Grade 11 Physics Notes
Grade 11 Physics - Vectors
Chapter 1: Vectors - Introduction
Mechanics studies the models physicists use to describe motion.
This includes describing what motion is and how various factors affect it.
Mathematics, especially vectors, is the language used to articulate these scientific models.
Objectives:
Differentiate between vectors and scalars.
Effectively indicate direction.
Determine the resultant of multiple vectors.
Resolve vectors into components.
John von Neumann: Sciences primarily create mathematical models with verbal interpretations to describe observed phenomena; justification is based on the model's effectiveness.
Mathematics is presented as the most successful global language.
Scalars vs. Vectors
Problem: Finding a friend who traveled 2 km in a straight line from school.
Without direction, the friend could be anywhere on a circle with a 2 km radius.
Knowing the direction (e.g., 2 km North) simplifies the search.
Scalar: Quantity with only magnitude (e.g., distance - 2 km).
Vector: Quantity with both magnitude and direction (e.g., displacement - 2 km due North).
Definitions
Scalar: A quantity that has only magnitude.
Vector: A quantity that has both magnitude and direction.
Distance: The actual path length covered by an object.
Displacement: Change in position in space, including length and direction of the straight line from start to finish.
Position is relative to a reference point (where position is zero, can be positive or negative).
Examples:
Distance and displacement are scalar and vector examples, respectively.
Other vector examples: velocity, acceleration, force, momentum.
Indicating Direction
Direction is always defined relative to a frame of reference (e.g., North, horizontal, or vertical line).
Methods to indicate direction:
Bearing: Measured clockwise from North.
Determining Resultants (Adding Vectors)
Adding vectors is different from adding scalars because direction matters.
Possibilities:
Vectors in a straight line.
Vectors at an angle to each other.
Resultant: Single vector with the same effect as the original vectors acting together.
Adding vectors in a straight line:
Same direction: Simply add magnitudes.
Opposite directions: Assign opposite signs and add.
Adding vectors at an angle:
Triangle method (for two vectors): Join tail of one vector to the head of the other; the resultant connects the tail of the first to the head of the last.
Parallelogram method (for two vectors): Draw vectors from the same point, complete the parallelogram, and the diagonal from the starting point is the resultant.
Polygon method (for more than two vectors): Add vectors tail to head; the resultant connects the tail of the first vector to the head of the last.
Examples
Determining the resultant of two vectors using a scale drawing and calculation.
Resolving Vectors into Components
Components: Vectors that add to give a resultant.
The process involves finding the original components from a known resultant.
Problem: A girl's resultant displacement is 20 km on a bearing of 40°. She walked East then North. Find the original components.
Graphical Method: Scale drawing to determine components by constructing a line perpendicular to the horizontal axis from the end of the displacement vector.
Calculation Method: Trigonometric functions can be used if components are perpendicular.
Vertical component = 20 \sin(50^ ) = 15.32 km
Horizontal component = 20 \cos(50^ ) = 12.86 km
Review 1.3
A vector with magnitude 30 has a bearing of 260°. The magnitude of the vertical component is: 30 \sin(10^ ).
Chapter 2: Force as a Vector
Introduction
Force is explored as a vector, building on knowledge from the previous chapter.
Objectives:
Understand the vector nature of force.
Know what an equilibrant is.
Use the triangle method for three forces in equilibrium.
Determine components of objects on slopes.
Fundamental Forces in Nature:
Electromagnetic force: Interaction between magnetism and electricity (Faraday).
Gravitational force: Exists between objects with mass.
Strong nuclear force: Holds atomic nuclei together.
Weak nuclear force: Responsible for radioactive decay.
Scientists seek a Theory of Everything (TOE) to describe all four forces under a common mechanism.
The Vector Nature of Force
Force is a push or pull, having magnitude and direction (vector).
Force can be represented by an arrow with length indicating magnitude and direction indicating direction.
Resultant force:
Single force that can replace all other forces, having the same effect.
Mathematically, vector addition for forces is the same as for other vectors.
Equilibrant of a Force
When vectors add up to zero, the resultant is zero, leading to a closed geometrical figure.
Equilibrium: Forces that add up to zero resultant. Each force is an equilibrant of the others.
Equilibrant: Force that keeps other forces in equilibrium; it is equal in magnitude to the resultant but acts in the opposite direction.
The Triangle Rule for Three Forces in Equilibrium
Forces in equilibrium form a closed geometrical figure.
Three forces in equilibrium form a triangle (triangle rule).
Triangle Rule: If three forces are in equilibrium, they can be represented in both magnitude and direction by the three sides of a triangle taken in order.
Objects on a Slope
Resolving vectors into components is useful for analyzing situations such as objects on slopes.
Weight (F_g) is always perpendicular to the ground, not the slope.
F_g = mg (g = 9.8 m/s^2).
Weight is measured in N, mass in kg.
Chapter 3: Velocity and Acceleration
Introduction
Mechanics involves mass, distance, time, and force.
This chapter focuses on the use of distance and time in motion models.
Objectives:
Differentiate between speed, velocity, and acceleration.
