Introduction to Engineering Mechanics

• Engineering Mechanics is a branch of engineering science that applies principles of mechanics to engineering problems.
• Statics deals with forces and their effects on bodies at rest.
• Dynamics deals with forces and their effects on bodies in motion, further divided into Kinetics and Kinematics.
• Kinetics considers the forces causing motion.
• Kinematics analyses motion without regard to the forces involved.

Force

• Force is an agent that produces, destroys, or tends to do so to the motion of a body.
• A force can change the motion of a body, retard it, balance existing forces, or cause internal stresses.
• Determining the effects of force requires knowing its magnitude, direction, nature (push/pull), and point of application.
• The SI unit of force is the Newton (N), with kilo-newton (kN) as a larger unit.

Resultant Force

• A single force that produces the same effect as all forces acting on a body.
• The parallelogram law of forces states that if two forces acting simultaneously on a particle are represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through their point of intersection.
• The resultant R of two forces P and Q acting at an angle 0 is given by R = sqrt(P² + Q² + 2PQ cos θ).
• The angle α, that the resultant makes with the force P is given by tan α = (Q sin θ) / (P + Q cos θ).
• The triangle law of forces states that if two forces, acting simultaneously on a particle, are represented in magnitude and direction by two sides of a triangle taken in order, their resultant is represented in magnitude and direction by the third side of the triangle taken in opposite order.
• The polygon law of forces states that if multiple forces acting simultaneously on a particle are represented in magnitude and direction by the sides of a polygon taken in order, their resultant is represented by the closing side of the polygon taken in the opposite order.
• The resultant of multiple intersecting forces can be found by resolving all the forces horizontally and vertically.
• R = sqrt((ΣH)² + (ΣV)²), with ΣH and ΣV being the sum of resolved parts in the horizontal and vertical directions, respectively.
• The angle α the resultant makes with the horizontal in this case is given by tan α = ΣV / ΣH.
• If the resultant of forces acting on a particle is zero, the particle is in equilibrium, and the set of forces are known as equilibrium forces.
• An equilibrant is a force that brings a set of forces into equilibrium, equal in magnitude but opposite in direction to the resultant force.
• A number of forces acting on a particle will be in equilibrium when ΣH = 0 and ΣV = 0.

System of Forces

• A system of forces is formed when two or more forces act on a body.
• Collinear forces act along the same line.
• Coplanar forces act in the same plane.
• Concurrent forces meet at one point.
• Coplanar concurrent forces meet at one point and act in the same plane.
• Coplanar non-concurrent forces do not meet at one point but act in the same plane.
• Non-coplanar concurrent forces meet at one point but do not act in the same plane.
• Non-coplanar non-concurrent forces do not meet at one point and do not act in the same plane.

Lami's Theorem

• Lami's theorem states that if three coplanar forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
• P/sin α = Q/sin β = R/sin γ, where P, Q, R are the forces, and α, β, γ are the angles opposite to them.

Moment of a Force

• The moment of a force is the turning effect produced by the force on a body.
• It equals the force multiplied by the perpendicular distance from the axis of rotation to the line of action of the force.
• Mathematically, Moment = P x l, where P is the force and l is the perpendicular distance.
• The unit of moment is Newton-metre (N-m).

Varignon's Principle of Moments

• Varignon's principle states that if a number of coplanar forces acting simultaneously on a particle are in equilibrium then the algebraic sum of their moments about any point is equal to the moment of their resultant force about the same point.

Parallel Forces

• Parallel forces have lines of action that are parallel to each other.
• Like parallel forces act in the same direction.
• Unlike parallel forces act in opposite directions.

