ME-247

Slide-1
• Text Books
• 1. Vector Mechanics for Engineers: Dynamics – Ferdinand P.
Beer, E Russell Jr. Johnston (10th Edition)
• 2. Engineering Mechanics Dynamics – R.C. Hibbeler. (14th
Edition)

• Syllabus
• Kinematics of particles; Kinetics of particles: Newton’s second
law; energy and momentum method; System of particles;
Kinematics of rigid bodies; Plane motion of rigid bodies: forces
and acceleration; Energy and momentum methods; Velocity and
acceleration in mechanism.
 Mechanics

Rigid Bodies Deformable Bodies Fluids

Statics Dynamics

Kinematics Kinetics
• Kinematics : It is the study of the motion of the bodies
(displacement, velocity, acceleration and time) without reference
which causes the motion (i.e regardless of forces)

• Kinetics : It is the study of bodies with reference to the force

which cause the motion. It is study of the relationship between
the forces acting on a body, the mass of the body and the
motion of the body.
• Statics
An object will remain in its original state of motion (rest or moving
at constant velocity in a straight line) if there is no unbalanced
force acting on it

𝑭𝑹 = 𝟎
• Kinematics : It is the study of the motion of the bodies
(displacement, velocity, acceleration and time) without reference
which causes the motion (i.e regardless of forces)
• Kinetics : It is the study of bodies with reference to the force
which cause the motion. It is study of the relationship between
the forces acting on a body, the mass of the body and the
motion of the body.
• It is concerned with how motion is resulted from external forces
acting on the object.

𝑭𝑹 = ma
 𝒅𝒔 𝒅𝒗
𝒗= a=
𝒅𝒕 𝒅𝒕

𝒅𝟐 𝒕
a= F=ma
𝒅𝒕𝟐

ads= vdv s=rθ

 𝑑𝑠 𝑑𝑠 𝑑𝑣
 𝑣= dt= = ads=vdv
𝑑𝑡 𝑣 𝑎

𝑑𝑣
 a= ads= vdv Fds= mads = mvdv
𝑑𝑡
𝑠2 𝑣2
𝑠1
𝐹𝑑𝑠 =m 𝑣1
𝑣𝑑𝑣

𝑠2 1 2 1
𝑠1
𝐹𝑑𝑠= 2
m𝑣2 - 2m𝑣12

Principle of Work and Energy

 F=ma

 s=rθ
 𝑑𝑣 𝑑𝑣 𝑡2 𝑣2
 a= F=m 𝑡1
𝐹𝑑𝑡 =m 𝑣1
𝑑𝑣
𝑑𝑡 𝑑𝑡

 F=ma
𝒕𝟐
𝒕𝟏
𝑭𝒅𝒕 = m𝒗𝟐 -m𝒗𝟏


Principle of linear impulse and momentum

• A Free body diagram is a simple diagram of a system
that is free of any contact or constrain

• It demonstrates the magnitudes and directions of all

the external forces acting upon the system.
• How to construct a FBD?

• Step -1 : Identify the problem

• Step 2 : Isolate the object
• Step 3 : Sketch all the external forces.
• 𝐹= ma

𝑭𝒙 = ma

𝑭𝒚 = ma

• 2D
• 𝐹= ma

𝑭𝒙 = ma
𝑭𝒚 = ma
𝑭𝒛 = ma

• 3D
• When a particle moves along a curved path, it is sometimes
convenient to describe its motion using coordinates system 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.
• The center of curvature, O always lies on the concave side of
the curve. The radius of curvature, r is defined as the
perpendicular distance from the curve to the center of curvature
at that point.

• If the particle moves along a path expressed as y= f(x).

• The radius of curvature, r at any point on the path can be
calculated from
•
•
• In this method, the velocity and acceleration of a particle
moving in a curved path is resolved into two components

 Along the tangent to the path (in the direction of motion),

 And normal to the path (towards the inside of the path),

Tangential Components : 𝐹𝑡, 𝑎𝑡

Normal Components : 𝐹𝑛 , 𝑎𝑛
 𝑎𝑡 is the change in speed and 𝑎𝑛 is the change in the direction
of the velocity
• Suppose a particle is moving in a 3-D curved path.
• Let us divide the curved path into small segments.
• Each of the small segments is a part of a circle.
• Let us draw to axis, one along the tangent to the curve and one
normal to the curve.
 𝑑𝑣
• Tangential Acceleration, 𝑎𝑡 =
𝑑𝑡
𝑣2
• Normal Acceleration, 𝑎𝑛 =
𝑟
• Total Acceleration, a = 𝑎𝑡 2+𝑎𝑛 2
• 𝑎𝑡 is the change in speed and 𝑎𝑛 is the change in the direction of the
velocity
• 𝑎𝑛 is also known as centripetal acceleration and acts towards the
center.

