CHAPTER 2

KINETICS OF A PARTICLE: FORCE AND ACCELERATION

2.1 Newton’s Laws of Motion

Today’s Objectives:Students will be able to:a) Write the equation of motion

for an accelerating body.b) Draw the free-body and

kinetic diagrams for an accelerating body.

In-Class Activities:• Applications• Newton’s laws of motion• Newton’s law of

gravitational attraction• Equation of motion for a

particle or system of particles

APPLICATIONS

The motion of an object depends on the forces acting on it.

Knowing the drag force, how can we determine the acceleration or velocity of the parachutist at any point in time?

A parachutist relies on the atmospheric drag resistance force to limit his velocity.

APPLICATIONS (continued)

A freight elevator is lifted using a motor attached to a cable and pulley system as shown.

How can we determine the tension force in the cable required to lift the elevator at a given acceleration?

Is the tension force in the cable greater than the weight of the elevator and its load?

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

MASS AND WEIGHT

It is important to understand the difference between the mass and weight of a body!

Mass is an absolute property of a body. It is independent of the gravitational field in which it is measured. The mass provides a measure of the resistance of a body to a change in velocity, as defined by Newton’s second law of motion (m = F/a).

The weight of a body is not absolute, since it depends on the gravitational field in which it is measured. Weight is defined as

W = mgwhere g is the acceleration due to gravity.

UNITS: SI SYSTEM VS. FPS SYSTEM

SI system: In the SI system of units, mass is a base unit and weight is a derived unit. Typically, mass is specified in kilograms (kg), and weight is calculated from W = mg. If the gravitational acceleration (g) is specified in units of m/s2, then the weight is expressed in newtons (N). On the earth’s surface, g can be taken as g = 9.81 m/s2.

W (N) = m (kg) g (m/s2) => N = kg·m/s2

FPS System: In the FPS system of units, weight is a base unit and mass is a derived unit. Weight is typically specified in pounds (lb), and mass is calculated from m = W/g. If g is specified in units of ft/s2, then the mass is expressed in slugs. On the earth’s surface, g is approximately 32.2 ft/s2.

m (slugs) = W (lb)/g (ft/s2) => slug = lb·s2/ft

2.2 The 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.

INERTIAL FRAME OF REFERENCE

This equation of motion is only valid if the acceleration is measured in a Newtonian or inertial frame of reference. What does this mean?

For problems concerned with motions at or near the earth’s surface, we typically assume our “inertial frame” to be fixed to the earth. We neglect any acceleration effects from the earth’s rotation.

For problems involving satellites or rockets, the inertial frame of reference is often fixed to the stars.

2.3 Equation of Motion for a System of Particles

The equation of motion can be extended to include systems of particles. This includes the motion of solids, liquids, or gas systems.

As in statics, there are internal forces and external forces acting on the system. What is the difference between them?

Using the definitions of m = mi as the total mass of all particles and aG as the acceleration of the center of mass G of the particles, then maG = miai .

The text shows the details, but for a system of particles: F = maG

where F is the sum of the external forces acting on the entire

system.

KEY POINTS

1) Newton’s second law is a “Law of Nature”--experimentally proven and not the result of an analytical proof.

2) Mass (property of an object) is a measure of the resistance to a change in velocity of the object.

3) Weight (a force) depends on the local gravitational field. Calculating the weight of an object is an application of F = ma, i.e., W = m g.

4) Unbalanced forces cause the acceleration of objects. This condition is fundamental to all dynamics problems.

PROCEDURE FOR THE APPLICATION OF THE EQUATION OF MOTION

1) Select a convenient inertial coordinate system. Rectangular, normal/tangential, or cylindrical coordinates may be used.

2) Draw a free-body diagram showing all external forces applied to the particle. Resolve forces into their appropriate components.

3) Draw the kinetic diagram, showing the particle’s inertial force, ma. Resolve this vector into its appropriate components.

4) Apply the equations of motion in their scalar component form and solve these equations for the unknowns.

EXAMPLE

Given: A crate of mass m is pulled by a cable attached to a truck. The coefficient of kinetic friction between the crate and road is k.

Find: Draw the free-body and kinetic diagrams of the crate.

Plan: 1) Define an inertial coordinate system.2) Draw the crate’s free-body diagram, showing all external forces applied to the crate in the proper directions.3) Draw the crate’s kinetic diagram, showing the inertial force vector ma in the proper direction.

EXAMPLE (continued)

1) An inertial x-y frame can be defined as fixed to the ground.Solution:

3) Draw the kinetic diagram of the crate:

The crate will be pulled to the right. The acceleration vector can be directed to the right if the truck is speeding up or to the left if it is slowing down.

2) Draw the free-body diagram of the crate:

The weight force (W) acts through the crate’s center of mass. T is the tension force in the cable. The normal force (N) is perpendicular to the surface. The friction force (F = uKN) acts in a direction opposite to the motion of the crate.

y

x

W = mgT

30°

NF = uKN

ma

2.4 Equations of Motion: Rectangular Coordinates

Today’s Objectives:Students will be able to apply Newton’s second law to determine forces and accelerations for particles in rectilinear motion.

In-Class Activities:

• Applications

• Equations of motion using

rectangular (Cartesian)

Coordinates

APPLICATIONS

If a man is pushing a 100 lb crate, how large a force F must he exert to start moving the crate?

What would you have to know before you could calculate the answer?

APPLICATIONS (continued)

Objects that move in any fluid have a drag force acting on them. This drag force is a function of velocity.

