Chapter 6: Momentum

Momentum and Impulse

Momentum is a measure of the motion of an object and is defined as the product of its mass and velocity. Impulse is the change in momentum resulting from a force applied over a time interval.

• Momentum (\( \vec{p} \)): Vector quantity. Formula:

• Impulse (\( \vec{J} \)): Vector quantity. Formula:

• Units: Momentum: kg·m/s; Impulse: N·s (equivalent to kg·m/s)

Conservation of Momentum

The total momentum of a closed system remains constant if no external forces act on it.

• Law of Conservation of Momentum:

• Applies to all types of collisions and explosions.

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

• Partially Elastic/Inelastic: Some kinetic energy is lost, but objects may not stick together.

Newton’s Laws and Impulse

• Newton’s 2nd Law: relates force to the rate of change of momentum (impulse).

• Newton’s 3rd Law: In collisions, forces between objects are equal and opposite, leading to equal and opposite impulses.

Impulse in Practice

• Impulse is the same if the change in momentum is the same, regardless of how force and time are distributed (e.g., catching a ball with bare hands vs. with a glove).

Example: Elastic and Inelastic Collisions

• Two carts on a track colliding and sticking together (inelastic) vs. bouncing apart (elastic).

Chapter 7: Energy

Work

Work is done when a force causes displacement. It is a scalar quantity.

• Formula:

• Units: Joule (J)

• Positive Work: Force and displacement in same direction.

• Negative Work: Force and displacement in opposite directions.

• Zero Work: Force perpendicular to displacement or no displacement.

Power

• Definition: Rate at which work is done.

• Formula:

• Units: Watt (W)

Kinetic and Potential Energy

• Kinetic Energy (KE):

• Potential Energy (PE): (gravitational)

Elastic Energy and Hooke’s Law

• Elastic Potential Energy:

• Hooke’s Law:

Energy Conservation in a Pendulum

• Potential Energy Highest: At the endpoints (maximum height)

• Kinetic Energy Highest: At the lowest point (maximum speed)

Work-Energy Theorem

• Theorem: Net work done equals change in kinetic energy:

• Application: "Sad" and "happy" balls (energy loss in inelastic vs. elastic collisions)

Conservation of Energy in a Roller Coaster

• Total mechanical energy (KE + PE) is conserved if no non-conservative forces (like friction) act.

Chapter 8: Rotational Motion

Rotational Displacement

• Definition: Angle through which an object rotates.

• Scalar/Vector: Vector (has direction)

• Units: Radian (rad), degree (°), revolution (rev)

• Conversions:

Rotational (Angular) Velocity

• Definition: Rate of change of angular displacement.

• Formula:

• Units: rad/s

Rotational (Angular) Acceleration

• Definition: Rate of change of angular velocity.

• Formula:

• Units: rad/s²

Relation Between Linear and Angular Velocity

Centripetal Force and Acceleration

• Centripetal Acceleration:

• Centripetal Force:

• Both are vectors, directed toward the center of the circle.

Tangential Velocity

• Definition: Linear speed along the edge of a rotating object.

• Formula:

• Direction: Tangent to the circle at any point.

Examples of Circular Motion

• Centripetal force and acceleration always point toward the center of the circle.

Sources of Centripetal Force

• Can be provided by tension (string), normal force (banked curve), friction (car tires), or gravity (planets).

Torque

• Definition: A measure of the tendency of a force to rotate an object about an axis.

• Formula:

• Units: N·m

• Vector: Yes

Center of Mass vs. Center of Gravity

• Center of Mass: Point representing the average position of mass.

• Center of Gravity: Point where the gravitational force can be considered to act.

Rotational Inertia (Moment of Inertia)

• Definition: Resistance to change in rotational motion.

• Formula:

• Units: kg·m²

• Vector: Scalar (but direction of rotation is vectorial)

Torque and Newton’s Second Law

• Rotational analog:

Angular Momentum

• Definition: Product of rotational inertia and angular velocity.

• Formula:

• Units: kg·m²/s

• Vector: Yes

Conservation of Angular Momentum

• In the absence of external torque, total angular momentum remains constant.

• Example: Figure skater spinning faster by pulling in arms.

Chapter 9: Gravity

Gravitational Force

• Newton’s Law of Universal Gravitation:

• G (Gravitational Constant): N·m²/kg²

• g (Acceleration due to gravity): m/s² on Earth’s surface

• Gravitational force is always attractive.

Weight vs. Weightlessness

• Weight: Force of gravity on an object ()

• Weightlessness: Apparent absence of weight, e.g., in free fall or orbit.

Ocean Tides

• Caused by the gravitational pull of the Moon (and Sun) on Earth’s oceans.

• Tides are not significant in small bodies of water like ponds or pools due to their size.

Spring and Neap Tides

• Spring Tides: Occur when Sun, Moon, and Earth are aligned; highest tides.

• Neap Tides: Occur when Sun and Moon are at right angles; lowest tides.

• Best time to collect clams: During spring tides (lowest low tides).

Gravitational Field

• Region of space around a mass where another mass experiences a force of gravity.

• Field Strength:

Chapter 10: Projectile and Satellite Motion

Projectile Motion

• Definition: Motion of an object thrown into the air, subject only to gravity.

• Characteristics: Parabolic trajectory, independent horizontal and vertical motions.

Horizontal and Vertical Components

• Horizontal Velocity: Constant (no horizontal force if air resistance is neglected)

• Vertical Velocity: Changes due to gravity (acceleration downward)

Fast-Moving Projectiles

• Satellites and space probes are examples; their speed allows them to "fall around" Earth.

Energy in Orbits

• Circular Orbit: Kinetic and potential energy are constant; total energy is negative.

• Elliptical Orbit: Kinetic and potential energy vary, but total energy remains constant.

Escape Velocity

• Definition: Minimum speed needed to escape a gravitational field.

• Formula: (for Earth’s surface)

• Earth: About 11.2 km/s

• Solar System: Higher, depends on position and Sun’s gravity

Kepler’s Three Laws

• 1st Law (Law of Orbits): Planets move in ellipses with the Sun at one focus.

• 2nd Law (Law of Areas): A line joining a planet and the Sun sweeps out equal areas in equal times.

• 3rd Law (Law of Periods): (square of orbital period proportional to cube of semi-major axis)

Chapter 12: Solids

Amorphous vs. Crystalline Solids

• Crystalline: Atoms arranged in a regular, repeating pattern (e.g., salt, diamond).

• Amorphous: Atoms lack long-range order (e.g., glass, plastic).

Mass Density vs. Weight Density

• Mass Density (\( \rho \)): ; units: kg/m³; scalar

• Weight Density: ; units: N/m³; scalar

Elasticity and Hooke’s Law

• Elasticity: Ability of a material to return to its original shape after deformation.

• Hooke’s Law: (force proportional to extension/compression)

Tension vs. Compression

• Tension: Force that stretches a material.

• Compression: Force that squeezes a material.

Table: Comparison of Key Properties

Additional info: Some explanations and formulas have been expanded for clarity and completeness, and tables have been constructed to summarize key comparisons and properties.