KINETICS of PARTICLES
Newton’s 2nd Law &
The Equation of Motion
Lecture VIII
a
F m
 a
F m


Or

Newton’s 2nd Law &
The Equation of Motion
 Kinetics is the study of the relations between the unbalanced forces and the changes in
motion that they produce.
 Newton’s 2nd law states that the particle will accelerate when it is subjected to unbalanced
forces. The acceleration of the particle is always in the direction of the applied forces.
 Newton’s 2nd law is also known as the equation of motion.
 To solve the equation of motion, the choice of an appropriate coordinate systems depends on
the type of motion involved.
 Two types of problems are encountered when applying this equation:
 The acceleration of the particle is either specified or can be determined directly from
known kinematic conditions. Then, the corresponding forces, which are acting on the
particle, will be determined by direct substitution.
 The forces acting on the particle are specified, then the resulting motion will be
determined. Note that, if the forces are constant, the acceleration is also constant and is
easily found from the equation of motion. However, if the forces are functions of time,
position, or velocity, the equation of motion becomes a differential equation which must
be integrated to determine the velocity and displacement.
 In general, there are three general approaches to solve the equation of motion: the direct
application of Newton’s 2nd law, the use of the work & energy principles, and the impulse and
momentum method.

Newton’s 2nd Law &
The Equation of Motion (Cont.)
Kinetic
Diagram
Free-body
Diagram
=
ma
F2
F1
FR = F
P P
Note: The equation of motion has to be applied in such
way that the measurements of acceleration are made from
a Newtonian or inertial frame of reference. This coordinate
does not rotate and is either fixed or translates in a given
direction with a constant velocity (zero acceleration).

Newton’s 2nd Law &
The Equation of Motion (Cont.)
0




y
x
x
F
ma
F
2
2
2
2
F
j
i
F
a
j
i
a


















y
x
y
x
y
x
y
x
y
y
x
x
F
F
F
F
a
a
a
a
ma
F
ma
F
Rectilinear
Motion
Curvilinear
Motion
Rectangular
Coordinates
n-t
Coordinates
Polar
Coordinates
2
2
2
2
F
e
e
F
a
e
e
a


















n
t
n
n
t
t
n
t
n
n
t
t
n
n
t
t
F
F
F
F
a
a
a
a
ma
F
ma
F
2
2
2
2
F
e
e
F
a
e
e
a


























F
F
F
F
a
a
a
a
ma
F
ma
F
r
r
r
r
r
r
r
r

Newton’s 2nd Law &
The Equation of Motion
Exercises

Exercise # 1
3/1: The 50-kg crate is projected along the floor with an
initial speed of 7 m/s at x = 0. The coefficient of kinetic
friction is 0.40. Calculate the time required for the crate to
come to rest and the corresponding distance x traveled.

Exercise # 2
3/2: The 50-kg crate of Prob. 3/1 is now projected down an
incline as shown with an initial speed of 7 m/s. Investigate
the time t required for the crate to come to rest and the
corresponding distance x traveled if (a)  = 15° (b)  = 30°.

Exercise # 3
3/4: During a brake test, the rear-engine car is stopped from
an initial speed of 100 km/h in a distance of 50 m. If it is
known that all four wheels contribute equally to the braking
force, determine the braking force F at each wheel. Assume
a constant deceleration for the 1500-kg car.

Exercise # 4
3/17: The coefficient of static friction between the flat bed of
the truck and the crate it carries is 0.30. Determine the
minimum stopping distance s which the truck can have
from a speed of 70 km/h with constant deceleration if the
crate is not to slip forward.
Problem 3/17

Exercise # 5
3/54: The hollow tube is pivoted about a horizontal axis through
point O and is made to rotate in the vertical plane with a constant
counterclockwise angular velocity 
.
= 3 rad/s. If a 0.l-kg particle is
sliding in the tube toward O with a velocity of 1.2 m/s relative to the
tube when the position  = 30° is passed, calculate the magnitude N
of the normal force exerted by the wall of the tube on the particle at
this instant. Problem 3/54

Exercise # 6
3/71: A small object A is held against the vertical side of the rotating
cylindrical container of radius r by centrifugal action. If the coefficient
of static friction between the object and the container is ms, determine
the expression for the minimum rotational rate 
.
= w of the container
which will keep the object from slipping down the vertical side.