An Overview of Mechanics

Statics: the study of bodies in equilibrium

Dynamics: the study of bodies in motion 1. Kinematics – concerned with the geometric aspects of motion

2. Kinetics - concerned with the

forces causing the motion

Mechanics: the study of how bodies react to forces acting on them

Rigid bodyParticle

KINEMATICS OF A PARTICLE

SUMMARY OF KINEMATIC RELATIONS:RECTILINEAR MOTION

• Differentiate position to get velocity and acceleration.

v = ds/dt ; a = dv/dt or a = v dv/ds

• Integrate acceleration for velocity and position.

• Note that so and vo represent the initial position and velocity of the particle at t = 0.

Velocity:

t

o

v

vo

dtadv s

s

v

v oo

dsadvvor t

o

s

so

dtvds

Position:

CONSTANT ACCELERATION

The three kinematic equations can be integrated for the special case

when acceleration is constant (a = ac) to obtain very useful equations.

A common example of constant acceleration is gravity; i.e., a body

freely falling toward earth. In this case, ac = g = 9.81 m/s2

downward. These equations are:

ta v v co

yieldst

oc

v

v

dtadvo

2coo

s

t(1/2)a t v s s yieldst

os

dtvdso

)s - (s2a )(v v oc

2o2 yieldss

sc

v

v oo

dsadvv

RECTANGULAR COMPONENTS: POSITION

It is often convenient to describe the motion of a particle in terms of its x, y, z or rectangular components, relative to a fixed frame of reference.

The position of the particle can be defined at any instant by the position vector

r = x i + y j + z k .

The x, y, z components may all be functions of time, i.e.,x = x(t), y = y(t), and z = z(t) .

The magnitude of the position vector is:The direction of r is defined by the unit vector: ur = (1/r)r

222 zyxr

RECTANGULAR COMPONENTS: VELOCITY

Since the unit vectors i, j, k are constant in magnitude and direction, this equation reduces to v = vxi + vyj + vzk

where vx = = dx/dt, vy = = dy/dt, vz = = dz/dtx y z•••

The velocity vector is the time derivative of the position vector:

v = dr/dt = d(xi)/dt + d(yj)/dt + d(zk)/dt

The magnitude of the velocity vector is

The direction of v is tangent to the path of motion.

222zyx vvvv

RECTANGULAR COMPONENTS: ACCELERATION

The direction of a is usually not tangent to the path of the particle.

The acceleration vector is the time derivative of the velocity vector (second derivative of the position vector):

a = dv/dt = d2r/dt2 = axi + ayj + azk

where ax = = = dvx /dt, ay = = = dvy /dt,

az = = = dvz /dt

vx x vy y

vz z

• •• ••

•••

•

The magnitude of the acceleration vector is 222zyx aaaa

KINEMATIC EQUATIONS: HORIZONTAL MOTION

Since ax = 0, the velocity in the horizontal direction remains constant (vx = vox) and the position in the x direction can be determined by:

x = xo + (vox)(t)

Projectiles

KINEMATIC EQUATIONS: VERTICAL MOTION

Since the positive y-axis is directed upward, ay = -g. Application of the constant acceleration equations yields:

vy = voy – g(t)

y = yo + (voy)(t) – ½g(t)2

vy2 = voy

2 – 2g(y – yo)

For any given problem, only two of these three equations can be used.

NORMAL AND TANGENTIAL COMPONENTS

When a particle moves along a curved path, it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used.

In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle).

The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion.The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.

NORMAL AND TANGENTIAL COMPONENTS (continued)

The positive n and t directions are defined by the unit vectors un and ut, respectively.

The center of curvature, O’, always lies on the concave side of the curve.The radius of curvature, , is defined as the perpendicular distance from the curve to the center of curvature at that point.

The position of the particle at any instant is defined by the distance, s, along the curve from a fixed reference point.

VELOCITY IN THE n-t COORDINATE SYSTEM

The velocity vector is always tangent to the path of motion (t-direction).

