Physics Study Guide

GENERAL PHYSICS 1 STUDY NOTES

LESSON 1: FORCE AND MOTION

THEORY OF EVERYTHING

A comprehensive theory unifying the four fundamental forces:
- Gravitational Force (weakest)
- Weak Nuclear Force
- Electromagnetic Force (EMF)
- Strong Nuclear Force (strongest)
This theory is derived from the Grand Unified Theory, which includes magnetic, electrical, and nuclear forces.

FORCE

Defined as a push/pull exerted by one object on another.
Vector Quantity: It possesses both magnitude and direction.

FOUR FUNDAMENTAL FORCES

Gravitational
Electromagnetic
Weak Nuclear
Strong Nuclear

TYPES OF FORCES

Contact Forces
- Applied
- Spring
- Drag (air friction)
- Frictional
- Normal
Field Forces
- Gravitational
- Magnetic
- Electrostatic

LAWS OF MOTION

NET FORCE

Defined as the resultant force, which is the vector sum of all forces acting on a body.
If the net force (ΣF) is not equal to zero (ΣF ≠ 0), the forces are unbalanced.

1ST LAW OF MOTION: INERTIA

Describes the behavior of an object at rest or in motion.
Key Points:
- An object at rest remains at rest.
- An object in motion remains in motion unless acted upon by an unbalanced force.
Inertial Frame of Reference: The observation of motion is relative to the point of view of the observer.

2ND LAW OF MOTION: ACCELERATION

States that the acceleration of a body is directly proportional to the net force acting upon it and inversely proportional to its mass.
Formula:
$F = ma$
SI Unit of Force: Newton (N)
Dyne: A smaller unit of force.
- $1 ext{ N} = 1 ext{ kg} imes ext{m/s}^2$
- $1 ext{ dyne} = 1 ext{ g} imes ext{m/s}^2$
- $1 ext{ N} = 10^5 ext{ dynes}$

MASS AND WEIGHT

Mass:
- Defined as the amount of matter in a body.
- It is a scalar quantity and remains constant.
Weight:
- Defined as the measure of the force of gravity acting on a body.
- It is a vector quantity and directed towards the center of the Earth.
- Dependence: Weight depends on the local acceleration due to gravity.
- Formula:
  $W = mg$

3RD LAW OF MOTION: INTERACTION

Often summarized as “Every action has an equal and opposite reaction.”
Key Concept: When body A exerts a force on body B, body B exerts an equal but opposite force on body A.
- These forces form an action-reaction pair.
- Notably, they act on different bodies and thus do not cancel each other out.

FIRST CONDITION OF EQUILIBRIUM

Defined via the Law of Inertia, where equilibrium occurs.
Types of Equilibrium:
- Static Equilibrium: Object at rest.
- Dynamic Equilibrium: Object moving at constant velocity.
Condition for Equilibrium:
$ΣF = 0$ (Net forces/summation of forces equals zero)

FREE-BODY DIAGRAM

A diagram illustrating the magnitude and direction of all forces acting on an object, excluding forces exerted by itself.

ACCELERATING SYSTEM OF MASSES

Atwood Machine:
- The first lab apparatus that verified Newton’s laws of motion.
- Utilizes cables and ropes effectively to transmit force.
- Forces:
 $(m2 - m1)g = (m2 + m1)a$

IMPULSE-MOMENTUM THEOREM

IMPULSE (I):
- Defined as the product of force and the time interval during which the force acts on an object.
- Formula:
  $I = F∆t$
- This concept connects to Newton’s 2nd law of motion.
MOMENTUM (P):
- Defined as the product of mass and velocity of an object.
- Formula:
  $p = m∆v$
Relational Formulas:
- Change in momentum:
  $I = p - p_0$
- Force in terms of impulse:
  $F∆t = m∆v$
- Alternative form:
  $F = \frac{mv - mv_0}{∆t}$

FRICTION

A force that resists the motion between materials in contact.
- It exists in all types of materials.
Types of Friction:
- Static Friction:
- Acts when an object is stationary.
- Formula:
 $Fs = µs mg$
- Kinetic Friction:
- Acts when an object is in motion.
- Formula:
 $Fk = µk mg$

LAWS ON FRICTION

Static friction is greater in magnitude than kinetic friction.
Friction acts in the opposite direction of motion of surfaces in contact.
The contact area and speed of sliding do not affect friction.
Friction is proportional to the normal force.
Friction depends on the nature and condition of the surfaces.

