Lecture 2: Newton's First Law

Introduction to Newton’s First Law of Motion

• Conceptual Overview: Newton's first law of motion describes the property of inertia. Inertia is defined as the resistance that objects have to changes in their motion. This includes resistance to:

  • Speeding up.

  • Slowing down.

  • Changing direction.

• Formal Statement: Newton’s first law states that "an object in motion tends to stay in motion, and an object at rest tends to stay at rest, unless acted upon by an outside force."

• Fundamental Principles:

  • All objects possessing mass will continue to move at a constant speed in the same direction unless acted upon by an outside force.

  • If an object of mass is not moving, it will remain at rest unless pushed or pulled by an outside force.

  • This law is frequently referred to as the Law of Inertia because it describes the inherent property of matter.

The Relationship Between Inertia and Mass

• Definition of Inertia: Inertia describes how difficult it is to change the motion of an object. This relates to how hard it is to get a resting object to move, or how hard it is to alter the velocity of a moving object.

• Experimental Demonstration of Inertia:

  • Setup: Three objects of similar shape and size but different materials were tested: a piece of Styrofoam, a clay brick, and a piece of lead.

  • Procedure: Each object was moved back and forth as many times as possible within a $5\,s$ interval to measure the difficulty of changing its motion.

  • Results:

    • Styrofoam: Very easy to move; low difficulty.

    • Clay Brick: Significantly more difficult to move than the Styrofoam.

    • Lead: Extremely difficult to move; required two hands to pick up.

  • Conclusion: Since the size (volume) and shape were the same, the difference in difficulty (inertia) must be due to the difference in mass. Therefore, mass is what causes inertia.

Demonstrations and Real-World Applications of Inertia

• The Tablecloth Demonstration:

  • Scenario: Objects (a plate, silverware, a flower in a vase, and a glass of blue water) sit on a table.

  • Physics involved: Gravity pulls the objects down, while a support force counteracts gravity to hold them up.

  • Result: When the tablecloth is removed quickly, no significant horizontal force is introduced. Gravity and support forces remain constant. The objects stay put (at rest) due to their inertia.

  • Observation: The silverware, having the least mass of all the objects, was slightly more affected than the heavier items, illustrating that lower mass results in lower inertia.

• The Ballistic Cart Demonstration:

  • Scenario: A cart moving at a constant speed contains a small yellow ping-pong ball. It passes a trigger that launches the ball vertically into the air.

  • Physics involved: Because the ball and cart were moving at the same speed when together, they continue moving at that same horizontal speed even when separated, provided no outside force (like air resistance) acts on them.

  • Result: The ball lands back inside the cart because their horizontal motion remains synchronized due to inertia.

• Objects in Relative Motion (The Train Example):

  • If you throw a ball up in a train moving at a constant velocity, you will catch it.

  • Scenario 1 (Speeding up): If the train accelerates while the ball is in the air, the ball continues at its initial speed, landing behind the point where it was thrown.

  • Scenario 2 (Slowing down): If the train decelerates while the ball is in the air, the ball continues at its initial speed, landing ahead of where it was thrown.

• Vehicle Safety (Seatbelts):

  • When driving, your body moves at the same speed as the car.

  • If the car stops abruptly (due to a collision or hard braking), your body’s inertia causes it to continue moving forward at the previous speed.

  • Seatbelt Function: They apply a force to the body to ensure it slows down with the car, preventing the occupant from being thrown forward or through the windshield.

Properties of Mass and Weight

• Definition of Mass ($m$): Mass is a property of physical objects relating to their resistance to changes in motion (inertia). It relates to how much "stuff" or matter is in an object.

• Units: Mass is measured in kilograms ($kg$).

• Invariance of Mass: Mass is a characteristic that does not change based on location. The mass of an object is the same on Earth, the Moon, Jupiter, or in space.

  • Measured Brick Masses:

    • Styrofoam: $0.04\,kg$

    • Clay Brick: $2\,kg$

    • Lead: $12\,kg$

• Human Mass: While humans can change their mass (via diet, exercise, or cutting hair), a constant mass value remains the same regardless of the local gravitational force.

• Mass vs. Weight:

  • Mass: Relates to inertia and resistance to motion changes.

  • Weight: A force that exists due to the pull of gravity on a mass.

