Newton's First Law, Inertia, and Free Fall: module 2 lecture 4

Acceleration of Freely Falling Objects

Definition and Characteristics

When an object is in free fall, it moves under the influence of gravity alone, with no other forces (such as air resistance) acting on it. The acceleration experienced by such an object is called the acceleration due to gravity, denoted by g.

• Direction: The acceleration is always directed downward, toward the center of the Earth.

• Magnitude: The standard value at Earth's surface is $g = 9.80\ \mathrm{m/s^2}$, but for simplicity in calculations, it is often approximated as $g = 10\ \mathrm{m/s^2}$.

• Variation: The value of g decreases with altitude and varies slightly with geographic location.

Equations of Motion for Free Fall

• Neglecting Air Resistance: For most introductory problems, air resistance is ignored.

• One-Dimensional Motion: If upward is taken as positive, then $a = -g$ for objects in free fall.

• From Rest: If an object is dropped from rest, its initial velocity $v_0 = 0$.

Key Equations:

• Velocity after time $t$: $v = gt$

• Distance fallen after time $t$: $d = \frac{1}{2}gt^2$

• Speed acquired: $\text{Speed} = \text{acceleration} \times \text{time} = gt$

• Distance as average velocity times time: $d = \left(\frac{0 + gt}{2}\right)t = \frac{1}{2}gt^2$

Example: If an object falls for 2 seconds, the distance covered is $d = \frac{1}{2} \times 10 \times (2)^2 = 20\ \mathrm{m}$.

Conceptual Questions on Free Fall

• Example Question: If you toss a coin straight up while riding in a van moving at constant speed, where will the coin land?

  • Answer: Back in your hand. (The coin retains the horizontal velocity of the van.)

Review of Motion Concepts

Key Terms

• Speed: The rate at which an object covers distance. Scalar quantity.

• Instantaneous Speed: The speed at a specific instant in time.

• Average Speed: Total distance divided by total time.

• Velocity: Speed with a specified direction. Vector quantity.

• Acceleration: The rate of change of velocity.

• Scalar: A quantity with magnitude only (e.g., speed, distance).

• Vector: A quantity with both magnitude and direction (e.g., velocity, acceleration).

Inertia and Newton's First Law of Motion

Observations and Everyday Examples

Everyday experiences, such as skating, illustrate the concept of inertia:

• At rest, a skater remains stationary unless pushed.

• In motion, a skater continues moving at constant speed and direction unless acted upon by an external force.

Inertia: The tendency of an object to resist changes in its state of motion. Mass is a measure of inertia; more mass means more inertia.

Example: When a car suddenly stops, your body tends to keep moving forward due to inertia. Seat belts provide the external force needed to stop you safely.

Newton's First Law of Motion (Law of Inertia)

• Statement (Version 1): An object free of external influences moves in a straight line and covers equal distances in equal times.

• Statement (Version 2): An object free of external influences moves at a constant velocity. (A motionless object is moving at a constant velocity of zero.)

• Implication: Objects at rest stay at rest, and objects in motion stay in motion with constant velocity unless acted upon by a net external force.

Historical Context: Aristotle, Copernicus, Galileo, and Newton

• Aristotle: Classified motion as 'natural' (objects seek their natural place) and 'violent' (caused by external forces).

• Copernicus: Proposed the heliocentric model, with Earth moving around the Sun.

• Galileo: Demonstrated that, in the absence of air resistance, all objects fall at the same rate regardless of mass. Introduced the concept of inertia.

• Newton: Formulated the laws of motion and universal gravitation, formalizing the concept of inertia as the First Law.

Force and Equilibrium

Definition of Force

• Force: A push or pull that can change the motion of an object. It is a vector quantity, having both magnitude and direction.

• Unit: The SI unit of force is the newton (N).

• Types of Forces:

  • Contact Forces: Require physical contact (e.g., friction, tension, normal force).

  • Field Forces: Act at a distance (e.g., gravity, electromagnetic forces).

Net Force and Mechanical Equilibrium

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

• Mechanical Equilibrium: When the net force on an object is zero ($\sum F = 0$), the object is in equilibrium.

• Types of Equilibrium:

  • Static Equilibrium: Object at rest.

  • Dynamic Equilibrium: Object moving at constant velocity in a straight line.

Examples and Applications

• Example 1: If you pull a box with 10 N and a friend pulls in the opposite direction with 5 N, the net force is 5 N in your direction.

• Example 2: If friction between a crate and the ground is 75 N (opposing motion), you must apply a force of 75 N in the direction of motion to keep the crate moving at constant speed (dynamic equilibrium).

Key Points about Force and Motion

• Force is required to change the motion (i.e., to accelerate an object), not to maintain constant velocity.

• On a frictionless surface, no force is needed to keep an object moving at constant velocity.

Sample Questions

• Question: After a ball leaves your hand when thrown upward, is there any force pushing it upward? (Neglect air resistance.)

  • Answer: No. Only gravity acts downward after release.

• Question: How much force is required to keep an object moving at constant velocity on a frictionless surface?

  • Answer: 0 N (no force required).

Summary Table: Types of Forces

Type of Force	Description	Examples
Contact Force	Requires physical contact between objects	Friction, tension, normal force
Field Force	Acts at a distance, no contact required	Gravity, electromagnetic force

What Did We Learn?

• The importance of Newton's laws in understanding motion.

• Newton's First Law (Law of Inertia) and its implications for objects at rest and in motion.

• The concept of force, including its types and effects.

• The conditions for static and dynamic equilibrium.

Additional info: Some explanations and context have been expanded for clarity and completeness, including historical context and explicit definitions of key terms.