Dynamics

Chapter 1Introduction to Dynamics

What is Dynamics?

Dynamics is the study of systems in which the motion of the object is changing (accelerating)• Dynamic – Changing

We will be exploring these motions within the context of several different analysis techniques:• Kinematics – position, velocity and acceleration• Forces• Energy• Momentum

221

r

MGMFg M1 and M2 – Masses of the two objects [kg]

G – Universal gravitational constant G = 6.67x10-11 N m2/kg2 or G = 3.439x10-8 ft4/(lb s4)r – distance separating the center of mass of the two objects [m]Fg – Gravitational force between the two objects [N]

This force is often very small unless you are using at least one very large mass!

rr

MGMFg

3

21

Newton’s Law of Gravitation

This is an example of Newton’s 3rd Law.

For objects near the surface of the Earth, r = RE and M2 = ME.

2E

E

R

GMga Wmg

R

GMmF

E

Eg

2

g = 9.81 m/s2 = 32.2 ft/sec2 near the surface of the Earth

Up until this point we have made the assumption that g is constant, but this is not true. It varies with altitude as we have discussed previously, but it also varies with latitude. This is because the Earth is not a perfect sphere and we have to take into account the variations in it’s radius.

We must also consider our reference frame when analyzing a dynamic system. Is the reference frame stationary or rotating with the system?

For all of our discussions of g so far we have assumed that the reference frame was fixed at the center of the Earth and therefore was stationary relative to the motion of the Earth. This is called absolute acceleration.

We can also examine the variations in g relative to a rotating reference frame, such as one attached to the surface of the Earth. This is called a relative acceleration. The following equation will enable you to determine the relative acceleration.

𝑔=9.780327 (1+0.005279 𝑠𝑖𝑛2𝛾+0.000023𝑠𝑖𝑛4𝛾+⋯ )𝑚𝑠2

Where g is the latitude at which the reference frame is attached.

Chapter 2Kinematics of Particles

Kinematics is the study of motion. It describes the motion of a particle without reference to forces. We typically use position, velocity and acceleration to describe the motion and link simultaneous motions in different coordinate directions through time.

The motion can be constrained (restricted to follow a specific path) or unconstrained (no path restrictions).

It is also important to choose the appropriate type of coordinate system for the path. Choosing an appropriate coordinate system can simplify the solution. We will primarily focus on the following common coordinate systems.

We will also explore the use of path variables, a reference frame defined relative to the path. Typical path variables used are normal components and tangential components.

We will discuss each of these coordinate systems in detail. Remember: any motion can be described in terms of each of these coordinate systems and can be converted between coordinate systems. Proper choice of coordinate system will simplify the solution to a problem.

• Rectangular/Cartesian (x, y, z) – best for motion along a straight line path

• Cylindrical (r, q, z) – best for motion along a curve

• Spherical (R, q, f) – best for motion along a curve

• Normal (n) – Normal (perpendicular) to the path at a specific point

• Tangential (t) – Tangential (parallel) to the path at a specific point

Rectilinear Motion

Rectilinear motion – used to describe motion along a path using rectangular coordinates.

A particle moves from one position (point P), whose location is known relative to a reference point (Point O), to a second position (point P’), whose position is also know relative to a known reference (point O). The particle changes positions through a distance Ds from a known location point P a distance s from O, and a second location point P’ a distance s’ from O. All of this motion occurs within a time interval Dt.

Ds is called the displacement. The distance traveled in a particular direction (+s direction).

The symbol s is used to describe the motion along the path and can correspond to any combination of coordinate directions.

Velocity and AccelerationVelocity – how fast you change positions

– Rate of change of position

Acceleration – how fast your velocity changes – Rate of change of velocity

𝑣=∆𝑠∆𝑡→𝑣= lim

∆ 𝑡→ 0

∆ 𝑠∆ 𝑡→v=

𝑑𝑠𝑑𝑡

v=𝑑𝑠𝑑𝑡

= �̇�

A symbol with a dot above it represents the time derivative of that quantity. The number of dots indicates the number of time derivatives.

𝑎=∆𝑣∆ 𝑡→𝑎= lim

∆ 𝑡→0

∆𝑣∆ 𝑡→a=

𝑑𝑣𝑑𝑡

a=𝑑𝑣𝑑𝑡

=�̇�=�̈�

We can also develop another useful equation by using time as a common quantity between velocity and acceleration.

v=𝑑𝑠𝑑𝑡

a=𝑑𝑣𝑑𝑡

→𝑑𝑡=𝑑𝑠𝑣

→adt=𝑑𝑣

→𝑑𝑣=𝑎𝑑𝑠𝑣

→𝑣𝑑𝑣=𝑎𝑑𝑠 𝑜𝑟 �̇� 𝑑 �̇�= �̈�𝑑𝑠

Graphical Interpretation

What can we obtain from figure (a)?

What can we obtain from figure (b)?

What can we obtain from figure (c)?

The slope of a tangent line at a point will provide information about the instantaneous velocity.

The slope of a tangent line at a point will provide information about the instantaneous acceleration.

The area under the curve for a segment of time dt will provide information about position.

The area under the curve for a segment of time dt will provide information about velocity.

What can we obtain from figure (a)?

What can we obtain from figure (b)?

The area under the curve for a segment of position ds will provide information about velocity.

𝑆𝑙𝑜𝑝𝑒𝑜𝑓 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒=𝑑𝑣𝑑𝑠

Using a similar right triangle we can also define the slope of the hypotenuse as

∴ 𝑑𝑣𝑑𝑠

=𝐶𝐵𝑣

→𝑣𝑑𝑣=𝐶𝐵𝑑𝑠

Comparing to requires that = a

Analytical IntegrationConstant Acceleration: – The equations are only valid if the acceleration is constant!

a=𝑑𝑣𝑑𝑡

𝑣𝑑𝑣=𝑎𝑑𝑠

𝑣=𝑑𝑠𝑑𝑡

For convenience let us look at the condition where t0 = 0 for each of the expressions above.

𝑣=𝑣0+𝑎𝑡

𝑣2=𝑣02+2a (𝑠−𝑠0 )

𝑠−𝑠0=𝑣0 𝑡+12𝑎𝑡2

These expressions should look familiar as the constant acceleration equations derived in your introductory physics course.

→∫𝑣0

𝑣

𝑑𝑣=𝑎∫𝑡 0

𝑡

𝑑𝑡→𝑣−𝑣0=𝑎 (𝑡− 𝑡0 ) →𝑣=𝑣0+𝑎 (𝑡− 𝑡0 )

→∫𝑣0

𝑣

𝑣 𝑑𝑣=𝑎∫𝑠0

𝑠

𝑑𝑠 →12

(𝑣2−𝑣02 )=𝑎 (𝑠−𝑠0 ) →𝑣2=𝑣02+2a (𝑠−𝑠0 )

→∫𝑠0

𝑠

𝑑𝑠=∫𝑡 0

𝑡

𝑣 𝑑𝑡 →∫𝑠0

𝑠

𝑑𝑠=∫𝑡 0

𝑡

[𝑣0+𝑎 (𝑡− 𝑡0 ) ]𝑑𝑡

→𝑠−𝑠0=𝑣0 ( 𝑡−𝑡 0 )+ 12𝑎 (𝑡2−𝑡 02 )−𝑎 𝑡0 (𝑡−𝑡 0 )