Technical Mechanics: Comprehensive Study Notes on Kinematics and Dynamics

Introduction to Technical Mechanics

• Technical Mechanics includes Kinematics and Dynamics.

• Kinematics is the study of motion as a phenomenon of change of position without considering the forces and masses involved.

• Dynamics considers both the forces and the masses influencing motion.

Fundamental Concepts of Kinematics (22.06.2026)

• Kinematics is the branch of mechanics that investigates the motion of bodies or material points without taking mass into account.

• Motion is the change of position of a body relative to another body.

• All motion is tracked relative to a fixed coordinate system (motionless reference frame).

• There are two basic tasks in kinematics:

  • Direct laws of motion.

  • Indirect laws of motion.

Definitions and Parameters

• Material Point (Materijalna tačka): A body that can be treated as having infinitely small dimensions compared to the path it travels.

• Motionless Body: A body relative to which we track the motion of another body.

• Path ($s$): The length that a body travels during a certain time interval.

• Speed ($v$): The distance traveled per unit of time.

Types of Linear Motion

Uniform Linear Motion

• Motion where the velocity remains constant over time ($v = \text{const}$).

Variable Linear Motion

• Motion where the velocity changes at every moment of time. It is categorized into uniformly accelerated and uniformly decelerated motion.

Uniformly Accelerated Motion

• This is motion where the velocity increases over time by a constant amount (acceleration is a constant value).

• Two cases are distinguished: with and without initial velocity ($v_0$).

• Case 1: With initial velocity ($v_0 \neq 0$)

  • Acceleration ($a$) is constant ($a = \text{const}$).

  • Final velocity: $v = v_0 + a \times t$

  • Path: $s = v_0 \times t + \frac{a \times t^2}{2}$

  • Velocity-distance relationship: $v^2 = v_0^2 + 2 \times a \times s$

• Case 2: Without initial velocity ($v_0 = 0$)

  • Acceleration ($a$) is constant ($a = \text{const}$).

  • Final velocity: $v = a \times t$

  • Path: $s = \frac{a \times t^2}{2}$

  • Velocity-distance relationship: $v^2 = 2 \times a \times s \Rightarrow v = \sqrt{2 \times a \times s}$

Uniformly Decelerated Motion

• This is motion where the velocity decreases at every moment of time by a constant amount.

• In this case, there must always be an initial velocity ($v_0$).

• Equations:

  • Acceleration ($a$) is constant ($a = \text{const}$).

  • Final velocity: $v = v_0 - a \times t$

  • Path: $s = v_0 \times t - \frac{a \times t^2}{2}$

  • Velocity-distance relationship: $v^2 = v_0^2 - 2 \times a \times s$

• For reaching a complete stop ($v = 0$):

  • Time to stop: $t = \frac{v_0}{a}$

  • Stopping distance: $s = \frac{v_0^2}{2 \times a}$

Practical Problems in Linear Kinematics

High-Speed Train Problem

• A high-speed train travels for $2\,h$ and $45\,min$ to cover a path of $180\,km$. Determine the average velocity of the train.

• Time ($t$) = $2\,h + 45/60\,h = 2.75\,h$

• Path ($s$) = $180\,km$

• Velocity ($v$) = $\frac{s}{t} = \frac{180\,km}{2.75\,h} \approx 65.45\,km/h$

• Conversion to meters per second: $v = \frac{65.45 \times 1000\,m}{3600\,s} \approx 18.18\,m/s$

Velocity-Time ($v-t$) Diagram Construction

• Plotting a graph for uniformly accelerated motion with $v_0 = 4\,m/s$ and $a = 2\,m/s^2$:

  • At $t = 0\,s$: $v = 4\,m/s$

  • At $t = 1\,s$: $v = 4 + 2 \times 1 = 6\,m/s$

  • At $t = 2\,s$: $v = 4 + 2 \times 2 = 8\,m/s$

  • At $t = 3\,s$: $v = 4 + 2 \times 3 = 10\,m/s$

Circular Motion (23.06.2026)

• Circular motion is motion where points of a body describe concentric circles around an axis of rotation.

• Velocities differ depending on the distance from the center of rotation.

• Angular Velocity ($\omega$): Calculated using rotations per minute ($n$):

  • $\omega = \frac{\pi \times n}{30}\,\text{rad/s}$

• Linear Velocity ($v$): Related to angular velocity and radius ($r$):

  • $v = r \times \omega$

Uniformly Accelerated Circular Motion

• Angular velocity increases by a constant amount of angular acceleration ($\epsilon$).

• Case 1: With initial angular velocity ($\omega_0 \neq 0$)

  • $\epsilon = \text{const}$

  • $\omega = \omega_0 + \epsilon \times t$

  • Angle describing the path ($\phi$): $\phi = \omega_0 \times t + \frac{\epsilon \times t^2}{2}$

  • $\omega^2 = \omega_0^2 + 2 \times \epsilon \times \phi$

• Case 2: Without initial angular velocity ($\omega_0 = 0$)

  • $\epsilon = \text{const}$

  • $\omega = \epsilon \times t$

  • $\phi = \frac{\epsilon \times t^2}{2}$

  • $\omega^2 = 2 \times \epsilon \times \phi$

Uniformly Decelerated Circular Motion

• Angular velocity decreases by a constant value of angular deceleration.

