CN04 Dynamics

1. Overview of Dynamics

• Definition of Dynamics: Study of forces and their effects on motion, particularly what causes an object to change its velocity (accelerate).

• Force: A push or pull acting on an object.

  • Forces are vectors: They have both magnitude and direction, represented visually by arrows.

2. Key Concepts

2.1 Free Body Diagrams

• Free Body Diagram (FBD): A diagram showing all the forces acting on an object.

• Net Force: The vector sum of all the forces acting on an object.

• Equilibrium: An object is in equilibrium if the net force acting on it is zero ($F_{net} = 0$), meaning it is either stationary or moving at constant velocity.

2.2 Newton’s Laws of Motion

• Newton’s 1st Law (Inertia): An object at rest or moving with constant velocity will remain in that state unless acted upon by an external force.

• Newton’s 2nd Law: $F_{net} = m \times a$

  • Acceleration $a$ is directly proportional to the net force and inversely proportional to mass.

• Newton’s 3rd Law (Action-Reaction): For every action, there is an equal and opposite reaction.

3. Newton's First Law

3.1 Details on First Law

• The law explains inertia; thus an object will stay still or continue moving unless acted upon by an external force.

• Common examples: A mass on a frictionless surface will continue indefinitely at constant speed unless a force (like friction) acts upon it.

3.2 Applications

• Free Body Diagram: To identify forces acting on objects at rest or in motion.

• Example: A book sliding off a bus seat when the bus stops.

4. Newton's Second Law

4.1 Formula and Implications

• Formula: $F_{net} = m \times a$

• Units of Force: Newtons (N) or $kg \times m/s^2$

• If 4 times the force is applied:

  • The acceleration will be 4 times greater.

4.2 Problem Solving with Newton’s 2nd Law

1. Draw a sketch of the problem.

2. Draw a Free Body Diagram for each object individually.

3. Select a coordinate system that simplifies calculations.

4. Resolve forces into x and y components.

5. Apply Newton's 2nd Law separately to x and y components.

6. Solve for unknowns.

7. Validate results.

5. Forces in Detail

5.1 Types of Forces

• Gravitational Force: Acts on objects due to their mass, calculated as: $F_g = m \times g$

  • Where $g = 9.8 m/s^2$ is acceleration due to gravity.

• Normal Force: The support force exerted upon an object in contact with a surface, perpendicular to the surface.

  • Normal force may vary based on the situation; it does not always equal weight ($F<em>N \neq F</em>g$).

• Tension: Force transmitted through a cord or rope, directed along the length of the cord.

5.2 Mechanical Equilibrium

• For equilibrium, forces must balance such that:

  • The vector sum of forces equals zero.

  • Example: A block on a frictionless table remains in place if no net force acts upon it.

6. Forces on an Incline

6.1 Inclined Plane Problems

• When analyzing objects on an incline:

  • Forces acting down the incline: $F_{gravity, parallel} = mg \sin(\theta)$

  • Normal force: $F_N = mg \cos(\theta)$

6.2 Critical Angle

• Static friction stops the object from sliding until the angle exceeds a critical angle, at which point it will slide down.

7. Friction

7.1 Types of Friction

• Static Friction: Acts when surfaces are not in relative motion; it has a maximum value before an object moves.

• Kinetic Friction: Acts when surfaces slide against each other.

• Coefficient of friction is dimensionless and varies with surface material.

7.2 Effects of Surface Area

• Surface area does not affect the coefficient of friction significantly; rather, it's dependent on the texture and materials of the surfaces.

8. Sample Problems

8.1 Example Problems

• Calculate acceleration when a force is applied to an object of mass m on a frictionless incline.

• Determine the tension in a cord supporting a mass hanging vertically.

8.2 Special Cases

• Block on a frictionless incline vs. an accelerated truck bed: The box will remain stationary unless sufficient friction exists.

9. Conclusion

• Understanding forces, motion, and the laws governing them provides foundational knowledge in dynamics, crucial for fields such as physics and engineering.

• Problems relating to tension, friction, and forces on inclined planes are common applications of these principles.

10. Formula Summary

Here is a summary of the key formulas in dynamics:

• Equilibrium Condition: $F_{net} = 0$

  • Description: The net force acting on an object in equilibrium (at rest or constant velocity) is zero.

• Newton's Second Law: $F_{net} = m \times a$

  • Description: The net force on an object is equal to its mass times its acceleration. This is a vector equation.

• Gravitational Force (Weight): $F_g = m \times g$

  • Description: The force of gravity acting on an object, where $g$ is the acceleration due to gravity ($9.8 m/s^2$).

• Normal Force on a Horizontal Surface: $F<em>N \neq F</em>g$

  • Description: The normal force is the support force perpendicular to a surface. It does not always equal the gravitational force.

• Gravitational Force Component Parallel to an Incline: $F_{gravity, parallel} = mg \sin(\theta)$

  • Description: The component of gravitational force acting parallel to an inclined surface, causing an object to slide down.

• Normal Force on an Incline: $F_N = mg \cos(\theta)$

  • Description: The normal force acting perpendicular to an inclined surface.