The magnitude is determined by taking the time derivative of the path function, s(t).

v = vut where v = s = ds/dt.

Here v defines the magnitude of the velocity (speed) andut defines the direction of the velocity vector.

ACCELERATION IN THE n-t COORDINATE SYSTEM

Acceleration is the time rate of change of velocity:a = dv/dt = d(vut)/dt = vut + vut

. .

Here v represents the change in the magnitude of velocity and ut represents the rate of change in the direction of ut.

..

. a = vut + (v2/)un = atut + anun.

After mathematical manipulation, the acceleration vector can be expressed as:

ACCELERATION IN THE n-t COORDINATE SYSTEM (continued)

There are two components to the acceleration vector:

a = at ut + an un

• The normal or centripetal component is always directed toward the center of curvature of the curve. an = v2/

• The tangential component is tangent to the curve and in the direction of increasing or decreasing velocity.

at = v or at ds = v dv.

• The magnitude of the acceleration vector is a = [(at)2 + (an)2]0.5

SPECIAL CASES OF MOTION

There are some special cases of motion to consider.

2) The particle moves along a curve at constant speed. at = v = 0 => a = an = v2/.

The normal component represents the time rate of change in the direction of the velocity.

1) The particle moves along a straight line. => an = v2/ a = at = v

.

The tangential component represents the time rate of change in the magnitude of the velocity.

SPECIAL CASES OF MOTION (continued)

3) The tangential component of acceleration is constant, at = (at)c.In this case,

s = so + vot + (1/2)(at)ct2

v = vo + (at)ct

v2 = (vo)2 + 2(at)c(s – so)

As before, so and vo are the initial position and velocity of the particle at t = 0. How are these equations related to projectile motion equations? Why?

4) The particle moves along a path expressed as y = f(x).The radius of curvature, at any point on the path can be calculated from

= ________________]3/2(dy/dx)21[ 2d2y/dx

POSITION (POLAR COORDINATES)

We can express the location of P in polar coordinates as r = rur. Note that the radial direction, r, extends outward from the fixed origin, O, and the transverse coordinate, is measured counter-clockwise (CCW) from the horizontal.

VELOCITY (POLAR COORDINATES)

The instantaneous velocity is defined as:v = dr/dt = d(rur)/dt

v = rur + rdur

dt.

Using the chain rule:dur/dt = (dur/d)(d/dt)

We can prove that dur/d = uθ so dur/dt = uθ

Therefore: v = rur + ruθ

Thus, the velocity vector has two components: r, called the radial component, and r called the transverse component. The speed of the particle at any given instant is the sum of the squares of both components or

v = (r 2r )2

ACCELERATION (POLAR COORDINATES)

After manipulation, the acceleration can be expressed as

a = (r – r2)ur + (r + 2r)uθ

.. . .. . .

.

The magnitude of acceleration is a = (r – r2)2 + (r + 2r)2

The term (r – r2) is the radial acceleration or ar.

The term (r + 2r) is the transverse acceleration or a

..

.. . .

. . ... ..

The instantaneous acceleration is defined as:

a = dv/dt = (d/dt)(rur + ruθ). .

CYLINDRICAL COORDINATES

If the particle P moves along a space curve, its position can be written as

rP = rur + zuz

Taking time derivatives and using the chain rule:

Velocity: vP = rur + ruθ + zuz

Acceleration: aP = (r – r2)ur + (r + 2r)uθ + zuz

.. . .. . .

. . .

..

DEPENDENT MOTION

In many kinematics problems, the motion of one object will depend on the motion of another object.

The blocks in this figure are connected by an inextensible cord wrapped around a pulley. If block A moves downward along the inclined plane, block B will move up the other incline.

The motion of each block can be related mathematically by defining position coordinates, sA and sB. Each coordinate axis is defined from a fixed point or datum line, measured positive along each plane in the direction of motion of each block.