• Always involves an initial velocity ($\omega_0$).

• Equations:

  • $\omega = \omega_0 - \epsilon \times t$

  • $\phi = \omega_0 \times t - \frac{\epsilon \times t^2}{2}$

  • $\omega^2 = \omega_0^2 - 2 \times \epsilon \times \phi$

Components of Acceleration in Circular Motion

• Total Acceleration ($a$): The vector sum of normal and tangential components.

  • $a = \sqrt{a_n^2 + a_t^2}$

• Normal (Centripetal) Acceleration ($a_n$): directed toward the center of rotation.

  • $a_n = r \times \omega^2$

• Tangential Acceleration ($a_t$): directed along the tangent of the path.

  • $a_t = r \times \epsilon$

Dynamics and Newton's Laws

• Dynamics studies the motion of rigid bodies or material points considering force and mass.

• First Law of Dynamics: Every body will remain in a state of rest or uniform linear motion unless acted upon by a external force.

• Second Law of Mechanics (Dynamics): The force required to change the state of rest or motion is equal to the product of mass ($m$) and acceleration ($a$).

  • $F = m \times a\,\text{(N)}$

  • Weight ($G$) is a force: $G = m \times g$

  • Local gravitational acceleration ($g$) is approximately $9.81\,m/s^2$

• Third Law of Mechanics (Dynamics): Two material points act on each other with forces of equal magnitude but opposite directions (Action and Reaction).

Work, Power, and Energy

Mechanical Work ($A$)

• Linear motion: The product of force and distance.

  • $A = F \times s$

• Rotational motion: The product of active rotation moment ($M$) and rotation angle ($\phi$).

  • $A = M \times \phi$

Power ($N$)

• Work performed per unit of time.

  • $N = \frac{A}{t} = \frac{M \times \phi}{t} = M \times \omega$

Physical Quantities of Motion

• Quantity of Motion (Momentum) ($K$): Product of mass and velocity.

  • $K = m \times v$

• Impulse of Force: Product of force and time.

  • $I = F \times t$

Energy Laws

• Energy cannot be created or destroyed, only transformed from one form to another.

• Mechanical Energy: The sum of kinetic energy ($E_k$) and potential energy ($E_p$).

  • $E = E_k + E_p$

• Kinetic Energy ($E_k$): Half the product of mass and the square of velocity.

  • $E_k = \frac{m \times v^2}{2}$

• Potential Energy ($E_p$): Product of weight and the height ($h$) at which the body is located.

  • $E_p = G \times h = m \times g \times h$

Advanced Principles in Dynamics

Dalamberov princip (D'Alembert's Principle)

• Refers to a system of material points or a rigid body where internal and external forces, including inertial forces, act. The set of external forces and inertial forces is in equilibrium.

• For a system in equilibrium: $\Sigma F_i^{spolj} + \Sigma F_i^{in} = 0$

• Linear motion: Sum of all external forces is zero.

• Circular motion: Sum of the moments of all external forces is zero: $\Sigma M_i + \Sigma M_i^{in} = 0$

Steinerova teorema (Steiner's Theorem)

• Applies to moments of inertia for parallel axes.

• The moment of inertia for an axis ($J$) is equal to the sum of the moment of inertia for a parallel axis passing through the center of gravity ($J_c$) and the product of the mass and the square of the distance ($d$) between the axes.

  • $J = J_c + m \times d^2$

Practical Problems in Dynamics and Energetics

Pump Station Calculation

• Determine the power of a pump station that lifts $Q = 360\,m^3$ of water to a height of $H = 30\,m$ in $t = 1\,h$, given efficiency $\eta = 0.75$.

• Volume ($V$) = $360\,m^3$; density of water ($\rho$) suggests $1\,m^3 = 1000\,kg$.

• Mass ($m$) = $360,000\,kg$

• Weight ($G$) = $m \times g = 360,000 \times 9.81 \approx 3,531.6 \times 10^3\,N$

• Work ($A$) = $G \times H = 3,531,600 \times 30 = 105,948 \times 10^3\,Nm$

• Theoretical Power ($N_{th}$) = $\frac{A}{3600\,s} \approx 29,430\,W$

• Real Power ($N$) = $\eta \times N_{th} = 0.75 \times 29,430 \approx 22,072.5\,W$

Free Fall Kinetic Energy Calculation

• Determine the kinetic energy of a body in free fall weighing $G = 5\,N$ after $t = 4\,s$ of falling.

• Mass ($m$) = $\frac{G}{g}$

• Velocity ($v$) = $g \times t$

• Kinetic Energy ($E_k$) = $\frac{m \times v^2}{2} = \frac{(\frac{G}{g}) \times (g \times t)^2}{2} = \frac{G \times g \times t^2}{2}$

• $E_k = \frac{5 \times 9.81 \times 4^2}{2} = \frac{5 \times 9.81 \times 16}{2} = 5 \times 9.81 \times 8 = 392.4\,J$