statics and dynamics

13

© Copyright 2008, S.B. Biggers, for Clemson University students currently in ME 201 only, no distribution

permitted without permission of author.

Chapter 1: Introduction to Newton’s Laws

1-1 Governing Equations for Dynamics and Statics

Prior to Sir Isaac Newton’s formulation of his three famous laws in 1687, scientists were not

completely successful in explaining the motion or lack of motion of bodies. Newton’s laws

allow us to be successful in both areas as long as the velocities are much less than the speed of

light and the bodies are much larger than individual atoms. In such extreme cases, Einstein’s

relativistic mechanics, published in 1905, and other complex theories of physics would have to

be used. Therefore, in the vast majority of cases, Newton’s laws serve perfectly well as the

foundation for applications of mechanics to practical situations. Newton’s second law states that

the sum of all forces acting on a body equals the product of its mass times its acceleration.

GF = m a (1.1)

This governing equation will be the starting point for our study of Dynamics. Later we will see

that the acceleration in this equation must be the acceleration of the center of gravity (CG) of the

body, thus the subscript “G” on the acceleration. This equation will be integrated in two special

ways to yield the governing equations of two additional methods, i.e. the work-energy method

(Chapter 9) and the impulse-momentum method (Chapter 10). When the body is restrained so

that it cannot move, or at least has a zero acceleration (i.e., constant velocity), Newton’s first law

is the result. This simplified version of the second law says simply that if the sum of all forces

acting on a body equals zero, the body is either (a) in static equilibrium, i.e. has no motion

at all, or (b) moves with constant velocity. In either case, equation (1.1) reduces to

F = 0 (1.2)

which is Newton’s first law. It is simply a special case of the second law and it forms the

starting point for our study of Statics. These two simple but important laws clearly show the

close relationship of Dynamics and Statics. These equations refer to lack of motion or motion

Statics

vs.

Dynamics

en.wikipedia.org/wiki/Image:

Space_Shuttle_Transit.jpg en.wikipedia.org/wiki/Image:

KSC-95EC-0911.jpg

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along a straight or curved line, motion referred to as translation, which we will define more

precisely later. We will show how to modify these equations to describe rotational motion or

lack of such rotation as follows:

G GM = I (1.3)

and

GM = 0 (1.4)

The terms in these equations will be defined later as required, but for now, equations (1.3 – 1.4)

can be interpreted as Newton’s Second and First Laws for rotational motion or lack of such

motion, respectively. The term IG represents the mass property that is a function of the size and

shape of the body. We will learn how to compute this property in Chapter 4. Motion involving

both translation and rotation is referred to as general motion and will require satisfaction of the

dynamic form of both the translational and rotational equations (1.1 and 1.3). Static

equilibrium implies the lack of all acceleration and therefore requires that the zero forms of

both equations (1.2 and 1.4) are satisfied. You will see that pure translation requires satisfaction

of the dynamic form of the translational equation (1.1) along with the zero form (static form) of

the rotational equation (1.4). Conversely, you will see that pure rotational motion about the

center of gravity of a rigid body requires satisfaction of equations (1.2 and 1.3).

Newton also developed a third law that is equally important in Dynamics and in Statics. This

states that when two separate bodies contact each other or attract each other, the

interacting forces are equal in magnitude and opposite in direction. You can easily feel the

static version of this by pushing two fingers together. The figure shows the pushing forces at the

point of contact. Each portion of each finger must react these forces to prevent motion. At the

top each finger feels only pushing all along its length. At the bottom, each finger feels pushing

and bending along its length. These cut-away-drawings of the fingers inside the circles are “free-

body-diagrams” (FBDs) which are the most important single tool that will be used in this course.

You may be surprised to learn that the foundation for this integrated course has now been stated

in terms of equations (1.1 – 1.4) comprising the static form (zero form) and the dynamic form

(nonzero form) of Newton’s Second Law for translation and rotation. We still need to define the

Newton’s 2nd

& 3rd

Laws: Equal & opposite contact forces, reactions to contact forces

F F F F

F F

F F

M M

photos: Biggers

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terms in the rotational equations. We also must define the physical point on the body known as

the “center of gravity” or “center of mass”. Along with Newton’s Third Law and the tool we call

FBDs, we are well on our way to being able to deal with rigid body mechanics.

In Dynamics the governing kinetics equations (1.1 and 1.3) introduced above will be

supplemented with additional equations of kinematics which describe the relationships

between translational and rotational motions and how they relate to time and position. In

Dynamics we will sometimes use integrated forms of Newton’s Second Law to formulate

problems in terms of Work-Energy and/or Impulse-Momentum principles. However, with

only a very few exceptions, Newton’s Second Laws for translation and rotation can be used,

along with kinematics, for any dynamic situation. In some cases, however, the other methods

may lead to simpler formulations and quicker solutions. The zero form of Newton’s Second

Laws may always be used for static analysis.