Solve relative velocity problems.
Solve ticker tape problems.
Sir Isaac Newton (1642-1727) developed calculus to better model natural phenomena.
Defining Quantities
Uniform Speed/Velocity: Relates distance covered over time.
Speed: Rate at which distance is covered.
Velocity: Rate of displacement (position), a vector quantity.
Mathematically: v = s / t, where v is speed/velocity, s is change in distance/displacement, and t is change in time.
Unit of measurement: m/s.
Uniform Acceleration: Constant rate of change in velocity.
Need to define a quantity that describes the rate of change.
Focus on both average and instantaneous velocity.
Acceleration: Rate of change of velocity, a vector quantity. a = v / t, where a is acceleration, v is change in velocity, and t is the time interval. Unit: m/s².
Average speed: Total distance traversed divided by the total time.
Average velocity: Total displacement divided by the total time.
Instantaneous speed: The magnitude of the velocity.
Instantaneous Velocity: The true velocity at a specific point in time.
Mathematical Descriptions
Average Velocity = \frac{total displacement}{total time}
Average Velocity = \frac{u + v}{2} where u is initial velocity, and v is final velocity.
Average Velocity = Instantaneous velocity in the middle of the time interval.
Relative Velocity
Motion is always relative to a frame of reference.
The earth is typically used as the frame of reference.
If all the velocities in a problem happened simultaneously, and the goal is to calculate an unknown velocity, the components can add to form a resultant.
Strategies for solving relative velocity problems:
Calculate displacements for each leg of the trip if the velocities and times of different parts of the trip is known.
Calculate resultant displacement.
Calculate resultant velocity by dividing resultant displacement by the total time.
Ticker Tape Problems
Ticker tapes and timers are used to investigate motion.
Ticker timer makes dots on the tape at regular intervals.
Analyzing dot patterns reveals the motion of the object pulling the tape.
Constant speed: Equally spaced dots.
Constant acceleration: Spaces increase by the same amount for successive intervals.
Frequency: Typically 50 Hz, creating dots every 0.02 seconds.
Direction: Must be specified to determine speeding up or slowing down.
The relationship between frequency and period is: frequency = \frac{1}{period}. e.g. 50 = \frac{1}{0.02}
Chapter 4: Graphs and Equations of Motion
Introduction
Exploration of relationships between displacement, time, velocity, and acceleration.
Relationships displayed through equations of motion and graphically.
Objectives:
Know and use equations of motion.
Solve problems using equations of motion.
Understand the relationships between different graphs.
Draw graphs for constant velocity and constant acceleration.
Equations of Motion
s = v\Delta t describes uniform velocity; no acceleration involved.
Equations needed for uniform acceleration incorporating initial velocity, final velocity and average velocity.
Symbols:
\Delta x (s) = displacement
v_i (u) = initial velocity
v_f (v) = final velocity
\Delta t = change in time
a = acceleration
Equations:
vf = vi + a\Delta t (excludes \Delta x)
vf^2 = vi^2 + 2a\Delta x (excludes \Delta t)
\Delta x = vi \Delta t + \frac{1}{2}a\Delta t^2 (excludes vf)
\Delta x = (\frac{vi + vf}{2}) \Delta t (excludes a)
Solving Problems
List given quantities first to identify the appropriate equation.
When dealing with gravitational acceleration, use g as the symbol with a value of 9.8 m/s².
When we write downv_i = 15m/s, we are taking the direction of motion to be positive. Seeing that the acceleration is in the opposite direction, we need to give it a negative sign owing to its vector. *
Relationships between Graphs
Definitions used to identify relationships between different graphs. Same for vectors and scalars.
v = \frac{\Delta x}{\Delta t} (Value of velocity at a given time is the gradient of the displacement-time graph at that time).
a = \frac{\Delta v}{\Delta t} (Value of acceleration at a given time is the gradient of the velocity-time graph at that time).
Change in displacement is equal to area under the v vs t graph for a certain change in time.
Graphs for Uniform Velocity
Constant positive velocity means zero acceleration.
Acceleration-time graph: Acceleration is zero for all times.
Velocity-time graph: Zero gradient, indicating positive velocity values.
Displacement-time graph: Constant positive gradient.
Graphs for Uniform Acceleration
Constant positive acceleration.
Acceleration-time graph: Constant positive acceleration.
Velocity-time graph: Constant positive gradient, indicating positive velocity values.
Displacement-time graph: Increasing positive gradient.
Chapter 5: Newton's Laws
Introduction
Describes the effect of force and mass on motion and explores the workings and secrets of nature.
Objectives:
Apply Newton's First, Second, and Third Laws.
Apply Newton's Universal Gravitational Law.
Solve vertical projectile motion problems.
Newton's First Law
Explains what would happen when a sheet of paper is pulled quickly from under a glass, and what happens to a motorist not wearing a seat belt.
Inertia: Property of matter to maintain its state of motion or rest.
Frictional force: The force that is needed to change the motion if it's above the object's inertia.
Mass: Quantitative measure of inertia.
