasing in time if ω increases in time.
Linear acceleration. In the same manner, the acceleration of the object is defined as:
which shows a leading term aθ in the acceleration tangential to the orbit related to the angular acceleration of the object (supposing ω to vary in time) and a second term aR directed inward from the object toward the center of rotation, called the centripetal acceleration.
In any given situation, the most useful coordinates may be determined by constraints on the motion, or by the geometrical nature of the force causing or affecting the motion. Thus, to describe the motion of a bead constrained to move along a circular hoop, the most useful coordinate may be its angle on the hoop. Similarly, to describe the motion of a particle acted upon by a central force, the most useful coordinates may be polar coordinates. Polar coordinates are extended into three dimensions with either the spherical polar or cylindrical polar coordinate systems. These are most useful in systems exhibiting spherical or cylindrical symmetry respectively.
In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually i, j, k are unit vectors in the x-, y-, and z-directions.
The position vector, r, the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way:
This coordinate system expresses only planar motion. It is based on three orthogonal unit vectors: the vector i, and the vector j which form a basis for the plane in which the objects we are considering reside, and k about which rotation occurs. Unlike rectangular coordinates, which are measured relative to an origin that is fixed and non-rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied.
The position, velocity, and acceleration vectors of a given point can be expressed using these coordinate systems, but we have to be a bit more careful than we do with fixed frames of reference. Since the frame of reference is rotating, the unit vectors also rotate, and this rotation must be taken into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at angular rate ω in the counterclockwise direction (that is, Ω = ω k using the right hand rule) then the derivatives of the unit vectors are as follows:
Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.
Position is straightforward:
It is just the distance from the origin in the direction of each of the unit vectors.
Velocity is the time derivative of position:
By the product rule, this is:
Which from the identities above we know to be:
where vrel is the velocity of the particle relative to the rotating coordinate system.
Acceleration is the time derivative of velocity.
We know that:
Consider the part. has two parts we want to find the derivative of: the relative change in velocity (), and the change in the coordinate frame
Next, consider . Using the chain rule:
So all together:
And collecting terms:
A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:
An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass,
For the case of an object that does not tip or turn, this reduces to v = R ω.
This is the case where bodies are connected by some cord that remains in tension and cannot change length. The constraint is that the sum of all components of the cord is the total length, and accordingly the time derivative of this sum is zero. See Kelvin and Tait and Fogiel. A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. An equilibrium problem (not kinematic) of this type is the catenary.
The following concepts and definitions are based on Gurtin (1972) and Truesdell and Noll (1992). These definitions are useful both for the linear and the nonlinear theory of elasticity.
We usually denote a body by the symbol . A body is essentially a set of points in Euclidean space. For mathematical definition see Truesdell and Noll (1992)
A configuration of a body is denoted by the symbol . A configuration of a body is just what the name suggests. Sometimes a configuration is also referred to as a placement.
Mathematically, we can think of a configuration as a smooth one-to-one mapping of a body into a region of three-dimensional Euclidean space.
Thus, we can have a reference configuration and a current configuration .
A one-to-one mapping is also called a homeomorphism.
A deformation is the relationship between two configurations and is usually denoted by . Deformations include both volume and shape changes and rigid body motions.
For a continuous body, a deformation can be thought of as a smooth mapping from one configuration () to another (). The inverse mapping should be possible.
This means that
For the inverse mapping to exist, we require that the Jacobian of the deformation is positive, i.e., .
The deformation gradient is usually denoted by and is defined as
In index notation
For a deformation to be allowable, we must be able to invert . That is why we require that . Otherwise, the body may undergo deformations that are unphysical.
The displacement is usually denoted by the symbol .
The displacement is defined as a vector from the location of a material point in one configuration to the location of the same material point in another configuration.
The definition is
In index notation
The gradient of the displacement is denoted by .
The displacement gradient is given by
In index notation,
The finite strain tensor () is also called the Green-St. Venant Strain Tensor or the Lagrangian Strain Tensor.
This strain tensor is defined as
In index notation,
In the limit of small strains, the Lagrangian finite strain tensor reduces to the infinitesimal strain tensor ().
This strain tensor is defined as
In index notation,
Therefore we can see that the finite strain tensor and the infinitesimal strain tensor are related by
If , then
For small strains, and
For small deformation problems, in addition to small strains we can also have small rotations (). The infinitesimal rotation tensor is defined as
In index notation,
If is a skew-symmetric tensor, then for any vector we have
The vector is called the axial vector of the skew-symmetric tensor.
In our case, is the skew-symmetric infinitesimal rotation tensor. The corresponding axial vector is the rotation vector defined as
The change in volume () during a finite deformation is given by
The volume change during an infinitesimal deformation () is given by
The quantity is called the dilatation.
A volume change is isochoric (volume preserving) if
Let be a displacement field. The displacement gradient tensor is given by . Let the skew symmetric part of the displacement gradient tensor (infinitesimal rotation tensor) be
Let be the axial vector associated with the skew symmetric tensor . Show that
The axial vector of a skew-symmetric tensor satisfies the condition
for all vectors . In index notation (with respect to a Cartesian basis), we have
Since eijk = − eikj, we can write
Therefore, the relation between the components of and is
Multiplying both sides by epij, we get
Recall the identity
Using the above identity, we get
Now, the components of the tensor with respect to a Cartesian basis are given by
Therefore, we may write
Since the curl of a vector can be written in index notation as
where indicates the p-th component of the vector inside the square brackets.
Let be a displacement field. Let be the strain field (infinitesimal) corresponding to the displacement field and let be the corresponding infinitesimal rotation vector. Show that
The infinitesimal strain tensor is given by
Also recall that
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Kinematics is the branch of physics dealing with the motion of particles or bodies. It defines movement at the level of position, velocity and acceleration, without incorporating masses and forces.
A proper study of the subject requires an understanding of 2D and 3D vectors. Of course you also need to review basic algebra, trignometry and calculus. A proper understanding of kinematics require an understanding of vector calculus too.
Displacement is the change in position with respect to the frame of reference. In the case of a car moving on a road, the displacement is the change in position with respect to the road. This is important because the Earth itself is rotating and orbiting at considerable speed, which you do not notice due to inertia. Since omnipotence is not possible, all observations are made from a specific frame of reference. Displacement is usually expressed as a length measurement. Velocity is the change in displacement with respect to a change in time. The velocity of an object is again relative to the frame of reference. Acceleration is the change in velocity with respect to a change in time. Therefore, we know that, for functions for displacement, velocity, and acceleration, respectively d, v, t :
Using the above can create the basic kinematics formulas(subject to acceleration being constant):
Motion is by far the simplest application of Kinematics, and uses the formulas with no adaptation.
Gravity affects most all earthly motion, and therefore is discussed first. Gravity pulls all objects the same distance from the earth with equal acceleration, regardless of mass, as discovered by Galileo in his famous test on the Leaning Tower of Pisa. In this section we will assume that friction and air resistance are negligible. The Gravitational constant, 'g', is the is equal to 9.8ms-2, and is the acceleration done by the earth's gravity at sea level. Therefore, a falling object fall accelerating at rate g. When using this with the Kinematics formulas, g replaces a.
Show that the force (F) of a body is directly proportional to its momentum