1-2 Force of Gravity

Newton formulated his universal law of gravitation when he discovered that any two masses

have equal and opposite attractive forces on each other. This is actually a quantitative version of

his third law. Our weight is simply the attractive force that the mass of the earth has on our body

mass. As discussed in the introduction, weight is a force and it will be measured in N or lb.

Newton’s law of gravitation states that the attractive force F

between two stationary masses m1 and m2 whose centers are a

distance r apart as shown at right is given by

(1.5)

where the gravitational constant G is given by

G = 6.673 x 10-11

m3/(kg-sec

2) in SI, and G = 3.439 x 10

-8 ft

4/(lb-sec

4) in USC.

If the attractive forces are not reacted by other forces, the masses will move toward each other

according to eq. (1.1). Which mass would move faster? If are in contact with each other,

Newton’s 3rd

Law would provide for a reaction force to prevent movement.

Since the earth is not spherical (radius larger at equator than poles), the distance r varies with

latitude, as well as elevation. Since the earth rotates, there is a slight tendency for bodies to be

thrown off the surface due to this rotation. This effect also varies from maximum at the equator

to minimum at the poles. Therefore, the total effective interactive force between any body and

the earth is a function of position (latitude and elevation). So the measured weight of any body

is maximized at the poles and minimized at the equator.

Since weight is also defined as mass times the so called “acceleration of gravity”, g, the

magnitude of g also varies in the same way as weight. The chart in Figure 1.1 summarizes the

acceleration of gravity at sea level at different locations on the earth according to the latest

F = (G) (m1 m2 )/( r2)

m1

F

m2

r

F

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International Gravitational Formula that

accounts for rotation and the shape of the

earth. If the earth did not rotate, the

values would be higher than these except

at the poles. Of course rotation is not a

factor at the poles. In this course, we

will always assume an average value of

g = 9.81 m/s2 in the SI system or 32.2

ft/s2 in the USC system, unless a

different value can be justified by

additional given information. Given the

values of g the poles (latitude=900) in

Figure 1.1 and the mass of the earth =

5.976 x 1024

kg, find the earth’s radius

at the poles as predicted by Newton. The

latest accepted value is 6,356,750 m.

1-3 Free Body Diagrams and Kinetic Diagrams

The most important step in formulating and solving dynamics and statics problems is to draw

simple sketches that represent each of the two sides of the equation stating Newton’s second

law. On the left hand side, we need to represent all forces acting on the body so they can be

summed. The sketch that shows the body and all known and unknown forces acting on it is

called the free body diagram (FBD). The earlier pictures of

fingers showed FBDs. The magnitude and direction of known

forces are shown. Unknown forces are shown with an assumed

direction and an identifying name. The term free simply

means the body has been cut free from all supports or

connections to other bodies and those unknown reaction or

connection forces are shown with names and directions as they

are known to act or are assumed to act on the body. If a

support prevents translation or rotation in a particular direction,

there is a reaction force or moment, respectively, opposite to

that direction. The kinetic diagram (KD) is drawn to

represent the right hand side of Newton’s second law when

modeling a dynamic condition. It shows the magnitude and

directions of known accelerations or the assumed directions

and identifying names of unknown accelerations. In static

equilibrium, the kinetic diagram is simply a zero and can be

omitted.

FBD – As an example for constructing FBDs, consider the

crane used to lift the large boat in and out of the water as

shown in Figure 1.2. The main cable wraps around a single

pulley. If you want to find the force in the main cable, you

T1 T2

FBD of boat,

frame, pulley www.boatsales.uk.net

12000 lb

0 0.26 0.52 0.79 1.05 1.31 1.579.77

9.78

9.79

9.8

9.82

9.83

9.849.84

9.77

g( )

2

0

0 15 30 45 60 75 90

(equator) Latitude, degrees (poles)

m/sec2

Figure 1.1 Effective “Acceleration”

(or Intensity) of Gravity.

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need to cut through the cable on each side of the pulley to expose the internal forces in the

cables. The FBD of the boat, pulley, and spreader frame after cutting them free from the crane is

shown. Note that if we only cut the main cable on one side of the pulley, we would not have

completely “freed” the body from the crane. The boat weighs 12,000 lb and its distributed

weight is replaced by its total weight concentrated at the center of gravity of the boat. Of course,

now you know that the weight is simply the attractive force the mass of the earth has on the mass

of the boat.

Let us simplify the model of the system by neglecting the comparatively small mass of the

spreader frame, cables, pulley, etc. If the system is static or if the pulley mass is neglected, the

tension at each cut in the main cable is the same, i.e. T1 = T2 acting upward. The observation

that the two tension forces are the same seems to be a trivial observation. However, this

observation will not become completely clear until we consider the dynamics of a rotating

pulley. In the dynamic case, the tensions will not be the same if the rotating pulley has

significant mass and is accelerating. There will be much more on this later.