DEPENDENT MOTION (continued)

In this example, position coordinates sA and sB can be defined from fixed datum lines extending from the center of the pulley along each incline to blocks A and B.

If the cord has a fixed length, the position coordinates sA and sB are related mathematically by the equation

sA + lCD + sB = lT

Here lT is the total cord length and lCD is the length of cord passing over arc CD on the pulley.

DEPENDENT MOTION (continued)

The negative sign indicates that as A moves down the incline (positive sA direction), B moves up the incline (negative sB direction).

Accelerations can be found by differentiating the velocity expression. Prove to yourself that aB = -aA .

The velocities of blocks A and B can be related by differentiating the position equation. Note that lCD and lT remain constant, so dlCD/dt = dlT/dt = 0

dsA/dt + dsB/dt = 0 => vB = -vA

KINETICS OF A PARTICLE

NEWTON’S LAWS OF MOTION

The motion of a particle is governed by Newton’s three laws of motion.

First Law: A particle originally at rest, or moving in a straight line at constant velocity, will remain in this state if the resultant force acting on the particle is zero.

Second Law: If the resultant force on the particle is not zero, the particle experiences an acceleration in the same direction as the resultant force. This acceleration has a magnitude proportional to the resultant force.

Third Law: Mutual forces of action and reaction between two particles are equal, opposite, and collinear.

NEWTON’S LAWS OF MOTION (continued)

The first and third laws were used in developing the concepts of statics. Newton’s second law forms the basis of the study of dynamics.

Mathematically, Newton’s second law of motion can be written

F = ma

where F is the resultant unbalanced force acting on the particle, and a is the acceleration of the particle. The positive scalar m is called the mass of the particle.

Newton’s second law cannot be used when the particle’s speed approaches the speed of light, or if the size of the particle is extremely small (~ size of an atom).

NEWTON’S LAW OF GRAVITATIONAL ATTRACTION

F = G(m1m2/r2)

where F = force of attraction between the two bodies, G = universal constant of gravitation ,

m1, m2 = mass of each body, and r = distance between centers of the two bodies.

When near the surface of the earth, the only gravitational force having any sizable magnitude is that between the earth and the body. This force is called the weight of the body.

Any two particles or bodies have a mutually attractive gravitational force acting between them. Newton postulated the law governing this gravitational force as

EQUATION OF MOTION

The motion of a particle is governed by Newton’s second law, relating the unbalanced forces on a particle to its acceleration. If more than one force acts on the particle, the equation of motion can be written

F = FR = ma

where FR is the resultant force, which is a vector summation of all the forces.

To illustrate the equation, consider a particle acted on by two forces.

First, draw the particle’s free-body diagram, showing all forces acting on the particle. Next, draw the kinetic diagram, showing the inertial force ma acting in the same direction as the resultant force FR.

The equation of motion, F = m a, is best used when the problem requires finding forces (especially forces perpendicular to the path), accelerations, velocities or mass. Remember, unbalanced forces cause acceleration!

Three scalar equations can be written from this vector equation. The equation of motion, being a vector equation, may be expressed in terms of its three components in the Cartesian (rectangular) coordinate system as

F = ma or Fx i + Fy j + Fz k = m(ax i + ay j + az k)

or, as scalar equations, Fx = max , Fy = may , and Fz = maz .

EQUATIONS OF MOTION: RECTANGULAR COORDINATES

NORMAL & TANGENTIAL COORDINATES

When a particle moves along a curved path, it may be more convenient to write the equation of motion in terms of normal and tangential coordinates.

The normal direction (n) always points toward the path’s center of curvature. In a circle, the center of curvature is the center of the circle.

The tangential direction (t) is tangent to the path, usually set as positive in the direction of motion of the particle.

EQUATIONS OF MOTION

This vector equation will be satisfied provided the individual components on each side of the equation are equal, resulting in the two scalar equations: Ft = mat and Fn = man .