ANGLE OF REPOSE (θ_r)

Defined as the angle at which an object is on the verge of motion down a slope.
Starts moving if the ramp angle equals θ_r.
Formula:
$θr = tan^{-1}(µs)$

ANGLE OF UNIFORM SLIP (Ø)

Occurs when the inclination of the plane is reduced to less than θ_r.
Conditions:
- The object moves at constant velocity if the angle of ramp is Ø.
Formula:
$Ø = tan^{-1}(µ_k)$

FRICTION AND BANKING OF ROADS

Designed speed must not exceed a certain limit to reduce skidding.
Speed Formulas:
- Given angle:
 $v_{max} = rgtan(θ)$
- Given the coefficient of static friction:
 $v{max} = rgµs$

MOTION IN A VISCOUS FLUID

Stoke’s Law of Resistance:
- Governs small object movement through air at speeds ≤ 25m/s.
- Formula:
  $F_D = Bv$
Newton’s Law of Resistance:
- Applies when speeds are greater than or equal to 25m/s or less than or equal to 325m/s.
- Formula:
  $F_D = Cv^2$
Terminal Velocity:
- A falling object accelerates due to gravity until the drag force equals the object’s weight.
- This results in no further acceleration; the object descends at a constant velocity.
- Formula:
  $F_D = mg$

FORMULA BANK

Friction:
- Static Friction:
 $Fs = µs mg$
- Kinetic Friction:
 $Fk = µk mg$
Forces:
- Newton’s 2nd law:
  $F = ma$
- Weight:
  $W = mg$
For Curved Roads:
- Speed with given angle:
 $v_{max} = rgtan(θ)$
- Speed with static friction coefficient:
 $v{max} = rgµs$
Impulse-Momentum Theorem:
- Impulse:
  $I = F∆t$
- Momentum:
  $p = m∆v$
- Impulse = Momentum:
  $F∆t = m∆v$
Angles Formulas:
- Angle of Repose:
 $θr = tan^{-1}(µs)$
- Angle of Uniform Slip:
 $Ø = tan^{-1}(µ_k)$
Viscous Fluid Law:
- Stoke's Law:
  $F_D = Bv$
- Newton's Law:
  $F_D = Cv^2$
- Terminal Velocity:
  $F_D = mg$

LESSON 2: WORK, ENERGY, AND POWER

WORK AS A DOT PRODUCT

Defined as the dot product of force and displacement in the direction of that force.
Formula:
$W = Fd ext{ cos} θ$
SI Unit: Newton meter (Nm) or Joules (J).

CASES OF WORK

Case 1: θ = 0° (Same direction)
- Calculation:
  $cos(0°) = 1$
- Work: Positive work done.
Case 2: 0° < θ < 90° (One component in the same direction)
- Work:
 $W = F_x d ext{ cos} θ$
Case 3: θ = 90° (Right Angle)
- Calculation:
  $cos(90°) = 0$
- Work: No work done.
Case 4: θ = 180° (Opposite direction)
- Calculation:
  $cos(180°) = -1$
- Work: Negative work.

WORK DONE BY SPRING

Work is defined as done during the compression/stretching from its normal length.
Force: $F = k∆x = k(x−x_0)$
- Where k is the force constant.
SI Unit of k: N/m
Formula for Work Done:
$W = \frac{1}{2}k(∆x)^2$

MECHANICAL ENERGY

Energy: The capacity to do work.
Expressed in Joules (J) or ergs.
Energy is considered a scalar quantity.

POTENTIAL ENERGY

General Definition: Represents energy possessed by a body due to its position or configuration.
Gravitational Potential Energy (GPE):
- Denoted as U_G, it is due to position relative to the ground.
- Formula:
  $U_G = mgh$
Elastic Potential Energy:
- Denoted as U_S, it arises from the configuration, commonly in elastic materials like springs.
- Formula:
  $U_S = \frac{1}{2}k(∆x)^2$

KINETIC ENERGY (KE)

Denoted as K, it is the energy possessed by an object due to its motion.
Formula:
$K = \frac{1}{2}mv^2$

WORK-ENERGY THEOREM

States that the work done on an object results in a change in its motion.
Formula:
$W = ∆K$
$W = K - K_0$
Expanded Formula:
$W = \frac{1}{2}m(v^2) - \frac{1}{2}m(v_0^2)$

POWER

Defined as the rate at which work is done.
SI Unit: Watt (W)
Formulas:
- $P = \frac{W}{t} = \frac{Fd ext{ cos }θ}{t} = Fv ext{ cos }θ$
- $1 W = 1 J/s$
- $1 hp = 746 W$

METABOLIC RATE

Represents how fast the body converts food into energy, specifically the rate of burning calories.
Factors Influencing Rate:
- Mass
- Age
- Gender
- Physical Activity
Energy Conversion Relation:
$1 ext{ calorie} = 4186 J$

