4.1 Applied Mechanics

4.1 Applied Mechanics

1. Particles, Rigid and Deformable Bodies

1. Particle:

   • Mass concentrated at a point.

   • No dimensions (idealization).

   • Only translational motion considered.

2. Rigid Body:

   • Collection of particles with fixed distances.

   • No deformation under applied loads.

   • Both translational and rotational motion.

3. Deformable Body:

   • Particles can move relative to each other.

   • Changes shape under load.

   • Studied in Strength of Materials/Solid Mechanics.

2. Statics and Equilibrium

1. Statics: Study of bodies at rest or moving with constant velocity.

2. Equilibrium Conditions:

   • For a particle: Sum of all forces = zero. $\sum \vec{F} = 0$

   • For a rigid body:

     • Sum of forces = zero: $\sum \vec{F} = 0$

     • Sum of moments about any point = zero: $\sum \vec{M} = 0$

3. Free Body Diagram (FBD):

   • Essential tool showing all external forces acting on a body.

   • Isolate the body from its surroundings.

4. Support Reactions:

   • Roller: Single reaction force perpendicular to surface.

   • Pin/Hinge: Two reaction forces (horizontal and vertical).

   • Fixed Support: Two forces + one moment reaction.

3. Friction

1. Dry Friction (Coulomb Friction):

   • Occurs between dry surfaces in contact.

   • Opposes impending or actual relative motion.

2. Friction Forces:

   • Static Friction ($f_s$): Acts before motion starts.

     • $f_s \leq \mu_s N$ where $\mu_s$ = coefficient of static friction.

   • Kinetic Friction ($f_k$): Acts during motion.

     • $f_k = \mu_k N$ where $\mu_k$ = coefficient of kinetic friction.

3. Characteristics:

   • $\mu_s > \mu_k$ (static > kinetic).

   • Friction force depends on normal force, not contact area.

   • Angle of Friction ($\phi$): $\tan \phi = \mu$.

4. Applications:

   • Wedges, belts, screws, brakes.

   • Limiting friction condition used in equilibrium analysis.

4. Newton's Laws

1. First Law (Inertia):

   • Body remains at rest or uniform motion unless acted upon by external force.

   • Defines inertial reference frames.

2. Second Law (Force-Acceleration):

   • $\sum \vec{F} = m\vec{a}$

   • Force produces acceleration proportional to mass.

   • Valid in inertial frames.

3. Third Law (Action-Reaction):

   • For every action, equal and opposite reaction.

   • Forces always occur in pairs between interacting bodies.

5. Work–Energy Theorem

1. Work Done by a Force:

   • $W = \vec{F} \cdot \vec{s} = F s \cos\theta$

   • Scalar quantity (Joules).

2. Kinetic Energy (KE):

   • Energy due to motion: $KE = \frac{1}{2}mv^2$

3. Work–Energy Theorem:

   • Net work done = change in kinetic energy.

   • $W_{net} = \Delta KE = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$

4. Conservative Forces:

   • Work independent of path (gravity, spring).

   • Potential energy (PE) can be defined.

5. Conservation of Mechanical Energy:

   • With only conservative forces: $KE_1 + PE_1 = KE_2 + PE_2$

6. Impulse–Momentum Principle

1. Linear Momentum:

   • $\vec{p} = m\vec{v}$

   • Vector quantity in direction of velocity.

2. Impulse:

   • $\vec{J} = \int \vec{F} dt$

   • Also: $\vec{J} = \vec{F}_{avg} \Delta t$

3. Impulse–Momentum Principle:

   • Impulse = change in momentum.

   • $\vec{J} = \Delta \vec{p} = m\vec{v}_2 - m\vec{v}_1$

4. Conservation of Linear Momentum:

   • If net external force = 0, total momentum is constant.

   • $\sum m\vec{v}_{initial} = \sum m\vec{v}_{final}$

5. Applications:

   • Impact/collision problems.

   • Variable force analysis.

   • Rocket propulsion.

6. Coefficient of Restitution (e):

   • $e = \frac{v_{2} - v_{1}}{u_{1} - u_{2}}$ (relative velocities).

   • $e = 1$: Perfectly elastic (KE conserved).

   • $e = 0$: Perfectly plastic (maximum KE loss).