Couple

• A couple consists of two equal and opposite forces whose lines of action are different.
• The arm of the couple is the perpendicular distance between the lines of action of the two forces.
• The magnitude of a couple is the product of one of the forces and the arm of the couple i.e. Moment of a couple = P × x.
• A couple produces a motion of rotation of the body on which it acts, i.e. does not produce any translatory motion

Centre of Gravity

• The center of gravity is the point through which the whole mass of the body can be assumed to act, irrespective of the body's position.
• For plane geometrical figures, the center of area is known as the centroid (center of gravity of the area).
• A uniform rod's center of gravity is at its middle point.
• A rectangle or parallelogram's center of gravity is at the intersection of its diagonals.
• A triangle's center of gravity is where its medians intersect.
• A semicircle’s center of gravity lies at a distance of 4r / 3π from its base, measured along the vertical radius.
• A hemisphere’s center of gravity lies at a distance of 3r / 8 from its base, measured along the vertical radius.
• The center of gravity of a trapezium with parallel sides a and b, lies at a distance of h(2a + 3b) / 3(a+b) measured from side b.
• The center of gravity of a right circular solid cone lies at a distance of h / 4 from its base, measured along the vertical axis.
• The center of gravity of a circular sector lies at a distance of (2r sin α) / 3α measure along the central axis.
• The center of gravity of a segment of a sphere lies at a distance of (3(2r-h)²) / (4(3r-h)) from the centre of the sphere measured along the height.

Moment of Inertia

• Moment of Inertia is the second moment of mass or area of a body, usually denoted by I.
• For a body of total mass m, composed of particles of masses m₁, m₂, m₃... at distances k₁, k₂, k₃... from a fixed line, I = m₁k₁² + m₂k₂² + m₃k₃² + ... = m k².
• The radius of gyration k is the distance from a reference point where the entire mass or area is assumed to be concentrated to produce the same value of I.
• In SI units, the mass moment of inertia is kg-m², and area moment of inertia is expressed in m⁴ or mm⁴.
• Parallel axis theorem: Ip= IG + m h² (mass) or IG+ a h² (area), where Ip is the moment of inertia about a parallel axis, IG is the moment of inertia about an axis through the center of gravity, and h is the distance between the two axes.
• For a thin disc of mass m and radius r, the moment of inertia about an axis through its center and perpendicular to the plane is I = m r²/2.
• For a thin disc of mass m and radius r, the moment of inertia about a diameter is I = m r²/4.
• For a thin rod of mass m and length l, the moment of inertia about an axis passing through its center of gravity and perpendicular to its length is I = m l²/12.
• For a thin rod of mass m and length l, the moment of inertia about a parallel axis through one end of the rod is I = m l²/3.
• For a rectangular section of width b and depth d, the moment of inertia about the x-axis (Ixx) is bd³/12, and about the y-axis (Iyy) is db³/12.
• For a hollow rectangular section, the moment of inertia Ixx = (BD³ - bd³) / 12 and Iyy = (DB³ - db³) / 12.
• For a circular section of diameter D, the moment of inertia about the x or y axis is πD⁴/64, and the polar moment of inertia is πD⁴/32.
• For a hollow circular section of outer diameter D and inner diameter d Ixx = Iyy = (π/64) [D⁴ - d⁴].
• The moment of inertia of a semicircular section ACB of radius r about the base AB is (1/2) M.I. of circular section = 0.393 r⁴.
• The moment of inertia about an axis passing through G and parallel to X-X axis is IG = 0.11 r⁴
• The moment of inertia of a triangular section of height h about an axis passing through the centre of gravity G and parallel to the base BC is (bh³)/36.
• The moment of inertia of a triangular section of height h about the base BC is, (bh³)/12.

Friction

• Friction is a force acting in the opposite direction to the motion of a body.
• Static friction is experienced by a body when at rest.
• Dynamic friction is experienced by a body when in motion (also called kinetic friction).
• Sliding friction occurs when a body slides over another body.
• Rolling friction occurs when balls or rollers are between two surfaces.

Limiting Friction

• Limiting friction is the maximum value of frictional force when a body just begins to slide over another surface.

Laws of Static Friction

• The force of friction opposes the direction the body tends to move.
• The magnitude of the force of friction is exactly equal to the force that tends to move the body.
• The magnitude of the limiting friction bears a constant ratio to the normal reaction between the two surfaces.
• The force of friction is independent of the area of contact between the two surfaces.
• The force of friction depends upon the roughness of the surfaces.