Force and Units

• Definition of Force ($F$): Simply put, a force is a push or a pull that causes an object to change its motion.

• Vector Nature: Force is a vector quantity, meaning it has both magnitude (strength) and direction.

• Notation: Represented by a capital letter $F$, usually in bold or with an arrow above it: $\mathbf{F}$ or $\vec{F}$.

• Units: The Newton ($N$).

  • A Newton is a derived unit comprised of base units:

  • $1\,N = 1\,kg \times \frac{m}{s^2}$

• Function: Forces are required to overcome inertia.

Mechanical Equilibrium

• Equilibrium Definition: An object is in mechanical equilibrium when the sum of all forces acting on it is equal to zero.

• Mathematical Notation: The Greek letter Sigma ($\Sigma$) denotes "the sum of."

  • $\Sigma F = 0$

• Net Force: The sum total of all forces is also called the net force. When the net force is zero, the object is in equilibrium.

• Result of Equilibrium: The object will have a constant velocity (constant speed along a straight line).

• Types of Mechanical Equilibrium:

  1. Static Equilibrium: The object is at rest ($v = 0$). For example, a cart sitting on a track where gravity pulling down is perfectly balanced by the support force from the track pushing up.

  2. Dynamic Equilibrium: The object moves at a constant, non-zero velocity. For example, a cart pushed until it travels at a constant speed on a low-friction track. Once the push stops, gravity and support forces are balanced, and if horizontal friction/drag are negligible, the cart remains in dynamic equilibrium.

Vectors and Scalars

• Scalar: A quantity containing magnitude but no direction.

  • Example: Mass is a scalar (e.g., $2\,kg$). It does not have a direction like "up" or "left."

• Vector: A quantity containing both magnitude and direction.

  • Representation: Denoted graphically with an arrow.

  • Length of Arrow: Represents the magnitude.

  • Arrowhead Position: Represents the direction.

• Directional Systems:

  • One-Dimensional/Cartesian: Using plus ($+$) for up/right and minus ($-$) for down/left.

  • Cardinal Directions: North, South, East, West.

  • Angles: While common in physics, trigonometry (sine, cosine, tangent) will not be used in this specific course context.

Vector Components and Calculations

• Components: Any vector can be broken down into horizontal and vertical parts.

• Example 1: Horizontal Vector:

  • Start: $(4, 0)$, End: $(0, 0)$.

  • Horizontal component: $-4$ (points left).

  • Vertical component: $0$.

  • Magnitude: $4$ (Magnitude is always positive).

• Example 2: Vertical Vector:

  • Start: $(-3, 0)$, End: $(-3, 2)$.

  • Horizontal component: $0$.

  • Vertical component: $+2$ (points up).

  • Magnitude: $2$.

• Example 3: Diagonal Vector:

  • Start: $(2, -1)$, End: $(6, 8)$.

  • Horizontal component: $6 - 2 = 4$.

  • Vertical component: $8 - (-1) = 9$.

  • Magnitude calculation using Pythagorean theorem:

  • $\text{Magnitude} = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97} \approx 9.85$

• Example 4: Diagonal Vector:

  • Start: $(6, 3)$, End: $(0, -2)$.

  • Horizontal component: $0 - 6 = -6$ (Left, length of $6$).

  • Vertical component: $-2 - 3 = -5$ (Down, length of $5$).

  • $\text{Magnitude} = \sqrt{(-6)^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} \approx 7.81$

  • Calculator Tip: Always use parentheses when squaring negative numbers: $(-6)^2$.

Vector Addition Methods

• Graphical Method:

  1. Draw the first vector.

  2. Place the tail of the second vector at the tip of the first vector.

  3. The sum (resultant) is the straight line drawn from the starting point of the first vector to the endpoint of the last vector.

  4. The order of drawing does not change the result.

• Numerical Method:

  1. Write out the components for each vector.

  2. Add all horizontal components together to find the resultant horizontal component.

  3. Add all vertical components together to find the resultant vertical component.

  4. Example: Adding $(4, 9)$ and $(-6, -5)$.

     • Horizontal: $4 + (-6) = -2$

     • Vertical: $9 + (-5) = 4$

     • Sum vector: $(-2, 4)$

     • $\text{Magnitude of sum} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47$