Dynamics Review

Question 1

particle

Answer 1

a point mass

Question 2

vector

Answer 2

a quantity having direction and magnitude.

a vector can change in time with both direction and magnitude.

Question 3

To describe the change with time, we can define the derivative of a vector A to be:

Answer 3

Question 4

“A vector dot” is tied to…

Answer 4

the frame of reference in which the derivative is taken.

a time derivative is completely dependent on the frame of reference.

Question 5

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?

Answer 5

Question 6

If F is the frame of reference, then the time derivavtive of A_vector w.r.t. F becomes…

Answer 6

Question 7

If we let {e1}_hat, {e2}_hat, and {e3}_hat are fixed in reference frame (are orthogonal basis), then

Answer 7

time rate of change of unit vectors are zero. Then left with:

Question 8

path

Answer 8

the locus of points of F that particle P occupies as t passes

Question 9

If O is the origin of F, then {r_op}_vector is

Answer 9

the position vector.

Question 10

The derivative of the position vector {r_op}_vector is

Answer 10

the velocity vector.

Question 11

The second derivative of the position vector {r_op}_vector is

Answer 11

the acceleration vector.

Question 12

The magnitude of {v_op}_vector is

Answer 12

the speed of particle P.

Question 13

Cylindrical coordinates are

Answer 13

radial, transverse, and unit vector k

Question 14

Compare the cylindrical and cartesian coordinates

Answer 14

{e_r}_hat and {e_theta}_hat are in the i-j plane.

theta goes from the cartesian i_vector to r_vector

Question 15

Define the cartesian {r_op}_vector

Answer 15

Question 16

Define the cylindrical {r_op}_vector

Answer 16

Question 17

Why is there no {e_theta}_hat term for the cylindrical r_vector equation?

Answer 17

Theta is implicit in the defition for {e_r}_hat.

Question 18

Define the cylindrical {v_op}_vector

Answer 18

Question 19

Decompose {e_r}_hat and {e_theta}_hat

Answer 19

Question 20

What is the time derivative of {e_r)_hat?

Answer 20

Note that {e_{<span>theta</span>})_hat is implicit in the definition of {e_{<span>r</span>})_hat

Question 21

Define the cylindrical coordinates for the acceleration vector

Answer 21

Question 22

What is the time derivative of {e_theta)_hat?

Answer 22

Question 23

What is the proof for:

Answer 23

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.

Question 24

Linear momentum of a particle is defined as…

Answer 24

the product of its mass and velocity

Question 25

Derivavtive of linear momentum and assumption...

Answer 25

assumption is that mass is not changing overtime (changes for the rocket problem) so m\_dot is zero

Question 26

Sum of all the forces acting on a particle is... **Hint:** Momentum Form of Euler's 1st Law

Answer 26

the time rate of change of the momentum of that particle.

Question 27

The _moment_ of force F exerted on a particle P out O is defined as

Answer 27

the position vector {r_op}\_vector cross Force

Question 28

Derivative of _moment_ of force F exerted on a particle P out O

Answer 28

Question 29

# Define _angular momentum_ or moment of momentum (Hint: time rate of change of angular momentum = moment)

Answer 29

Question 30

Drawing of relationship b/w vectors w.r.t. two different reference frames

Answer 30

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.

Question 31

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...

Answer 31

Question 32

Defining A\_vector w.r.t. body frame is

Answer 32

Question 33

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. )

Answer 33

the Transport Theorem or Basic Kinematic Equations (BKEs). Relates the derivative of *any* vector in two reference frames.

Question 34

If velocity relationship is in two frames (e.g. body and inertial), what is the velocity in the inertial frame?

Answer 34

Question 35

If relationship is in two frames (e.g. body and inertial), what is the acceleration in the inertial frame?

Answer 35

Question 36

Assuming no translation of O', take generaic expression for acceleration in the inertial frame and write the expression in cylindrical coordinates.

Answer 36