Here Ft & Fn are the sums of the force components acting in the t & n directions, respectively.

Since the equation of motion is a vector equation , F = ma,it may be written in terms of the n & t coordinates as

Ftut + Fnun = mat + man

Since there is no motion in the binormal (b) direction, we can also write Fb = 0.

NORMAL AND TANGENTIAL ACCERLERATIONS

The tangential acceleration, at = dv/dt, represents the time rate of change in the magnitude of the velocity. Depending on the direction of Ft, the particle’s speed will either be increasing or decreasing.

The normal acceleration, an = v2/, represents the time rate of change in the direction of the velocity vector. Remember, an always acts toward the path’s center of curvature. Thus, Fn will always be directed toward the center of the path.

Recall, if the path of motion is defined as y = f(x), the radius of curvature at any point can be obtained from

=[1 + ( )2]3/2

dydx

d2ydx2

SOLVING PROBLEMS WITH n-t COORDINATES

• Use n-t coordinates when a particle is moving along a known, curved path.

• Establish the n-t coordinate system on the particle.

• Draw free-body and kinetic diagrams of the particle. The normal acceleration (an) always acts “inward” (the positive n-direction). The tangential acceleration (at) may act in either the positive or negative t direction.

• Apply the equations of motion in scalar form and solve.

• It may be necessary to employ the kinematic relations:

at = dv/dt = v dv/ds an = v2/

EQUATIONS OF MOTION:CYLINDRICAL COORDINATES

This approach to solving problems has some external similarity to the normal & tangential method just studied. However, the path may be more complex or the problem may have other attributes that make it desirable to use cylindrical coordinates.

Equilibrium equations or “Equations of Motion” in cylindrical coordinates (using r, , and z coordinates) may be expressed in scalar form as:

Fr = mar = m(r – r2)

F = ma = m(r – 2r)

Fz = maz = mz

EQUATIONS OF MOTION (continued)

Note that a fixed coordinate system is used, not a “body-centered” system as used in the n – t approach.

If the particle is constrained to move only in the r – plane (i.e., the z coordinate is constant), then only the first two equations are used (as shown below). The coordinate system in such a case becomes a polar coordinate system. In this case, the path is only a function of

Fr = mar = m(r – r2)

F = ma = m(r – 2r)

TANGENTIAL AND NORMAL FORCES

If a force P causes the particle to move along a path defined by r = f (), the normal force N exerted by the path on the particle is always perpendicular to the path’s tangent. The frictional force F always acts along the tangent in the opposite direction of motion. The directions of N and F can be specified relative to the radial coordinate by using angle .

DETERMINATION OF ANGLE

The angle defined as the angle between the extended radial line and the tangent to the curve, can be required to solve some problems. It can be determined from the following relationship.

ddrr

drr d tan

If is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise.

KINETICS OF A PARTCLE

WORK AND ENERGY

WORK AND ENERGY

Another equation for working kinetics problems involving particles can be derived by integrating the equation of motion (F = ma) with respect to displacement.

This principle is useful for solving problems that involve force, velocity, and displacement. It can also be used to explore the concept of power.

By substituting at = v (dv/ds) into Ft = mat, the result is integrated to yield an equation known as the principle of work and energy.

To use this principle, we must first understand how to calculate the work of a force.

WORK OF A FORCE

A force does work on a particle when the particle undergoes a displacement along the line of action of the force.

Work is defined as the product of force and displacement components acting in the same direction. So, if the angle between the force and displacement vector is , the increment of work dU done by the force is

dU = F ds cos

By using the definition of the dot product and integrating, the total work can be written as

r2

r1

U1-2 = F • dr

WORK OF A FORCE (continued)

Work is positive if the force and the movement are in the same direction. If they are opposing, then the work is negative. If the force and the displacement directions are perpendicular, the work is zero.

