5. Studying External effect of
forces on a body such as
velocity, acceleration,
displacement etc.
Studying Internal effect of
forces on a body such as
stresses (internal resistance),
change in shape etc.
Rigid body mechanics
Deformable body mechanics
6. Statics
ï¶ Deals with forces and its effects
when the body is at rest
Dynamics
ï¶ Deals with forces and its effects when
the body is in moving condition
Truss Bridge IC Engine
7. Rigid body mechanics
Actual structures and machines are never rigid under the action of
external loads or forces.
But the deformations induced are usually very small which does not
affect the condition of equilibrium.
Negligible deformation (no deformation) under the action of forces.
Assuming 100% strength in the materials. Large number of particles
occupying fixed positions with each other.
8. Particle Mechanics
ï¶ Treating the rigid body as a particle which is negligible in
size when compared to the study involved. (very small
amount of matter which is assumed as a point in a space).
Example: studying the orbital motion of earth
9. Types of forces
Concurrent coplanar forces
Collinear forces
Non Concurrent coplanar
(Parallel)
Concurrent non-coplanar
10. Components of a Force
Plane ForcePlane Force
Space ForceSpace Force
11. Couple Two equal and opposite forces are acting at
some distance forming a couple
15. Parallelogram law:
Two forces acting on a particle can be replaced by the single
component of a force (RESULTANT) by drawing diagonal of the
parallelogram which has the sides equal to the given forces.
Parallelogram law cannot be proved mathematically . It is an
experimental finding.
16. The two vectors can also be added by head to tail by using triangle law.
Triangle law states that if three concurrent coplanar forces are acting at a
point be represented in magnitude and direction by the sides of a triangle,
then they are in static equilibrium.
17. Lamiâs Theorem states that if three concurrent coplanar
forces are acting at a point, then each force is directly
proportional to the sine of the angle between the other two
forces.
18. Lamiâs theorem considering only
the equilibrium of three forces
acting on a point not the stress
acting through a ropes or strings
The principle of transmissibility is
applicable only for rigid bodies not
for deformable bodies
19. F1
F2
F5
F4
F3
A B
E
D
C
Polygon Law of Forces
âIf many number of forces acting at a
point can be represented as a sides of
a polygon, then they are in
equilibriumâ
31. Theory of Dry Friction
Uneven distribution of friction force
and normal reaction in the surface.
Microscopic irregularities produces
reactive forces at each point of contact.
The distance âxâ is to avoid âtipping
effectâ caused by the force âPâ so that
moment equilibrium has been arrived
about point âOâ.
32.
33.
34. Limiting static frictional force: when this value is reached then the body will be in
unstable equilibrium since any further increase in P will cause the body to move.
At this instance, frictional force is directly proportional to normal reaction on the
frictional surface.
Where ÎŒs ï coefficient of static friction
When a body is at rest, the angle that the resultant force makes with normal reaction is
known as angle of static friction.
35. Where ÎŒk ï coefficient of kinetic friction
When a body is in motion, the angle that the resultant force makes with normal reaction
is known as angle of kinetic friction.
41. Trusses ï Stationary, fully constrained structures in
which members are acted upon by two equal and
opposite forces directed along the member.
Frames ï Stationary, fully constrained structures in
which atleast one member acted upon by three or more
forces which are not directed along the member.
Machines ï Containing moving parts, always contain
at least one multiforce member.
47. Internal and External Redundancy
External Redundancy ï More additional supports
Internal Redundancy
If m + 3 = 2j, then the truss is statically determinate structure
If m + 3 > 2j, then the truss is redundant structure (statically
indeterminate structure)
[more members than independent equations]
If m + 3 < 2j, then the truss is unstable structure (will collapse under
external load)
[deficiency of internal members]
ï¶ For statically determinate trusses, â2jâ equations for a truss with âj â joints is equal to
m+3 (âmâ two force members and having the maximum of three unknown support
reactions)
50. Zero Force Members
ï± These members are not useless.
ï± They do not carry any loads under the loading conditions shown,
but the same members would probably carry loads if the loading
conditions were changed.
ï± These members are needed to support the weight of the truss and
to maintain the truss in the desired shape.
52. When a particle moves along a curve other than a straight line, then the particle is in
curvilinear motion.
Curvilinear Motion.
Velocity of a particle is a vector tangent to the path of the particle
53. Acceleration is not tangent to the path of the particle
The curve described by the tip of v is called the
hodograph of the motion
54. Tangential and Normal Components
Tangential component of the acceleration is equal to the rate of change of the speed of
the particle.
Normal component is equal to the square of the speed divided by the radius of
curvature of the path at P.
55.
56.
57. Radial and Transverse components
The position of the particle P is defined by polar coordinates r and Ξ. It is then
convenient to resolve the velocity and acceleration of the particle into components
parallel and perpendicular to the line OP.
ïŒ Unit vector er defines the radial direction, i.e., the direction in which P would move if r
were increased and Ξ were kept constant.
ïŒ The unit vector eΞ defines the transverse direction, i.e., the direction in which P would
move if Ξ were increased and r were kept constant.
58. Where -er denotes a unit vector of sense opposite to that of er
Using the chain rule of differentiation,
Using dots to indicate differentiation with respect to t
59. To obtain the velocity v of the particle P, express the position vector r of P as the
product of the scalar r and the unit vector er and differentiate with respect to t:
Differentiating again with respect to t to obtain the acceleration,
60. The scalar components of the velocity and the acceleration in the radial and transverse
directions are, therefore,
In the case of a particle moving along a circle of center O, have r = constant and
61. Kinetics of Particles
Work Energy Method ï Work of a force & Kinetic energy of particle.