Laws of Dynamic or Kinetic Friction

• The force of friction always acts in a direction, opposite to that in which the body tends to move.
• The magnitude of the kinetic friction bears a constant ratio to the normal reaction between the two surfaces.
• For moderate speeds, the force of friction remains constant. But it decreases slightly with the increase of speed.

Coefficient of Friction

• Coefficient of friction (µ) is the ratio of limiting friction (F) to the normal reaction (RN) between two bodies.
• μ = F / RN.

Limiting Angle of Friction

• The limiting angle of friction (Φ) is the angle which the resultant reaction (R) makes with the normal reaction (RN) when the body is just about to move.
• tan Φ = μ.

Angle of Repose

• The angle of repose (α) is the angle of inclination of a plane to the horizontal at which the body just begins to move down the plane.
• α = Φ, where Φ is the angle of friction.

Minimum Force Required to Slide a Body on a Rough Horizontal Plane

• An effort P is applied at an angle θ to the horizontal
• P = W sin Φ / cos(θ – Φ)
• For P to be minimum, then Pmin = W sin 0

Effort Required to Move the Body Up an Inclined Plane

• The effort P required to move the body up an inclined plane, where α is the angle of inclination of the plane with the horizontal, is
• P = (W sin (α + φ)) / [sin (θ - (α + φ)]

Effort Required to Move the Body Down an Inclined Plane

• The effort P required to move the body down an inclined plane, where α is the angle of inclination of the plane with the horizontal, is
• P = (W sin (α – φ)) / [sin (θ - (α – φ)]

Efficiency of an Inclined Plane

• The efficiency is defined as the ratio of the effort required neglecting friction (⁰P) to the effort required considering friction (P) i.e., η = ⁰P / P

Screw Jack

• The screw jack works on a similar principle to that of an inclined plane
• tan α = p/ (πd), where α is the helix angle, p is the pitch of the thread and d is the mean diameter of the screw
• The effort P required at the circumference of the screw is equal to W tan( α ± φ) for raising or lowering the load

Lifting Machine

• Enables lifting a heavy load W by a small effort P
• Mechanical advantage (M.A.) = Load lifted (W) / Effort applied (P)
• Velocity ratio (V.R.) = Distance moved by effort (y) / Distance moved by load (x)
• Input of machine = Effort × Distance moved by effort = P × y
• Output of machine = Load × Distance moved by load = W × x
• Efficiency of machine = Output / Input = (W × x) / (P × y) = (W/P) / (y/x) = M.A. / V.R.
• An ideal machine has an efficiency of 100%.
• In a reversible machine the efficiency should be more than 50%.
• A non-reversible / self locking machine is that its efficiency should be less than 50%.
• Law of the machine: P=m W+C
• Maximum mechanical advantage: Max. M.A. = 1/m
• Maximum efficiency: Max. η = 1 / (m × V.R.)

Systems of Pulleys

• First system of pulleys: V.R. = 2ⁿ, where n is the number of pulleys.
• Second system of pulleys: V.R. = n.
• Third system of pulleys: V.R. = 2ⁿ – 1

Velocity Ratio of Lifting Machines

• Simple wheel and axle (diameters D and d): V.R. = D/d
• Differential wheel and axle (wheel diameter D, axle diameters d₁ and d₂): V.R. = 2D / (d₁-d₂)
• Weston's differential pulley block (pulley diameters D and d): V.R. = 2D / (D-d)
• Worm and worm wheel (handle length l, teeth T, drum radius r): V.R. = (lT) / r
• Simple screw jack (handle length l, pitch p): V.R. = (2πl) / p
• Differential screw jack V.R. = (2πl) / (PA-PB)
• Single purchase winch crab (handle length l, drum diameter D, gear teeth T₁ and T₂): V.R. = (2lT₁T₂) / D
• Double purchase winch crab (handle length l, drum diameter D, pinion teeth T₁, T₂, spur teeth T₃, T₄) V.R. = (2lT₁T₂) / (DT₃T₄)

Frame

• Made up of several bars riveted or welded together.
• A perfect frame is that which is composed of members just sufficient to keep it in equlibrium, when loaded, without any change in its shape, following the rule n = 2j-3
• An imperfect frame is one which does not satisfy the above equation (n = 2j - 3).
• The imperfect frame which has number of members (n) less than 2j – 3, is known as deficient frame.
• If the number of members are greater than 2j – 3, then the imperfect frame is known redundant frame.