We can also cut a free body of the pulley itself as shown below. Now the four tension forces T2

in the cables attached to the spreader frame appear in the new FBD. These forces all have the

same magnitude due to symmetry, but to get their direction and expression as vectors, the

geometry of the spreader frame is needed. We will return to this problem in Chapter 5 after

discussing ways to define vectors in 3-D space in the Chapter 2.

Next, we can cut a free body of only the spreader frame as shown.

Now the four equal (due to symmetry) tension forces T3 in the

nearly vertical straps surrounding the boat appear in the new FBD.

Finally, the FBD of the boat alone is shown in Figure 1.2 as

supported by the T3 forces in the straps.

Note that in all cases, tension forces “pull” on the body on which

they act. Internal forces on either side of a cut are always equal in

magnitude and opposite in direction (Newton’s Third Law). It is

very important to note that the internal forces are not shown on a

FBD unless there has been a cut made in the support (cable in this

instance) on that particular FBD. Then the internal force is shown

as it acts on the cut surface. There are many other FBDs that

could be cut and drawn from the original photo and later we will

do some of these. It is a good practice to actually indicate on the

original body exactly where the FBD is being cut free. This has

been done with the dashed colored lines on the photo in Figure

1.2. The FBD is only what is inside the dashed line and internal

forces are shown only where the dashed line cuts through a

supporting element. As your first Learning Exercise you will

practice drawing FBDs for a number of situations. You will be

expected to do this correctly as the first step in formulating the

governing equations using Newton’s Second Law for dynamics or

statics.

T1 T1

T2 T2 T2

T2

FBD of pulley

T2

T2 T2 T2

T3 T3 T3

T3

FBD of spreader

frame www.boatsales.uk.net

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After identifying the portion of the structure or machine that is to be modeled, it is a good

practice to actually encircle that region with a line to indicate where cuts must be made to free

the body. This is done in the example to follow. A complete FBD shows:

A simple sketch of the body that has been cut free and all forces acting on that particular

body including:

o The weight, located at the CG, with magnitude and direction shown

o The magnitude and direction of any other known forces tending to cause, restrict,

or prevent motion

o Any unknown forces in members that were cut to free the body but tending to

cause, restrict, or prevent motion – an assumed direction and a unique name

should be shown. They should always be shown at the actual location where they

are applied, not at some abstract point as you may did in your physics class.

A coordinate system identifying directions used in vector expressions

Important geometric information needed to solve the problem. If this information is too

extensive, sometimes it is best to show it on a separate diagram.

KD – The kinetic diagram is much simpler than the FBD. It represents the term Gm a on the

right hand side of Newton’s Second Law for forces and translation, and the term GI for

moments and rotation (to be examined later). Sometimes the KD is so simple students bypass

drawing it because it seems to be trivial. However, this tendency must be avoided in formulating

the governing equations for dynamics. Drawing the KD forces you to make a conscious decision

about which mass or masses you are considering, and which masses are actually accelerating.

You are also forced to make a first estimate or assumption of the direction(s) of the

acceleration(s) and to show the point (i.e. the CG) where the acceleration is measured. It aids in

making observations about relationships of motions of different points on the body and in

recording any assumptions about these accelerations.

A complete KD shows:

A simple sketch of the body under consideration, its mass, and the accelerations that

particular body may be experiencing including:

o The linear acceleration vector at the CG with the given, assumed, or observed

direction(s) shown and the vector or vector components named.

o The angular acceleration of the body with the given, assumed, or observed

direction(s) shown and the vector or vector component(s) named. In cases of 2-D

motion, only a vector component normal to the plane of motion will be present.

Any support or connection to another element that restricts the motion in any way, be that

zero motion or nonzero motion. (Note this is quite different from the FBD which must be

shown free from all supports, and shows forces not motion.)

Some students (and some texts) attempt to combine the FBD and KD into a single diagram.

This should never be done. Some examples of FBDs and KDs a successful student would

create for the boat lifting problem are shown in Figure 1.3. You will be expected to do this

correctly as the second step in formulating the governing equations using Newton’s Second

Law for dynamics.

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T1 T1

T2 T2 T2

T2

T2

T2 T2 T2

T3 T3 T3

T3

T1 T1

T1 T1

12000 lb

12000 lb

T3

T3

T3

T3

12000 lb

FBD of boat

FBD of spreader frame

FBD of pulley

FBD of boat,

frame, pulley

Figure 1.2 Free bodies cut from crane lifting boat. photo courtesy of: www.boatsales.uk.net

Coordinate System:

Right-Handed,

x-y is plane of motion,

applies to all diagrams.

y

x

z

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FBD of pulley

Figure 1.3 Student FBDs and KDs.