𝑑𝑠 𝑑𝑣
V= 𝑎𝑡 = 𝑎𝑡 ds = vdv
𝑑𝑡 𝑑𝑡
•
• When an object moves along a curved path under the influence
of gravity, then it follows a parabolic path and we can refer
the motion of the object as projectile motion.
• Suppose a rock is thrown from the edge of a cliff at some angle
with the horizontal.
• We can resolve the velocity into two components along x and y
direction.

 The vertical velocity will be greatest at the time of throw and

start to decrease at it approaches the peak, will be zero at the
pick position and will eventually start to increase negatively
and in the vertical direction this change will be due to
acceleration due to gravity.

 The horizontal velocity will remain constant if we neglect the

air resistance and hence no acceleration in horizontal direction.
• Let us Consider a duck is swimming in a river in a certain
direction.
• On the bank there is person standing there and watching the
duck and on the river there is another person in a boat who is
moving towards the duck.
• Here the motion of the duck looks different from the two
observers.
• To the observer on the boat the duck probably swims a much
slower than to the observer standing on the bank of the river
• Let us set up a fixed reference frame on the observer standing
on the bank.
• At any given point, the positions of the guy on the boat and of
the duck can be represented by position vectors 𝑟𝐴 and 𝑟𝐵
respectively.
• In order to see the motion of the duck from the eye’s of the guy
in the boat, we set up another reference frame fixed on him.
• This reference frame will move with him and it is parallel with
our fixed reference frame on earth.
• In the eye’s of the guy in the boat, the position of the duck is
represented by a position vector originated from him, 𝑟𝐵/𝐴
• 𝑟𝐵/𝐴 is the relative position of B relative to A.
• 𝑟𝐵/𝐴 = 𝑟𝐵 - 𝑟𝐴
• Relative Position :

𝒓𝑩/𝑨= 𝒓𝑩 - 𝒓𝑨 𝒓𝑩 = 𝒓𝑨 + 𝒓𝑩/𝑨

• Relative Velocity :

𝒗𝑩/𝑨= 𝒗𝑩 - 𝒗𝑨
𝒗𝑩 = 𝒗𝑨 + 𝒗𝑩/𝑨

• Relative Acceleration :

𝒂𝑩/𝑨 = 𝒂𝑩 - 𝒂𝑨 𝒂𝑩 = 𝒂𝑨 + 𝒂𝑩/𝑨
• Let us consider two object A and B that are connected by a
rope wrapped around a pulley.
• From the figure, we can see that if the object A moves upward,
then object B moves downward.
• As the length of the rope is constant, so the motion of the two
particles are dependent on each other.
• First we to select a fixed point to be the origin or a fixed datum
line. Then we can represent the position of the particle with
position vectors.
• Here 𝑆𝐴 and 𝑆𝐵 are both variables that change with time.
• Finally in the figure the length 𝑙𝑠𝑒𝑚𝑖 is constant although it
always moves.
• So, finally from the figure we get,

𝑆𝐴 + 𝑙𝑠𝑒𝑚𝑖 + 𝑆𝐵 = 𝑙𝑇
𝑑(𝑆𝐴 + 𝑙𝑠𝑒𝑚𝑖 + 𝑆𝐵 ) 𝑑(𝑙𝑇 )
=
𝑑𝑡 𝑑𝑡
𝑑(𝑆𝐴 ) 𝑑(𝑆𝐵 )
𝑑𝑡
+0+ 𝑑𝑡
=0

𝑉𝐴 + 𝑉𝐵 = 0
𝑉𝐴 = - 𝑉𝐵
• Differentiating velocity with respect to time, we get the
acceleration,

• 𝑉𝐴 = - 𝑉𝐵

𝑑(𝑉𝐴 ) 𝑑(𝑉𝐵 )
• =-
𝑑𝑡 𝑑𝑡

• 𝑎𝐴 = - 𝑎𝐵
Thank You