If the ship has an initial velocity vo and the magnitude of the opposing drag force at any instant is half the velocity, how long it would take for the ship to come to a stop if its engines stop?

EQUATION OF MOTION

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 .

PROCEDURE FOR ANALYSIS

• Free Body Diagram

Establish your coordinate system and draw the particle’s free body diagram showing only external forces. These external forces usually include the weight, normal forces, friction forces, and applied forces. Show the ‘ma’ vector (sometimes called the inertial force) on a separate diagram.

Make sure any friction forces act opposite to the direction of motion! If the particle is connected to an elastic spring, a spring force equal to ks should be included on the FBD.

PROCEDURE FOR ANALYSIS (continued)

• Equations of Motion

If the forces can be resolved directly from the free-body diagram (often the case in 2-D problems), use the scalar form of the equation of motion. In more complex cases (usually 3-D), a Cartesian vector is written for every force and a vector analysis is often best.

A Cartesian vector formulation of the second law is

F = ma or

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

Three scalar equations can be written from this vector equation. You may only need two equations if the motion is in 2-D.

PROCEDURE FOR ANALYSIS (continued)

The second law only provides solutions for forces and accelerations. If velocity or position have to be found, kinematics equations are used once the acceleration is found from the equation of motion.

• Kinematics

Any of the tools learned in Chapter 12 may be needed to solve a problem. Make sure you use consistent positive coordinate directions as used in the equation of motion part of the problem!

Plan: Since both forces and velocity are involved, this problem requires both the equation of motion and kinematics. First, draw free body diagrams of A and B. Apply the equation of motion . Using dependent motion equations, derive a relationship between aA and aB and use with the equation of motion formulas.

EXAMPLE

Given: WA = 10 lb WB = 20 lb voA = 2 ft/s k = 0.2

Find: vA when A has moved 4 feet.

EXAMPLE (continued)

Solution:

Free-body and kinetic diagrams of B: =

2T

WB mBaB

yy ma F

BBB amTW 2

BaT 2.3220220 (1)

Apply the equation of motion to B:

EXAMPLE (continued)

0 yy maF

lbWN A10

lbNFk 2

xx maF

AA amTF

AaT 2.32102 (2)

Apply the equations of motion to A:

=

WA

T

N

mAaA

F = mkN

Free-body and kinetic diagrams of A:y

x

EXAMPLE (continued)

Constraint equation:

sA + 2 sB = constant

or

vA + 2 vB = 0

Therefore

aA + 2 aB = 0

aA = -2 aB (3)

(Notice aA is considered

positive to the left and aB is

positive downward.)

A

B

sA

sB

Datums

Now consider the kinematics.

EXAMPLE (continued)

Now combine equations (1), (2), and (3).

lbT 33.7322

22 16.1716.17 sft

sftaA

Now use the kinematic equation:

)(222

oAAAoAA ssavv

)4)(16.17(22 22 Av

sftvA 9.11

2.5 Equations of Motion: Normal and Tangential Coordinates

Today’s Objectives:Students will be able to apply the equation of motion using normal and tangential coordinates.

In-Class Activities:• Applications• Equation of motion in n-t

coordinates

APPLICATIONS

Race tracks are often banked in the turns to reduce the frictional forces required to keep the cars from sliding at high speeds.

If the car’s maximum velocity and a minimum coefficient of friction between the tires and track are specified, how can we determine the minimum banking angle () required to prevent the car from sliding?

APPLICATIONS (continued)

Satellites are held in orbit around the earth by using the earth’s gravitational pull as the centripetal force – the force acting to change the direction of the satellite’s velocity.

Knowing the radius of orbit of the satellite, how can we determine the required speed of the satellite to maintain this orbit?

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/

EXAMPLE

Given:At the instant = 60°, the boy’s center of mass G is momentarily at rest. The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords.

Find: The boy’s speed and the tension in each of the two supporting cords of the swing when = 90°.

1) Since the problem involves a curved path and finding the force perpendicular to the path, use n-t coordinates. Draw the boy’s free-body and kinetic diagrams.

2) Apply the equation of motion in the n-t directions.3) Use kinematics to relate the boy’s acceleration to his speed.

Plan:

EXAMPLE (continued)

Solution:1) The n-t coordinate system can be established on the

boy at some arbitrary angle . Approximating the boy and seat together as a particle, the free-body and kinetic diagrams can be drawn.

T = tension in each cordW = weight of the boy

n

t

man

mat

Kinetic diagram

W

n

t

2T

Free-body diagram

=

EXAMPLE (continued)

2) Apply the equations of motion in the n-t directions.

Note that there are 2 equations and 3 unknowns (T, v, at). One more equation is needed.

Using an = v2/ = v2/10, w = 60 lb, and m = w/g = (60/32.2),

we get: 2T – 60 sin 60/32.2)(v2/10) (1)

(b) Ft = mat => W cos = mat

=> 60 cos = 60/32.2) at

Solving for at: at = 32.2 cos

(a) Fn = man => 2T – W sin = man

EXAMPLE (continued)

v dv = at ds where ds = d = 10 d3) Apply kinematics to relate at and v.

This v is the speed of the boy at = 90. This value can be substituted into equation (1) to solve for T.

2T – 60 sin(90°) = (60/32.2)(9.29)2/10

T = 38.0 lb (the tension in each cord)

=> v dv = 32.2 cos ds = 32.2 cos (10 d )

=> v dv = 32.2 cos d

=> = 32.2 sin => v = 9.29 ft/s

90

600

v

90

60

v2

2