KD of pulley

KD of boat,

frame, pulley

FBD of boat,

frame, pulley

KD of boat FBD of boat

Coord

System

FBD of frame KD of frame

www.boatsales.uk.net

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EXAMPLE 1.1 – Statics and Dynamics of Lifting: If the boat is held by the crane but not

moved up or down, its velocity is a constant (i.e. zero) and therefore the vertical acceleration of

the mass of the boat is zero. In fact, if the crane is lifting the boat at a constant vertical speed,

the same can be said. The case of no motion at all is clearly a case of static equilibrium. The

constant speed case is referred to as quasi-static since the governing equation and its solution are

the same as for the truly static case. Therefore, Newton’s Second Law for the zero acceleration

in the vertical direction (upward assumed positive and shown by the arrow at left side) gives

12 12000lb 0yF T= and 1 6000lbT .

The fact that T1 is positive means that the assumed direction shown on the FBD is CORRECT.

Using the same directions for the FBD but assuming downward (note arrow) is positive gives

12 12000lb 0yF T= and 1 6000lbT .

Here again we find T1 is positive and this again means that the assumed direction shown on the

FBD is CORRECT. The direction selected for positive in writing the equation does not matter

but the direction shown on the FBD matters a lot. You should make it a habit to show the

positive direction used when writing an equation as was done above to avoid sign errors.

What would the two equations above have given if you had drawn the FBD assuming the

cable force was pushing down on the cut in the cable? How would you interpret the result for

T1?

Define in your own words what we mean by acceleration and deceleration.

As your first quantitative Learning Exercise, you will be asked to use the diagrams in Figure 1.2

and the guide given for problem solving to model lifting the boat as it is being lifted up or

lowered.

From these simple examples, you will easily see how closely static and dynamic conditions are

related. The importance of showing the directions of forces and accelerations or decelerations

in the FBD and KD and using these in the governing equations will also be evident. Hopefully,

the effects of the direction and magnitude of velocity vector and changes in the velocity vector

will also have been noted.

1-4 Types of Motion: Pure Translation

We have already dealt with translation in the example above since it is the simplest type of

motion. Motion in a straight line is the simplest type of translation. If the boat above is lifted or

lowered vertically with no swinging, it is undergoing straight line translation. For a more

general definition, we say the body is moving in pure translation when it moves such that all

points on it travel along paths that have the same length, shape, and orientation. Figures 1.4 and

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1.5 illustrate such motion, first along a straight line and then along a curved line. Envision a

book lying flat on a horizontal desk and being moved as shown. Even though the book is a 3-D

body, this motion is called planar translation since it can be fully defined by two position

coordinates, say x and y, at any given time during the motion. The position coordinate system

can be aligned with the sides of the desk as shown or in any other direction in the plane of the

desk. Each and every point on a body moves along an identical path when the body translates

along a straight line or curved line.

y

x

Figure 1.4 Planar translation of a body along a straight line in the x-y plane.

G G

y

x Figure 1.5 Planar translation of a body along a curved line in the x-y plane.

G

G

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As previously mentioned, in dynamics we will often be particularly interested in studying the

motion of the point defined as the center of gravity, CG. We will use the symbol G to represent

the CG. We used the CG of the boat to define the location of its total weight in the FBDs. This

very important point will be precisely defined in a subsequent section and methods to find it will

be developed. For the time being, just think of it as the point on which any body can be

balanced, regardless of the body’s orientation in space. For simple rectangular shaped bodies

with uniform density, the CG is simply the geometric center. Some textbooks study dynamics

and statics of a “particle”, a body whose size and shape are unimportant, separately from rigid

bodies. In this text, we concentrate on rigid bodies since all real bodies have some size and

shape. However, a rigid body in pure translation can be fully modeled by tracking the motion of

its CG, which is identical to defining the motion of a particle located at the same point. This is

exactly what we did in the boat example.

(1) How is the motion of the book shown in Figure 1.6 different from that in Figure 1.5 even

though the CG moves along the same path? (2) How would you classify this motion? (3)

Sketch the paths of motion of the corners of the book and compare these to the paths the

corners took in Figure 1.5. (4) What differences do you see? Suppose the size of the book is

extremely small, in fact negligibly small, compared to the path of motion. In this case one might

approximate the motion of the book as the motion of a single point or particle having the mass of

the book. The particle approximating the book would most reasonably be located at the point

defined as the CG. (5) If this assumption is to be made, how would you classify the motion of

the approximation of the book in Figure 1.6 as a particle? (6) How would it compare to the

motion in Figure 1.5 if the book is treated as a particle?

y

x

Figure 1.6 Planar motion of a body along a curved line in the x-y plane.

G G

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