LAW OF CONSERVATION OF ENERGY

States that energy cannot be created or destroyed; it can only change form.
Formula:
$K1 + U1 = K2 + U2$
In classical mechanics, the sum of K and U in a closed system remains constant:
$\frac{1}{2}m(v1^2) + mg(h1) = \frac{1}{2}m(v2^2) + mg(h2)$

CONSERVATION OF ENERGY

Conservative Forces:
- Independent of path
- Work done in a round trip is zero
- Total energy remains constant and is recoverable
Non-Conservative Forces:
- Depends on path
- work done in round trips is not zero
- Energy dissipated as heat, thus not fully recoverable.

THREE TYPES OF EQUILIBRIUM

Stable Equilibrium:
- A slight displacement results in forces returning the object to its original position.
Unstable Equilibrium:
- A slight displacement results in forces moving the object further away from its original position.
Neutral Equilibrium:
- Any displacement results in forces keeping the object in the new position.

EQUILIBRIUM & POTENTIAL ENERGY DIAGRAMS

Represents a graph of potential energy on the vertical axis and position on the horizontal axis.
- Max Point: Represents unstable equilibrium.
- Min Point: Represents stable equilibrium.
- Saddle Point: Represents neutral equilibrium.

CONSERVATION OF MECHANICAL ENERGY

If no non-conservative forces are present, the total mechanical energy in an isolated system remains constant.
Formula:
$E_m = U + K = constant$

FORMULA BANK

Work as a Dot Product:
$W = F · d = Fd ext{ cos }θ$
Mechanical Energies:
- Gravitational Potential Energy:
  $U_G = mgh$
Elastic Potential Energy:
$U_S = \frac{1}{2}k(∆x)^2$
Kinetic Energy:
$K = \frac{1}{2}mv^2$
Work-Energy Theorem:
$W = K - K_0$
Power:
$P = \frac{W}{t}$
Conservation of Energy:
$K1 + U1 = K2 + U2$

IMPULSE, MOMENTUM, AND COLLISION

CENTER OF MASS

Defined as the average location of all masses in an object, or the point where you can balance the object.
- Geometric Center: Applicable for regular shapes.
- Center of Gravity: Used for irregular shapes, calculated via mass-weighted position.

MOMENT (τ)

Defined as the cross product of a quantity such as force and its distance to a point from a pivot.
Formula:
$τ = F r ext{ sin }θ$
To find center of gravity, use:
$(m1 + m2)x{cm} = m1x1 + m2x_2$

CALCULATING THE CENTER OF MASS

The center of mass can be calculated using the formula:
$(x{cm}, y{cm}, z{cm}) = \frac{ extstyle ext{sum of all } mi ext{ x}i}{M}, \frac{ extstyle ext{sum of all } mi ext{ y}i}{M}, \frac{ extstyle ext{sum of all } mi ext{ z}_i}{M}$
This formula is split and simplified as:
$x{cm} = \frac{ extstyle ext{sum of all } mi ext{ x}_i}{M}$
Where M is the total mass and all moments are summed in each component and divided by this total mass.

VELOCITY OF THE CENTER OF MASS

Can be calculated using a similar approach to the coordinates of the center of mass:
$v{cm} = \frac{m1v1 + m2v2 + … + mNvN}{m1 + m2 + … + mN}$

MOMENTUM

Defined as mass in motion, expressed as the product of mass and velocity.
Formula: $p = mv$
- An object with a greater mass will have high momentum, even at low speeds, and vice versa for objects with high velocity.
Force as Change in Momentum:
$F = \frac{p - p_0}{t}$

IMPULSE

Defined as the change in momentum.
Formula:
$I = ∆p$
Related to force:
$I = Ft$

CONSERVATION OF LINEAR MOMENTUM

The principle that momentum is conserved in a system.
- Internal Forces: Forces that particles in a system exert on one another (fundamental forces excluding gravity).
- External Forces: Forces applied by outside objects.
- Isolated System: A system with no external forces acting on it.
Formula for Conservation:
$mA v{Af} + mB V{Bf} = mA V{A0} + mB V{B0}$

COLLISION

Defined as an interaction of two or more masses that encounter each other.
Key Concept: Total momentum is always conserved during a collision.

COEFFICIENT OF RESTITUTION

Defined as the negative ratio of the relative velocities of the masses after and before the collision.
Formula:
$e = -\frac{(vA2 - vB2)}{(vA1 - vB1)}$

TYPES OF COLLISION

Elastic Collision: In this collision, objects