kinematics n : the branch of mechanics concerned with motion without reference to force or mass
Kinematics (Greek κινειν,kinein, to move) is a branch of dynamics which describes the motion of objects without consideration of the circumstances leading to the motion. In contrast, kinetics is concerned with the forces and interactions that produce or affect the motion. The simplest application of kinematics is to point particle motion (translational kinematics or linear kinematics). The description of rotation (rotational kinematics or angular kinematics) is more complicated. The state of a generic rigid body may be described by combining both translational and rotational kinematics (rigid-body kinematics). A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints. The kinematic description of fluid flow is even more complicated, and not generally thought of in the context of kinematics.
Translational or curvilinear kinematics is the description of the motion in space of a point along a trajectory. This path can be linear, or curved as seen with projectile motion. There are three basic concepts that are required for understanding translational motion:
Displacement (denoted by r below )is the shortest distance between two points: the origin and the displaced point. The origin is (0,0) on a coordinate system that is defined by the observer. Because displacement has both magnitude (length) and direction, it is a vector whose initial point is the origin and terminal point is the displaced point.
Velocity (denoted by υ below) is the rate of change in displacement with respect to time; that is the displacement of a point changes with time. Velocity is also a vector. For a constant velocity, every unit of time adds the length of the velocity vector (in the same direction) to the displacement of the moving point. Instantaneous velocity (the velocity at an instant of time) is defined as
- \boldsymbol v = \frac \ ,
Acceleration (denoted by a below) is the rate of change in velocity with respect to time. Acceleration is also a vector. As with velocity if acceleration is constant, for every unit of time the length of the acceleration vector (in the same direction) is added to the velocity. If the change in velocity (a vector) is known, the acceleration is parallel to it. Instantaneous acceleration (the acceleration at an instant of time) is defined as:
- \boldsymbol a = \frac \ ,
- \boldsymbol a = \frac \ ,
Integral relationsThe above definitions can be inverted by integration to find:
- \boldsymbol(t) =\boldsymbol_0+\ \int_0^t \ dt' \ \boldsymbol
- \boldsymbol(t) =\boldsymbol_0+\ \int_0^t \ dt' \ \boldsymbol (t')
- =\boldsymbol_0+\boldsymbol_0\ t +\ \int_0^t \ dt' \int_0^ \ dt
\ \boldsymbol (t)
- =\boldsymbol_0+\boldsymbol_0\ t +\ \int_0^t \ dt' \left(t-t'\right) \ \boldsymbol (t') \ ,
- =\boldsymbol_0+\boldsymbol_0\ t +\ \int_0^t \ dt' \int_0^ \ dt \ \boldsymbol (t)
Constant accelerationWhen acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the above equations can be simplified:: \boldsymbol v = \int_0^ \boldsymbol a dt = \boldsymbol_0 + \boldsymbol a t Those who are familiar with calculus may recognize this as an initial value problem. Because acceleration ( a ) is a constant, integrating it with respect to time (t) gives a change in velocity. Adding this to the initial velocity ( υ0) gives the final velocity ( υ ).: \boldsymbol r = \int_0^t \boldsymbol v dt = \int_0^t \left( \boldsymbol v_0 + \boldsymbol a t \right) dt = \boldsymbol v_0 t + \frac \boldsymbol a t^2 Using the above formula, we can substitute for υ to arrive at this equation, where r is displacement.: \boldsymbol r = \frac t By using the definition of an average, this equation states that average velocity times time equals displacement. Using Eq. (1) to find υ−υ0 and multiplying by Eq. (3) we find a connection between the final velocity at time t and the displacement at that time:
- \boldsymbol t = \left( \boldsymbol v - \boldsymbol _0 \right)\boldsymbol \frac t \ ,
- v^2= v_0^2 + 2 a s\ ,
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
- \boldsymbol_ = \boldsymbol_ - \boldsymbol_ \,\!
- \boldsymbol_ = \boldsymbol_A - \boldsymbol_B = \boldsymbol_A - \boldsymbol_O - \left(\boldsymbol_B-\boldsymbol_O\right) = \boldsymbol_-\boldsymbol_ \ .
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:
- \boldsymbol_ = \boldsymbol_ + \boldsymbol_ \ ,
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 relation 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 \boldsymbol_ relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity \boldsymbol_, each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity \boldsymbol_), the equation above gives:
- \boldsymbol_ = \boldsymbol_ + \boldsymbol_ \,\! .
To find \boldsymbol_ we simply rearrange this equation to obtain:
- \boldsymbol_ = \boldsymbol_ -\boldsymbol_ \,\! .
At velocities comparable to the speed of light, these equations are not valid. They are replaced by equations derived from Einstein's theory of special relativity.
Rotational 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 for convenience chosen as the z-axis..
Description of rotation then involves these three quantities:
Angular position: The oriented distance from a selected origin on the rotational axis to a point of an object is a vector r ( t ) locating the point. The vector r ( t ) has some projection (or, equivalently, some component) r\perp ( t ) on a plane perpendicular to the axis of rotation. Then the angular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r\perp ( t ) in a known rotation sense (typically given by the right-hand rule).
