Laws of Motion

A. Newton's First Law of Motion:

"An object remains in a state of rest or of uniform motion in a straight line unless acted upon by an external unbalanced force."

• Qualitatively defines Force: an external agency that alters the state of rest or uniform motion.
• Introduces Inertia: the inherent natural property of a body to resist any change in its velocity.
• Measure of Inertia: Mass is the quantitative measure of inertia ($M \propto \text{Inertia}$). A heavier body possesses greater inertia.

B. Three Manifestations of Inertia:

1. Inertia of Rest: When a stationary bus abruptly accelerates forward, passengers jerk backward because their feet move with the bus floor while their upper bodies tend to stay at rest.
2. Inertia of Motion: When a speeding car suddenly brakes, passengers lurch forward because their bodies tend to maintain their forward velocity. (Hence, seatbelts are mandatory!).
3. Inertia of Direction: When a car takes a sharp left curve, passengers lean toward the right because their inertia resists turning and tends to continue in a straight line.

A. Linear Momentum ($p$):

The total quantity of motion contained in a body, defined as the product of its mass and velocity:

SI unit: $\mathbf{\text{kg}\cdot\text{m/s}}$ or $\text{N}\cdot\text{s}$ (Vector quantity, direction identical to velocity).

B. Derivation of $F = ma$:

Newton's Second Law states: Rate of change of momentum is proportional to applied force:

Since $\frac{v - u}{t} = a$ (acceleration):

In the SI system, the unit of force (Newton) is chosen such that $k = 1$:

• One Newton ($1\text{ N}$): The force that acts on a mass of $1\text{ kg}$ to produce an acceleration of $1\text{ m/s}^2$: $$1\text{ N} = 1\text{ kg} \times 1\text{ m/s}^2 = 1000\text{ g} \times 100\text{ cm/s}^2 = \mathbf{10^5\text{ dynes}}$$

C. Impulse of a Force:

A cricket fielder pulls his hands backward while catching a ball to increase the contact time $\Delta t$. By increasing $\Delta t$, the retarding force $F = \frac{\Delta p}{\Delta t}$ is dramatically reduced, preventing pain and injury to the hands!

Statement:

"To every action, there is always an equal and opposite reaction."

where $\vec{F}_{AB}$ is force exerted on body $A$ by body $B$, and $\vec{F}_{BA}$ is force exerted on body $B$ by body $A$.

Crucial Characteristics:

• Forces always occur in matched pairs; single isolated forces cannot exist in the universe.
• Action and reaction act on TWO DIFFERENT BODIES. Therefore, they never cancel each other out!
• Examples: (1) Walking: foot pushes ground backwards (action), ground pushes foot forward (reaction); (2) Rocket propulsion: high-speed exhaust gases expelled downward (action) push rocket upward (reaction); (3) Recoil of gun: bullet propelled forward, gun recoils backward.

A. Universal Law of Gravitation:

where $G = 6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ is the Universal Gravitational Constant.

B. Mass vs Weight:

Problem 1: A force acts for $0.1\text{ second}$ on a body of mass $2\text{ kg}$ initially at rest. The force ceases to act, and the body moves through $4\text{ meters}$ in the next $2\text{ seconds}$. Find the magnitude of the applied force.

Solution:

1. After the force ceases to act, the body moves with constant velocity $v$:

2. During the first $0.1\text{ s}$, the body accelerated from $u = 0$ to $v = 2\text{ m/s}$:

3. Magnitude of Force:

Problem 2: A bullet of mass $20\text{ g}$ moving at $300\text{ m/s}$ penetrates a sandbag and is brought to rest in $0.05\text{ seconds}$. Calculate: (i) The impulse of the force, (ii) The average retarding force exerted by the sand.

Solution:

Mass $m = 20\text{ g} = 0.02\text{ kg}$, Initial velocity $u = 300\text{ m/s}$, Final velocity $v = 0$, Time $\Delta t = 0.05\text{ s}$.

(i) Impulse ($J$):

(ii) Average Retarding Force ($F$):