Newton's Laws Study Notes

Chapter 4: Newton's Laws

Kinematics:
- Definition: The study of how objects move.
- Introduced: Chapters 2-3.
- Described by: Kinematic Equations for constant acceleration.
Dynamics:
- Definition: The study of why objects move.
- Introduced: Chapters 4-5.
- Described by: Newton’s three laws.
- Key Point: Forces are responsible for acceleration.

Definition: In everyday language, a force is a push or a pull.
Examples of forces in physics (PHYS 2414):
1. Gravity
2. Friction (static and kinetic)
3. Normal force
4. Tension
5. Restoring (elastic or spring): described mathematically as $ F = -k \, riangle x$, where $k$ is the spring constant and $\triangle x$ is the displacement from equilibrium.
Important Note: A force always results from the interaction between two objects.
A force is a vector quantity, meaning:
- It has both magnitude and direction.

Statement: An object at rest will remain at rest, and an object in motion will continue moving at a constant velocity in a straight line unless acted upon by a net external force.
Clarifications:
- There may be multiple forces acting on an object, but if these sum to zero, the net external force is zero.
- Essential idea: The net force is defined as the vector sum of all forces acting on an object.
Key takeaway: Every object continues in its state of rest or uniform motion unless a net external force is applied.

Scenario: A starship with engines turned off in deep space
- Options provided for question on its motion.
Scenario: A rocket ship shutting off engines at different paths.
Scenario: A car moving at a constant velocity of 65 mph
- Implications on the forces acting on the car.

Definition: The resistance of an object to change in its state of motion.
Mass:
- Definition: A measure of an object’s inertia.
- Larger mass indicates larger inertia, hence requiring greater force for the same change in motion.
- SI Unit of mass: kilograms (kg)
- British Unit of mass: slugs.
Important Note:
- Change of motion translates to acceleration.

Statement: The acceleration $a$ of an object is directly proportional to the net force $ ext{F}$ acting on it and inversely proportional to its mass $m$.
Mathematical representation:
- a = \frac{\Sigma F}{m}
- Frequently written as:
- \Sigma F = ma
Vector Equation:
- Each component can be analyzed separately:
- \Sigma Fx = m ax
- \Sigma Fy = m ay
- \Sigma Fz = m az

SI Unit of Force: Newton (N) defined as: 1 \, ext{N} = 1 \, ext{kg} imes \frac{m}{s^2}
British Unit of Force: Pound (lb).
Relation between units:
- $1.0 \, ext{kg} = 2.2 \, ext{lb}$, although this may be misleading as kg is a mass unit and lb is a weight unit only on Earth.

Identifying the net force is crucial when utilizing Newton’s second law.
Draw a Free Body Diagram (FBD) to visualize all forces acting on the object.
Generally, most forces are contact forces; ask, "What is touching the object in question?"
Key Note:
- In PHYS 2414, gravity is the only non-contact force considered.
Follow these steps for problem-solving:
1. Think through the problem and sketch a diagram.
2. Outline a physics representation of the situation and define variables.
3. Choose an axis in the direction of acceleration.
4. Decompose the forces acting on the object accordingly.
5. Implement \Sigma F = ma for calculations.
6. Validate the final answer logically based on physical implications.

Equation: F_g = m a = mg
Analysis:
- The force of gravity acting on an object is defined as its weight.
- Mass ($m$) of the object remains constant while weight ($F_g$) changes with variations in gravitational force, as demonstrated by taking an object from Earth to the moon.

Example: An athlete releases a shot with a speed of 13 m/s over a distance of 2.8m.
- Average net force exerted can be determined using kinematic equations and Newton's second law.
A force exerted on a rest cart gives a final velocity influenced by friction.
- To achieve the same final speed with half the force, time must be adjusted according to the relation of force and time.

Definition: The force exerted by ropes, cables, etc., that can only pull, not push, directed along their length.
Example: A stationary object hanging from a string experiences gravitational pull down, balanced by the tension in the opposite direction.

Description: The upward force exerted by a surface that supports the weight of an object resting on it.
For a stationary object on a table, the normal force equals the gravitational force if the system is in equilibrium (net force = 0).

Free body diagrams can illustrate a problem involving tension in wires securing an object.
Analyzing the normal force when an object is stationary on a flat surface or subjected to external pulls illustrates variations based on forces.

Statement: For every action, there is an equal and opposite reaction.
Explanation: Forces exist in pairs that act on different objects, ensuring net force on a single object is not zero.

Definition:
- Internal Forces: Forces within a system affecting each other.
- External Forces: Forces acting on an object from outside the system.
Advantage: Analyzing each object in a system separately and employing separate FBDs can aid in understanding forces better.

Multiple forces acting in scenarios and how they interact provide insight into dynamics.
Understanding static vs. kinetic friction can be analyzed with FBDs and the force equations.
Problems involving inclined planes require decomposition of forces along axes utilizing trigonometric relations.

Kinetic Friction: Expressed as $Fk = \, \muk \, F_N$, always opposing direction of motion.
Static Friction is described as $Fs \leq \mus F_N$, demonstrating variability in frictional force necessary to counteract applied pushes or pulls relevant until a maximum threshold is reached.

Forces must be clearly analyzed with Newton's Laws as a framework. Understanding action-reaction pairs, the nature of friction, and net forces act as the foundation for studying motion and dynamics.