MEL Module 2: Engineering Mechanics

1. Introduction to Dynamics

Figure 1: Particle kinematics showing position, velocity, and acceleration vectors

What is Dynamics?

Dynamics is the branch of mechanics that deals with the motion of objects in response to forces. Unlike statics (which studies objects at rest), dynamics studies objects that are accelerating or moving with changing velocities.

F = ma (Newton's Second Law)

Key Concepts in Dynamics

Kinematics: Description of motion without considering forces
Kinetics: Analysis of motion considering the forces that cause it
Mass: Measure of an object's resistance to acceleration
Force: Push or pull that causes acceleration
Energy: Capacity to do work (kinetic, potential, etc.)
Momentum: Product of mass and velocity

2. Kinematics of Particles

Figure 2: Projectile motion showing trajectory and velocity components

2.1 Motion in One Dimension

x = x₀ + v₀t + ½at² (Position as function of time)

v = v₀ + at (Velocity as function of time)

1D Motion Calculator

Initial Position (m):

Initial Velocity (m/s):

Acceleration (m/s²):

Time (s):

Results:

Final Position: - m

Final Velocity: - m/s

2.2 Motion in Two Dimensions (Projectile Motion)

x(t) = x₀ + v₀cos(θ)t (Horizontal position)

y(t) = y₀ + v₀sin(θ)t - ½gt² (Vertical position)

Projectile Motion Calculator

Initial Speed (m/s):

Launch Angle (degrees):

Initial Height (m):

Results:

Maximum Height: - m

Range: - m

Time of Flight: - s

3. Kinetics: Newton's Laws of Motion

Figure 3: Relationship between force, mass, and acceleration

3.1 Newton's First Law (Law of Inertia)

An object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by a net external force.

3.2 Newton's Second Law (F = ma)

ΣF = ma (Vector equation)

Example 1: Force Calculation

Problem: A 1500 kg car accelerates from 0 to 100 km/h in 8 seconds. Calculate the average force applied.

Given: m = 1500 kg, Δv = 100 km/h = 27.78 m/s, Δt = 8 s

Calculate acceleration: a = Δv/Δt = 27.78/8 = 3.47 m/s²

Apply Newton's Second Law: F = ma = 1500 × 3.47 = 5205 N

3.3 Newton's Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction. Forces always occur in pairs.

Newton's Second Law Calculator

Mass (kg):

Acceleration (m/s²):

Result:

Force: - N

4. Work and Energy Principles

Figure 4: Energy transformation between kinetic and potential forms

4.1 Work

W = F·s·cos(θ) (Work done by constant force)

4.2 Kinetic Energy

KE = ½mv² (Translational kinetic energy)

4.3 Potential Energy

PE = mgh (Gravitational potential energy)

Energy Calculator

Mass (kg):

Velocity (m/s):

Height (m):

Results:

Kinetic Energy: - J

Potential Energy: - J

Total Mechanical Energy: - J

4.4 Conservation of Energy

KE₁ + PE₁ = KE₂ + PE₂ + W_non-conservative

Example 2: Energy Conservation

Problem: A 5 kg ball is dropped from 20 m height. Calculate its velocity just before hitting the ground.

Initial energy: PE₁ = mgh = 5 × 9.81 × 20 = 981 J, KE₁ = 0

Final energy: PE₂ = 0, KE₂ = ½mv²

Apply conservation: PE₁ = KE₂ → 981 = ½ × 5 × v²

Solve for velocity: v = √(2 × 981 / 5) = 19.8 m/s

5. Momentum and Impulse

Figure 5: Impulse-momentum relationship and force-time curves

5.1 Linear Momentum

p = mv (Linear momentum)

5.2 Conservation of Momentum

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (Before = After)

5.3 Impulse

J = F·Δt = Δp (Impulse-momentum theorem)

Momentum Calculator

Mass 1 (kg):

Velocity 1 (m/s):

Mass 2 (kg):

Velocity 2 (m/s):

Results:

Momentum of Object 1: - kg⋅m/s

Momentum of Object 2: - kg⋅m/s

Total Momentum: - kg⋅m/s

6. Circular Motion

Figure 6: Uniform circular motion showing centripetal force direction

6.1 Uniform Circular Motion

v = 2πr/T (Linear speed)

ω = 2π/T (Angular velocity)

a_c = v²/r = ω²r (Centripetal acceleration)

F_c = mv²/r (Centripetal force)

Circular Motion Calculator

Radius (m):

Linear Speed (m/s):

Mass (kg):

Results:

