PHY101 Exam-1 Study Guide: Foundations of Mechanics and Gravity

About Science

Scientific Hypothesis

A scientific hypothesis is a proposed explanation for a phenomenon that can be tested through experimentation or observation. For a hypothesis to be considered scientific, it must be testable and falsifiable, meaning there must be a possible outcome that could prove it wrong.

• Testable: Can be evaluated by experiments or observations.

• Falsifiable: There must be a conceivable result that would show the hypothesis is incorrect.

• Example: "All objects fall at the same rate in a vacuum" is a scientific hypothesis because it can be tested and potentially disproven.

Vectors and Scalars

Scalars vs. Vectors

Physical quantities are classified as scalars or vectors based on whether they have direction as well as magnitude.

• Scalar: Has magnitude only (e.g., speed, mass, temperature).

• Vector: Has both magnitude and direction (e.g., force, velocity, acceleration).

• Examples: Mass = scalar; Velocity = vector.

Forces as Vectors

All forces are vector quantities, meaning they have both magnitude and direction. Examples include friction, weight, air resistance, and the normal force.

Vector Addition

When combining vectors, their directions must be considered:

• Same direction: Add magnitudes.

• Opposite direction: Subtract magnitudes.

• Angled vectors: Use vector addition rules (e.g., parallelogram rule or Pythagorean theorem for perpendicular vectors).

• Resultant force: The single vector that has the same effect as all the individual forces combined.

• Range for two equal forces: The resultant can vary from 0 (opposite directions) to twice the magnitude (same direction).

Newton's First Law of Motion - Inertia

Newton's First Law (Law of Inertia)

Newton's First Law states that an object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Balanced forces: When all forces cancel, resulting in zero net force and no change in motion.

• Example: A hockey puck sliding on ice will keep moving at constant speed unless friction or another force acts on it.

Linear Motion

Net Force and Motion

The net force is the vector sum of all forces acting on an object. If the net force is zero, the object's velocity remains constant (it does not accelerate).

• Constant velocity: Occurs when net force is zero.

• Friction and constant velocity: When an object moves at constant velocity, the applied force equals the friction force.

Acceleration

Acceleration is the rate at which velocity changes with time.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Units: meters per second squared (m/s2).

• Turning motion: An object can accelerate by changing direction even if its speed is constant (e.g., a car turning a corner).

Speed vs. Velocity

• Speed: Scalar quantity; how fast an object moves.

• Velocity: Vector quantity; speed with direction.

Gravity and Free Fall

Free Fall

Free fall occurs when gravity is the only force acting on an object.

• Acceleration due to gravity: $g \approx 9.8\ \text{m/s}^2$ downward near Earth's surface.

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

• At the highest point: Velocity is zero, but acceleration is still downward ($g$).

Mass and Weight

Mass

Mass is the amount of matter in an object. It does not change with location.

Weight

Weight is the gravitational force acting on an object.

• Formula: $W = mg$

• Units: Newtons (N).

• Mass vs. Weight on Planets: Mass remains constant; weight changes with gravitational strength.

Newton's Second Law of Motion

Newton's Second Law

Newton's Second Law relates force, mass, and acceleration.

• Formula: $F = ma$

• Constant force: Produces constant acceleration.

• Effect of mass: Greater mass results in smaller acceleration for the same force.

Energy and Work

Work

Work is done when a force moves an object over a distance.

• Formula: $W = Fd$

• Units: Joules (J), where $1\ \text{J} = 1\ \text{N} \cdot 1\ \text{m}$

• Normal force: Does no work if perpendicular to motion.

Kinetic Energy

Kinetic energy is the energy of motion.

• Formula: $KE = \frac{1}{2}mv^2$

Energy Conversion

• Mechanical energy lost due to friction is converted into thermal energy.

• Many energy sources (wind, hydroelectric, fossil fuels) ultimately derive from solar energy.

Gravity

Newton's Law of Universal Gravitation

Describes the gravitational force between two masses.

• Formula: $F = \frac{GMm}{r^2}$

• G: Universal gravitational constant.

• m, M: Masses of the two objects.

• r: Distance between centers of mass.

Surface Gravity

• Formula: $g = \frac{GM}{R^2}$

• Surface gravity depends on a planet's mass and radius.

Planet Discovery

• Some planets were discovered by observing gravitational effects on the orbits of known planets.

Elasticity and Hooke's Law

Hooke's Law

Describes the behavior of elastic materials (e.g., springs).

• Formula: $F = kx$

• F: Force applied

• k: Spring constant

• x: Extension or compression from equilibrium

Summary Table of Key Formulas

Concept	Formula
Acceleration	$a = \frac{\Delta v}{\Delta t}$
Free fall distance	$d = \frac{1}{2}gt^2$
Weight	$W = mg$
Work	$W = Fd$
Kinetic Energy	$KE = \frac{1}{2}mv^2$
Gravitational Force	$F = \frac{GMm}{r^2}$
Surface gravity	$g = \frac{GM}{R^2}$
Hooke's Law	$F = kx$

Additional info:

• All formulas are presented in SI units.

• Understanding the distinction between mass and weight is crucial for solving problems involving gravity on different planets.

• Energy conservation and conversion are central themes in mechanics and thermodynamics.