In this method, there is no determination of acceleration.
This method relates force, mass, velocity and displacement.
Work of a Constant Force in Rectilinear Motion
64. Kinetic Energy of a particle
Consider a particle of mass m acted upon by a force F and moving along a path which is
either rectilinear or curved.
When a particle moves from A1 to A2 under the action of a force F, the work of the force F
is equal to the change in kinetic energy of the particle. This is known as the principle of
work and energy.
65. Dynamic Equilibrium Equation
ÎŁF - ma = 0
The vector -ma, of magnitude âmaâ and of direction opposite to that of the acceleration,
is called an inertia vector.
The particle may thus be considered to be in equilibrium under the given forces and the
inertia vector or inertia force.
When tangential and normal components are used, it is more convenient to represent
the inertia vector by its two components -mat and âman.
66. Principle of Impulse and Momentum
Consider a particle of mass m acted upon by a force F. Newtonâs second law can be
expressed in the form
where âmvâ is the linear momentum of the particle.
The integral in Equation is a vector known as the linear impulse, or simply the
impulse, of the force F during the interval of time considered.
Vectorial addition of initial
momentum mv1 and the impulse
of the force F gives the final
momentum mv2.
Definition: A force acting on a particle during a very short time interval that is large
enough to produce a definite change in momentum is called an impulsive force and the
resulting motion is called an impulsive motion.
67. When two particles which are moving freely collide with one another, then the total
momentum of the particles is conserved.
68. KINEMATICS OF RIGID BODIES ï Investigate the relations existing between the
time, the positions, the velocities, and the accelerations of the various particles
forming a rigid body.
Various types of rigid-body motionVarious types of rigid-body motion
Translation ï A motion is said to be a translation if any straight line inside the
body keeps the same direction during the motion.
Rectilinear translation
(Paths are straight lines)
Curvilinear translation
(Paths are curved lines)
69. Rotation about a Fixed Axis ï Particles forming the rigid body move in parallel
planes along circles centered on the same fixed axis called the axis of rotation.
The particles located on the axis have zero velocity and zero acceleration
Rotation and the curvilinear translation are not the same.
70. General Plane Motion ï Motions in which all the particles of the body move in
parallel planes.
Any plane motion which is neither a rotation nor a translation is referred to as a
general plane motion.
Examples of general plane motion :
71. Motion about a Fixed Point ï The three-dimensional motion of a rigid body
attached at a fixed point O, e.g., the motion of a top on a rough floor is known as
motion about a fixed point.
General Motion ï Any motion of a rigid body which does not fall in any of the
categories above is referred to as a general motion.
Example:
72. Translation (either rectilinear or curvilinear translation)
Since A and B, belong to the same rigid body, the derivative of rB/Ais zero
When a rigid body is in translation, all the points of the body have the same velocity and
the same acceleration at any given instant.
In the case of curvilinear translation, the velocity and acceleration change in direction
as well as in magnitude at every instant.
73. Rotation about a fixed axis
Consider a rigid body which rotates about a fixed axis AAâ
âPâ be a point of the body and ârâ its position vector
with respect to a fixed frame of reference.
The angle Ξ depends on the position of P
within the body, but the rate of change ŃČ is
itself independent of P.
The velocity v of P is a vector
perpendicular to the plane containing AAâ
and r.
74. The vector
It is angular velocity of the body and is equal
in magnitude to the rate of change of ŃČ with
respect to time.
The acceleration âaâ of the particle âPâ
α is the angular acceleration of a
body rotating about a fixed axis is a
vector directed along the axis of
rotation, and is equal in magnitude
to the rate of change of âÏâ with
respect to time
75. Two particular cases of rotation
Uniform Rotation ï This case is characterized by the fact that the angular
acceleration is zero. The angular velocity is thus constant.
Uniformly AcceleratedRotation ï n this case, the angular acceleration is constant
77. Absolute and relative velocity in plane motion
Any plane motion of a slab can be replaced by a translation defined by the motion of an
arbitrary reference point A and a simultaneous rotation about A.
The absolute velocity vB of a particle B of the slab is
78. The velocity vA corresponds to the translation of the slab with A, while the relative
velocity vB/A is associated with the rotation of the slab about A and is measured with
respect to axes centered at A and of fixed orientation
79. Consider the rod AB. Assuming that the velocity vA of end A is known, we propose to
find the velocity vB of end B and the angular velocity Ï of the rod, in terms of the
velocity vA, the length l, and the angle Ξ.
80. The angular velocity Ï of the rod in its rotation about B is the same as in its rotation
about A.
The angular velocity Ï of a rigid body in plane motion is independent of the reference
point.
82. For any body undergoing planar motion, there always exists a point in the plane of
motion at which the velocity is instantaneously zero. This point is called the
instantaneous center of rotation, or C. It may or may not lie on the body!
Instantaneous Centre
As far as the velocities are concerned, the slab seems to rotate about the instantaneous
center C.
If vA and vB were parallel and having same magnitude the instantaneous center C would be
at an infinite distance and Ï would be zero; All points of the slab would have the same
velocity.
If vA = 0, point A is itself is the instantaneous
center of rotation, and if Ï = 0, all the particles
have the same velocity vA.
83. Concept of instantaneous center of rotation
At the instant considered, the velocities of all the particles of the rod are thus the same as
if the rod rotated about C.