Centripetal Acceleration: - m/s²

Centripetal Force: - N

Period: - s

Frequency: - Hz

7. Rigid Body Dynamics

Figure 7: Rigid body rotation showing angular displacement and velocity

7.1 Rotational Motion

θ = ωt (Angular displacement)

α = Δω/Δt (Angular acceleration)

7.2 Moment of Inertia

I = Σmᵢrᵢ² (Discrete system)

I = ∫r²dm (Continuous body)

7.3 Rotational Kinetic Energy

KE_rot = ½Iω² (Rotational kinetic energy)

7.4 Torque

τ = Iα (Rotational equivalent of F = ma)

Rotational Motion Calculator

Mass (kg):

Radius (m):

Angular Velocity (rad/s):

Applied Torque (N⋅m):

Results:

Moment of Inertia: - kg⋅m²

Angular Acceleration: - rad/s²

Rotational Kinetic Energy: - J

8. Mechanical Vibrations

Figure 8: Simple harmonic motion showing displacement, velocity, and acceleration

8.1 Simple Harmonic Motion (SHM)

x(t) = A cos(ωt + φ) (Displacement)

v(t) = -Aω sin(ωt + φ) (Velocity)

a(t) = -Aω² cos(ωt + φ) (Acceleration)

8.2 Natural Frequency

f = (1/2π)√(k/m) (Natural frequency)

ω = √(k/m) (Angular natural frequency)

SHM Calculator

Mass (kg):

Spring Constant (N/m):

Amplitude (m):

Time (s):

Results:

Natural Frequency: - Hz

Displacement: - m

Velocity: - m/s

Acceleration: - m/s²

9. Real-World Applications

9.1 Automotive Engineering

Vehicle Dynamics: Understanding dynamics is crucial for designing safe and efficient vehicles. Key applications include:

Braking system design using impulse-momentum principles
Suspension systems utilizing spring-mass-damper models
Cornering forces in circular motion
Energy recovery in regenerative braking systems

9.2 Aerospace Engineering

Aircraft Dynamics: Flight dynamics involve complex interactions between aerodynamic forces and vehicle motion:

Projectile motion for trajectory planning
Circular motion in banking turns
Vibrations in wing structures
Energy management in launch systems

9.3 Robotics and Automation

Robot Kinematics: Understanding dynamics is essential for robot design and control:

Joint motion planning and control
Payload capacity and inertia considerations
Energy-efficient motion planning
Vibration suppression in precise positioning

10. Practice Problems

Problem 1: Projectile Motion

A baseball is hit with an initial speed of 40 m/s at an angle of 30° above the horizontal. Calculate:

Maximum height reached
Time of flight
Horizontal range

Solution Hint: Use the projectile motion equations with v₀ = 40 m/s, θ = 30°

Problem 2: Energy Conservation

A roller coaster car (mass = 500 kg) starts from rest at the top of a 25 m high hill. Calculate its speed at the bottom of the hill (ignore friction).

Solution Hint: Use conservation of energy: PE₁ = KE₂

Problem 3: Circular Motion

A car travels around a circular track with radius 100 m at a constant speed of 25 m/s. Calculate the centripetal force required if the car has a mass of 1200 kg.

Solution Hint: Use F_c = mv²/r

🌀 MEL Module 2: Engineering Mechanics - Dynamics

1. Introduction to Dynamics

What is Dynamics?

Key Concepts in Dynamics

2. Kinematics of Particles

2.1 Motion in One Dimension

Results:

2.2 Motion in Two Dimensions (Projectile Motion)

Results:

3. Kinetics: Newton's Laws of Motion

3.1 Newton's First Law (Law of Inertia)

3.2 Newton's Second Law (F = ma)

3.3 Newton's Third Law (Action-Reaction)

Result:

4. Work and Energy Principles

4.1 Work

4.2 Kinetic Energy

4.3 Potential Energy

Results:

4.4 Conservation of Energy

5. Momentum and Impulse

5.1 Linear Momentum

5.2 Conservation of Momentum

5.3 Impulse

Results:

6. Circular Motion

6.1 Uniform Circular Motion

Results:

7. Rigid Body Dynamics

7.1 Rotational Motion

7.2 Moment of Inertia

7.3 Rotational Kinetic Energy

7.4 Torque

Results:

8. Mechanical Vibrations

8.1 Simple Harmonic Motion (SHM)

8.2 Natural Frequency

Results:

9. Real-World Applications

9.1 Automotive Engineering

9.2 Aerospace Engineering

9.3 Robotics and Automation

10. Practice Problems