Speed

• The rate of change of distance with respect to its surrounding.
• It is a scaler quantity.

Velocity

• It is the rate of change of displacement with respect to its surrounding, in a particular direction.
• It is a vector quantity.

Acceleration

• It is the rate of change of velocity of a body.
• Positive when the velocity of a body increases with time and it is negative when the velocity decreases with time.
• The negative acceleration is also called retardation.

Equations of Linear Motion

• v = u + a t
• 𝑠 u t + (1/2) a t²
• v² = u² + 2 a s
• s = (u + v) t / 2
• Where: u, v, a, and s is Initial velocity, final velocity, acceleration, and displacement
• Substitute acceleration due to gravity(g) for ‘a’ in vertical motion

Newton's Law of Motion

• 1. A body continues in its state of rest or uniform motion unless acted upon by an external force.
• 2. The rate of change of momentum is proportional to the impressed force.
• 3. To every action, there is an equal and opposite reaction.

Mass, Weight and Momentum

• Mass is the quantity of matter, measured in kg.
• Weight is the force due to gravity on a mass, measured in N: W = mg.
• Momentum is the total motion, which is the product of mass and velocity.

D'Alembert's Principle

• For a rigid body acted upon by a set of forces P -inertia = 0 at equilibrium

Motion of a Lift

• Moving upwards: R - mg = ma or R = m(g + a)
• Moving downwards: m.g -R = m.a or R = m (g- a)

Motion of Two Bodies Connected by a string

• a = g(m1 - m2) /m1 +m2
• T = 2m1m2(g)/ m1+m2
• Bodies Rest over smooth surface
• W = m2.g/ m1+m2

Projectile

• A projectile moves under combined vertical and horizontal forces (gravity).
• Trajectory: The path traced.
• Velocity of projection: The initial velocity.
• Angle of projection: Angle with horizontal..
• Time of flight: Duration to reach and return.
• Range: Distance between projection and landing points.

Equation of the Path of Projectile

• y=x tan- g.x^2/2u^2cos²α

Angular Displacement

• The angle described from one point to another.
• Vector quantity

Angular Motion

• Equations for Angular Motion: • ω = ω0 + αt • θ = ω0t + (1/2) α t 2 • ω2 = ω02 + 2 αθ

Angular Velocity

– Rate of change of angular displacement.

Coefficient Of Friction

• µ = F/Rn, with the angle that is between normal reaction i.e., (rn)

Centrifugal Force

Force away to center from centroid with mass ‘Kg’, this force acting radially outward which is known equation Centrifugal force is m.w³.r

Simple Harmonic Motion

• Simple harmonic motion if it follows the condition of time or medium (1)-Its acceleration should know at the center points and position.
• This action is proportional to move this from that point

Amplitude

It is a maximum displacement of bodies which made them to mean to the position.

Pulleys, its Types And it Motion

1 System of pulleys- (V, R) = 2" 2 second system of the pulley- (V,R)= n 3 third system of the pulley- (V,R) = 2(-1)

Liffing Machines

The velocity of Ratio of the various lifting Machines are as followed, 1 The velocity Ratio of simplest wheel and axle with diameters D and respectively, the =

Collision of Two bodies

the impact between two or which more the number of bodies, which having different velocities alongside with state straight line, It is assumed that the point of impact line joining the center of the materials. – the material of two bodies may be perfectly elastic or perfectly inelastic.

Work

Whenever this ‘f’ acting on bodies to that it performs under a way that displacement (x) in the direction, of course, to the fore they said should be done and said to be done.