Definition:
Inertia: The property of an object that causes it to resist a change in its state of rest or uniform motion.
Newton’s First Law: A body will continue in a state of rest or uniform motion in a straight line unless acted upon by a resultant force. If \sum F = 0, then \Delta v = 0
Newton's Second Law
Determines what will happen to a changing object when force is implied.
Modified Atwood machine is used to investigate Second Law.
Constant resultant force causes constant acceleration on a constant mass.
Resultant force magnitude is proportional to acceleration for a given mass.
Acceleration is inversely proportional to inertial mass.
F = kma
Definition:
Newton’s Second Law: When a resultant force is applied to an object, the acceleration is directly proportional to the resultant force and inversely proportional to the mass of the object.
OR The net force acting on an object is equal to the rate of change of momentum.Linear momentum: The linear momentum of a body is the product of its mass and its velocity
Newton's Third Law
Describes the relationship between forces that two objects exert on each other.
Forces act on different objects, not one object only, as stated in two examples in the transcript.
Definition:
Newton’s Third Law: When object A exerts a force on object B, object B simultaneously exerts an oppositely directed force of equal magnitude on object A
Newton's Universal Gravitational Law
Presents a mathematical model for interaction between masses, specifically gravity.
Law describes that there is in fact gravity, not what causes it.
Vertical Projectile Motion
Vertical projectile motion refers to objects moving up or down, perpendicular to the ground.
Projectiles fall freely with gravitational acceleration “g” taken as 9,8 m \cdot s^{-2} near the surface of earth.
The sign of g does not change during the motion.
Chapter 6: Momentum
Introduction
Explores how collisions or explosions affect quantities like velocity or momentum.
Objectives:
Know the relationship between impulse and change in momentum.
Apply the principle of conservation of linear momentum.
Differentiate between elastic and inelastic collisions.
Introduction
Impulse and momentum are often used to increase the efficiency of car production and the amount of power people obtain in karate.
Defintion: Impulse is the product of the net force and the contact time
Conservation of Linear Momentum
*Definition: Principle of Conservation of Linear Momentum: The total linear momentum of an isolated system remains constant (is conserved)-
If two objects, moving at constant velocities were to collide in an isolated (closed) system (aka no external forces), they would exert equal and opposite forces on each other according to Newton’s Third Law.
Elastic and Inelastic Collisions
Linear momentum always conserved during collisions in an isolated system (Newton's Third Law).
Sometimes kinetic energy is conserved.
*Definition: An Elastic Collision is when both momentum and kinetic energy are conserved. An Inelastic Collision is only when momentum is conserved.
Chapter 7: Work, Energy, Power
Introduction
Energy's role in describing motion and its significance in daily life.
Calculating energy use, or rather, change between energy states.
Work
Work is done on an object when a force exerted on object moves in the direction of the force.*
If energy is gained by the object, work done on the object can be considered positive. If energy is lost by the object, work done on the object can be considered negative.
NOTE:Do not confuse this statement with the Work-energy theorem : The kinetic energy of the system is increased when Fnet is in the same direction as Δx and is decreased when Fnet is in the opposite direction of motion (Δx)
Definition*:Work-energy theorem: The work done by a resultant force on an object is equal to the change in its kinetic energy
Energy
*Definition: Energy: Energy is the ability to do work.
*To do work, there must be energy. When work is done, energy is transformed to another form and often transferred to another object.
*There are two types of energy: Kinetic energy and Potential energy.
Mechanical energy is defined as the sum of the gravitational potential energy and the kinetic energy of an object.
*Definition of Gravitational potential: energy that a body has by virtue of its height above the ground*
(Ep +Ek)i = (Ep +Ek)f From this definition one can derive that the potential energy of a falling object at the top is equal to the kinetic energy at the bottom during free fall
Vertical motion (upward) for a rocket that is powered up by the thrust of an engineForces: T = Fg + Fres + Ffriction OR Fres = T – Fg -
FrictionWork done: Work done by thrust (T x ∆x) = work done against gravity (Fg x ∆x) + work done to accelerate (Fres x ∆x) + work done against friction (Ffriction x ∆x).Energy (no friction): Ep (top) + Ek (top) = Ep (bottom) + Ek (bottom) Ep (top) - Ep (Bottom) = Ek (bottom) - Ek (top) ∆Ep = ∆EkHorizontal motionForces: T = Fres + Ffriction OR Fres = T - FfrictionWork done: Work done by thrust (T x ∆x) = work done to accelerate (Fres x ∆x) + work done against friction (Ffriction x ∆x)
Power:
Calculates rate at which work is being done
Power is J \cdot s^{-1} or watts(w) P= \frac{w}{t}
Definition; A watt (1 W) of power is delivered when 1 joule of energy is transferred per second. (1 W = 1 \frac{J}{s})When force acts on object and object is at contant velocity or accelerating where vavg can be calculate.P=\frac{w}{t} = P=\frac{F\cdot \Delta x}{t}
= F \cdot v\P=\frac{powerout}{powerin} \cdot 100 (Percent efficiency. )