If F is a function of position (a common case) this becomes

s2

s1

F cos dsU1-2

If both F and are constant (F = Fc), this equation further simplifies to

U1-2 = Fc cos s2 - s1)

WORK OF A WEIGHT

The work done by the gravitational force acting on a particle (or weight of an object) can be calculated by using

The work of a weight is the product of the magnitude of the particle’s weight and its vertical displacement. If y is upward, the work is negative since the weight force always acts downward.

- W (y2 - y1) = - W y- W dy = U1-2 = y2

y1

WORK OF A SPRING FORCE

When stretched, a linear elastic spring develops a force of magnitude Fs = ks, where k is the spring stiffness and s is the displacement from the unstretched position.

If a particle is attached to the spring, the force Fs exerted on the particle is opposite to that exerted on the spring. Thus, the work done on the particle by the spring force will be negative or

U1-2 = – [ 0.5k (s2)2 – 0.5k (s1)2 ] .

The work of the spring force moving from position s1 to position s2 is

= 0.5k(s2)2 - 0.5k(s1)2k s dsFs dsU1-2

s2

s1

s2

s1

SPRING FORCES

1. The equations just shown are for linear springs only! Recall that a linear spring develops a force according toF = ks (essentially the equation of a line).

3. Always double check the sign of the spring work after calculating it. It is positive work if the force put on the object by the spring and the movement are in the same direction.

2. The work of a spring is not just spring force times distance at some point, i.e., (ksi)(si). Beware, this is a trap that students often fall into!

It is important to note the following about spring forces:

KINEMATICS OF A PARTICLE

Conservative Forces and Potential Energy

CONSERVATIVE FORCE

A force F is said to be conservative if the work done is independent of the path followed by the force acting on a particle as it moves from A to B. In other words, the work done by the force F in a closed path (i.e., from A to B and then back to A) equals zero.

This means the work is conserved.

rF 0· d

x

y

z

A

BF

A conservative force depends only on the position of the particle, and is independent of its velocity or acceleration.

CONSERVATIVE FORCE (continued)

A more rigorous definition of a conservative force makes use of a potential function (V) and partial differential calculus, as explained in the texts. However, even without the use of the these mathematical relationships, much can be understood and accomplished.

The “conservative” potential energy of a particle/system is typically written using the potential function V. There are two major components to V commonly encountered in mechanical systems, the potential energy from gravity and the potential energy from springs or other elastic elements.

V total = V gravity + V springs

POTENTIAL ENERGY

Potential energy is a measure of the amount of work a conservative force will do when it changes position.

In general, for any conservative force system, we can define the potential function (V) as a function of position. The work done by conservative forces as the particle moves equals the change in the value of the potential function (the sum of Vgravity and Vsprings).

It is important to become familiar with the two types of potential energy and how to calculate their magnitudes.

POTENTIAL ENERGY DUE TO GRAVITY

The potential function (formula) for a gravitational force, e.g., weight (W = mg), is the force multiplied by its elevation from a datum. The datum can be defined at any convenient location.

+ W yVg _

Vg is positive if y is

above the datum and negative if y is below the datum. Remember, YOU get to set the datum.

ELASTIC POTENTIAL ENERGY

Recall that the force of an elastic spring is F = ks. It is important to realize that the potential energy of a spring, while it looks similar, is a different formula.

Notice that the potential function Ve always yields positive energy.

2

21

ksVe

Ve (where ‘e’ denotes an elastic spring) has the distance “s” raised to a power (the result of an integration) or

CONSERVATION OF ENERGY

When a particle is acted upon by a system of conservative forces, the work done by these forces is conserved and the sum of kinetic energy and potential energy remains constant. In other words, as the particle moves, kinetic energy is converted to potential energy and vice versa. This principle is called the principle of conservation of energy and is expressed as

2211 VTVT = Constant

T1 stands for the kinetic energy at state 1 and V1 is the potential energy function for state 1. T2 and V2 represent these energy states at state 2. Recall, the kinetic energy is defined as T = ½ mv2.