Angular velocity: The angular velocity ω is the rate at which the angular position θ changes with respect to time t:
- \mathbf = \frac
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.
Angular acceleration: The magnitude of the angular acceleration \alpha is the rate at which the angular velocity \omega changes with respect to time t:
- \mathbf = \frac
The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:
- \,\!\theta_f - \theta_i = \omega_i t + \frac \alpha t^2 \qquad
\theta_f - \theta_i = \frac (\omega_f + \omega_i)t
- \,\!\omega_f = \omega_i + \alpha t \qquad \alpha = \frac \qquad \omega_f^2 = \omega_i^2 + 2 \alpha (\theta_f - \theta_i)\ .
Here \,\!\theta_i and \,\!\theta_f are, respectively, the initial and final angular positions, \,\!\omega_i and \,\!\omega_f are, respectively, the initial and final angular velocities, and \,\!\alpha 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.
Point object in circular motionThis 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:
- \boldsymbol (t) = R \mathbf_R (t) \ ,
Linear velocity. The velocity of the object is then
- \boldsymbol (t) =\frac \boldsymbol (t) = R \frac\mathbf_R (t) \ .
- \frac\mathbf_R (t) = \boldsymbol \mathbf_R = \omega (t) \mathbf_ \ ,
- \boldsymbol (t) = R\omega (t) \mathbf_ \ .
Linear acceleration. In the same manner, the acceleration of the object is defined as:
- \boldsymbol (t) = \frac \boldsymbol (t) = R\frac\omega\mathbf_ \
- =\mathbf_ R\frac\omega + R\omega\frac\mathbf_
- =\mathbf_ R\frac\omega + R\omega \boldsymbol\mathbf_
- =\mathbf_ R\frac\omega - \mathbf_\omega^2R\
- =\mathbf_ + \mathbf_R \ ,
- =\mathbf_ R\frac\omega + R\omega \boldsymbol\mathbf_
- =\mathbf_ R\frac\omega + R\omega\frac\mathbf_
Coordinate systemsIn 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.
Fixed rectangular coordinates
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 (or s), the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way:
- \boldsymbol = x\ \hat + y \ \hat + z \ \hat \, \!
- \boldsymbol = \dot = \dot \ \hat + \dot \ \hat + \dot \ \hat \, \!
- \boldsymbol = \ddot = \ddot \ \hat + \ddot \ \hat + \ddot \ \hat \, \!
Note: \dot = \frac , \ddot = \frac
Two dimensional rotating reference frameseealso Centripetal force 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.
Derivatives of unit vectorsThe 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:
- \dot = \omega \hat \times \hat = \omega\hat
- \dot = \omega \hat \times \hat = - \omega \hat
Position, velocity, and accelerationGiven these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.
PositionPosition is straightforward:
- \boldsymbol = x \ \hat + y \ \hat
It is just the distance from the origin in the direction of each of the unit vectors.
VelocityVelocity is the time derivative of position:
- \boldsymbol = \frac = \frac + \frac
By the product rule, this is:
- \boldsymbol = \dot x \ \hat + x \dot + \dot y \ \hat + y \dot
Which from the identities above we know to be:
- \boldsymbol = \dot x \ \hat + x \omega \ \hat + \dot y \ \hat - y \omega \ \hat = (\dot x - y \omega) \ \hat + (\dot y + x \omega) \ \hat
- \boldsymbol= (\dot x \ \hat + \dot y \ \hat ) + (y \dot + x \dot) = \boldsymbol_ + \boldsymbol \times \boldsymbol
where vrel is the velocity of the particle relative to the rotating coordinate system.
AccelerationAcceleration is the time derivative of velocity.
We know that:
- \boldsymbol = \frac \boldsymbol
Consider the \stackrel \boldsymbol_ part. \boldsymbol_ has two parts we want to find the derivative of: the relative change in velocity (\boldsymbol_), and the change in the coordinate frame (\boldsymbol \times \boldsymbol_).
- \frac = \boldsymbol_ + \boldsymbol \times \boldsymbol_
Next, consider \stackrel (\boldsymbol \times\boldsymbol). Using the chain rule:
- \frac = \dot \times \boldsymbol + \boldsymbol \times \dot
- \dot=\boldsymbol=\boldsymbol_ + \boldsymbol \times \boldsymbol from above:
- \frac =
So all together:
- \boldsymbol = \boldsymbol_ + \boldsymbol \times \boldsymbol_ +
And collecting terms:
- \boldsymbol = \boldsymbol_ + 2(\boldsymbol \times \boldsymbol_) +
Three dimensional rotating coordinate frame(to be written)
Kinematic constraintsA kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:
Rolling without slipping
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,
- \boldsymbol_G(t) = \boldsymbol \times \boldsymbol_.
Inextensible cordThis 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, however they are defined, is the total length, and the time derivative of this sum is zero.
References and notes
- Java applet of 1D kinematics
- Flash animated tutorial for 1D kinematics
- Physclips: Mechanics with animations and video clips from the University of New South Wales
- KINEMATICS FOR HIGH SCHOOL AND IIT JEE LEVEL
- KMODDL: Kinematic Models for Design Digital Library, Cornell University Library
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