1.3. Dynamics

Newton’s Laws of Motion

Newton’s 3rd Law of Motion

• Definition of Force: A force is defined as a push or pull that causes a change in the motion of a body.

• Types of Forces: There are various types of forces, and they are categorized as vector quantities. This means they have both magnitude and direction, typically measured in Newtons (N).

• Statement of the Law: Newton’s Third Law states that for every action, there is an equal and opposite reaction. This implies that if body A exerts a force of magnitude F on body B, then body B will exert a force of magnitude -F on body A.

• Application Example: An example of Newton's 3rd Law is rocket propulsion. When a rocket accelerates hot exhaust gases downward, the reaction force acts on the rocket, propelling it upward.

Newton’s 2nd Law of Motion

• Free Body Diagrams: Free body diagrams visually represent a body and the forces acting on it. For instance, consider a book resting on a table.

• Mathematical Formulation: Newton’s second law can be mathematically formulated as ∑F = ma, which means the resultant force acting on a body in one direction is equal to the mass of the body multiplied by its acceleration in that direction.

  • Key Considerations: This law only applies when the body’s mass remains constant. Both the force and acceleration are vector quantities, meaning they have direction.

• Example Application: Analyzing the forces on a book, we find the resultant force and can determine the acceleration using the equation derived from Newton's second law.

Force and Motion Examples

Example 1: Horizontal and Vertical Forces

• Vertical Forces: The book remains in vertical equilibrium with a resultant force of 0N.

• Horizontal Forces: The book experiences a resultant horizontal force of 1N to the right, calculated from the forces acting on it (5N - 1N = 4N to the right).

• Mass Calculation: Using the approximation g = 10 N/kg, the mass of the book is calculated as 1 kg. Using this mass:

  • From Newton’s second law: 1N = 1 kg × a, we find acceleration (a) = 15 m/s².

Example 2: Complex Forces

• Weight of Body: When considering a body with a weight of 45N, its mass is 4.5kg.

• Force Application: By applying Newton’s second law, we can analyze forces (e.g. using a new notation approach). This leads to finding an acceleration of 0.713 m/s² to the right and 8.80 m/s² downward.

Momentum in Mechanics

Linear Momentum

• Definition and Equation: Linear momentum (p) is defined as p = mv, where m is mass and v is velocity. The units for linear momentum are kg·m/s.

• Vector Quantity: Momentum is also considered a vector quantity, pointing in the direction of the object's velocity. Force is related to momentum as it represents the rate of change of momentum.

Practical Example of Momentum Change

• Velocity Change of Block: A block's velocity changes from 2 m/s to 7 m/s over 2 seconds:

  • The change in momentum (∆p) is: ∆p = ∆t * F, leading to the formula and calculation of 5N as average force acting on the block.

Conservation of Momentum

• In closed systems (no external forces), total momentum remains conserved. Momentum can be transferred among bodies without changing the total.

• Collision Example: Consider two particles involved in a collision: The total momentum before the collision can be computed, equating it to total momentum after. Using the mass and speed of particles, the final velocity of one particle can be derived.

Collisions and Energy Conservation

Inelastic vs. Elastic Collisions

• Inelastic Collision: Kinetic energy is lost during the collision, illustrated through numerical examples showing the total kinetic energy before and after.

• Elastic Collision: In an elastic collision, there is no loss of kinetic energy. Example scenario involves calculating the speed of two particles post-collision using both conservation of momentum and conservation of energy principles.

  • The relation between speeds is deduced, leading to a determinative calculation of particles' velocities after collision with attention to significant figures.

Complex Collision Calculations

• When dealing with collisions without prior speed knowledge, momentum conservation allows for deriving speeds post-collision via quadratic equations. The physical relevance of solutions must be checked to deduce feasible outcomes.