Dynamics Review Flashcards
particle
a point mass
vector
a quantity having direction and magnitude.
a vector can change in time with both direction and magnitude.
To describe the change with time, we can define the derivative of a vector A to be:
“A vector dot” is tied to…
the frame of reference in which the derivative is taken.
a time derivative is completely dependent on the frame of reference.
Let {e1}_hat, {e2}_hat {e3}_hat be mutually perpendicular unit vectors and A1, A2, and A3 are secular units of A_vector. What is A_vector?
If F is the frame of reference, then the time derivavtive of A_vector w.r.t. F becomes…
If we let {e1}_hat, {e2}_hat, and {e3}_hat are fixed in reference frame (are orthogonal basis), then
time rate of change of unit vectors are zero. Then left with:
path
the locus of points of F that particle P occupies as t passes
If O is the origin of F, then {rop}_vector is
the position vector.
The derivative of the position vector {rop}_vector is
the velocity vector.
The second derivative of the position vector {rop}_vector is
the acceleration vector.
The magnitude of {vop}_vector is
the speed of particle P.
Cylindrical coordinates are
radial, transverse, and unit vector k
Compare the cylindrical and cartesian coordinates
{er}_hat and {etheta}_hat are in the i-j plane.
theta goes from the cartesian i_vector to r_vector
Define the cartesian {rop}_vector
Define the cylindrical {rop}_vector
Why is there no {etheta}_hat term for the cylindrical r_vector equation?
Theta is implicit in the defition for {er}_hat.
Define the cylindrical {vop}_vector
Decompose {er}_hat and {etheta}_hat
What is the time derivative of {er)_hat?
Note that {e<span>theta</span>)_hat is implicit in the definition of {e<span>r</span>)_hat
Define the cylindrical coordinates for the acceleration vector
What is the time derivative of {etheta)_hat?
What is the proof for:
projection of A onto its derivative vector is zero (no projection), but if A and its derivavtive vector were not orthogonal, there would be some non-zero projection.
Linear momentum of a particle is defined as…
the product of its mass and velocity
Derivavtive of linear momentum and assumption…
assumption is that mass is not changing overtime (changes for the rocket problem) so m_dot is zero
Sum of all the forces acting on a particle is…
Hint: Momentum Form of Euler’s 1st Law
the time rate of change of the momentum of that particle.
The moment of force F exerted on a particle P out O is defined as
the position vector {rop}_vector cross Force
Derivative of moment of force F exerted on a particle P out O
Define angular momentum or moment of momentum
(Hint: time rate of change of angular momentum = moment)
Drawing of relationship b/w vectors w.r.t. two different reference frames
Ex. orbit defining w.r.t earth fixed frame and at some point want to describe in the intertial frame.
Doesn’t have to be body frame and inertial frames - can be any two frames.
Time rate of change of one’s constant unit vector should be orthogonal.
Differentiating in inertial frame, time rate of change of body frame {b1}_hat, {b2}_hat, and {b3}_hat is…
Defining A_vector w.r.t. body frame is
Differentiate A_vector in body frame w.r.t. time in inertial frame is…
(Note: can differentiate something in one frame even when defined in another frame. )
the Transport Theorem or Basic Kinematic Equations (BKEs).
Relates the derivative of any vector in two reference frames.
If velocity relationship is in two frames (e.g. body and inertial), what is the velocity in the inertial frame?
If relationship is in two frames (e.g. body and inertial), what is the acceleration in the inertial frame?
Assuming no translation of O’, take generaic expression for acceleration in the inertial frame and write the expression in cylindrical coordinates.
