Introduction to Physics

Physics deals with the interaction of matter, force, and energy.

Classical Physics (Pre-1900)

• Mechanics - area of physics concerned with the relationships between force, matter, and motion among physical objects

• Heat and Thermodynamics - it studies the relations between heat, work, temperature, and energy.

• Optics - it studies how light moves through certain objects.

• Electricity and Magnetism - it studies the relationship between electric currents and magnets

• Wave Motion and Sounds - it studies how energy moves through air or water in the form of vibrations and sound

Modern Physics (Post-1900)

• Nuclear Physics - it studies the interactions of atomic nuclei and it’s constituents

• General Relativity - It describes how gravity is a geometric product of space and time

• Special Relativity - explains how speed, mass, time, and space are all interrelated.

• Quantum Mechanics - Quantum mechanics is the field of physics that explains how extremely small objects simultaneously have the characteristics of both particles

• Particle Physics - is the study of fundamental particles and forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons.

Measurement is the process of comparing something with standard.

There are two systems of measurement

• Metric System (MKS and CGS)

• English or Imperial System (FPS)

Physical Quantities

• Fundamental units/quantities - units independent of one another.

  Ex. Mass, amount of matter in an object.

  Length, how long an object is

• Derived Units/quantities - combination of fundamental units

  Ex. Weight changes based on factors like density, gravity, and mass

  Density changes based on an objects mass and volume, the more mass in a small space, the more dense. Or the less mass in a large space, the less dense it is.

Scientific Notation

is a conveniently and widely used method of expressing large and small numbers.

Ex. 5,000,000,000 = 5.00 × 10⁹

0.000000005 = 5.00 × 10^-9

Unit Conversion

a. 55 km to m

55 km x (1000 m / 1 km) = 55, 000 m

b. 12g to kg

12 kg x (1 kg / 1000 g) = 0.012 kg

c. 85 m/s to kph

85 m/s x (1 km / 1000 m) x (3600 s / 1 hr) = 306 kph

Uncertainty and Errors

Measurements always have some degree of uncertainty because of unavoidable errors.

Error is the deviation of a measured value from the expected or true value.

Uncertainty is a way of expressing this error.

Measured Value = True Value ± Uncertainty

Precision - closeness of a measurement to another measurement.

Accuracy - closeness of a measurement to the expected value of a physical quantity

Random Error - Results from unpredictable or inevitable changes during data measurement.

Example: When weighing a substance, the scale may show slightly different readings each time due to minor vibrations, air currents, or electrical interference, even if the substance's mass hasn't changed.

Systematic Errors - usually come from the measurement or in the design of the experiment itself.

Example: When measuring the length of a table multiple times with a ruler that is incorrectly marked, each measurement yields a consistent discrepancy. For instance, if the ruler is off by 1 cm, every measurement will be 1 cm shorter than the actual length, leading to biased results.

Percent Error = (| x - x_r | / x_r) x 100

• x_r = true or accepted value

• x = measured value

Percent Difference = (|x₁ - x₂| / x₁ + x₂ / 2) × 100

Variance - measures the squared deviation of each number in the set from the mean.

Mean (μ) = (Σx) / N

• Σx = sum of all values

• N = number of values.

Variance (σ²) = Σ (xi - μ)² / N

• σ² = variance

• xi = each value

• μ = mean

• N = number of values

Standard Deviation (σ) = √(Σ(xi - μ)² / N)

Uncertainty indicates the range of values within which the measurement is asserted to lie with some level of confidence

Absolute Uncertainty = ± (Maximum Value - Minimum Value) / 2

Ex. Resistance of a wire 25.00 Ohm ± 0.05 Ohm

Range of Resistance 24.95 - 25.05 Ohm

25.05 - 24.95 / 2

= 0.05

Relative Uncertainty = (Absolute Uncertainty / Measured Value) × 100%

Ex. 0.05 Ohm / 25.00 Ohm x 100 = 0.2%

A zero variance means that all measurements are identical

Less variance means more precision

Standard Deviation is how spread out measurements from the average

Least Count refers to the smallest value that can be read from any measuring device.

Ex. Rulers in metric usually have 0.1 cm as the Least Count

Rule 1: If data are to be added, add their absolute uncertainties.

Ex. Perimeter = sum of all the sides = 2(L +W)

= 2(6.5m ± 0.1m + 3.4m ± 0.2m)

= 2(9.9m ± 0.3m)

= 19.8m

Rule 2: If data are to be multiplied or divided, add their relative uncertainties.

Ex. The area of a rectangle = L x W

Relative uncertainty for length = 1.54%

Relative uncertainty for width = 5.88%

Sum = 7.42%

= ((7.42%)(22.1m²) / 100%) = 1.64m² - absolute uncertainty.

Vector Addition

Graphical - associated with illustrations, drawings, and pictures

Analytical - associated with computation and analysis

Scalar Quantity - Magnitude

From the Latin word Scala - “steps or ladders”

Vector Quantity - Magnitude + Direction

From the Latin word Veherre - “To Carry”

Things to consider when adding vectors:

• Magnitude and direction

The term used to describe the sum of two or more vectors is called Resultant Vector or V_r

Vector addition has two properties:

• Commutative - The order of adding vectors maybe changed without affecting the resultant. This property can be represented as A + B = B + B

• Associative - when the addition of three or more numbers, the total/sum will be the same, even when the grouping of addends are changed. This property can be represented as; A+(B+C) = (A+B)+CT

The Graphical Methods:

• Parallelogram Method - This method is useful for getting the resultant of two vectors, but cannot be applied beyond two vectors.

STEPS:

1. Using a suitable scale, draw the arrows representing the vectors from a common point.

2. Construct a parallelogram using the Two vectors.

3. Draw diagonal, this represents the resultant.

4. Measure the length of the diagonal, from the scale used.

5. Determine the direction using a protractor.

• Head-to-Tail Method - This method is quite simple and useful, as it can be used for 3 or more vectors.

The Analytical Methods:

• Component Method

• Sine-Cosine Method

Component Method Example 1:

• d₁ = 10m, E

• d₂ = 15m, 30^o N of E

• d_1x = 10m

• d_1y = 0m

• d_2x = (cos(30^o))(15) = 13m

• d_2y = (sin(30^o))(15) = 7.5m

• Σ_x = 10m + 13m = 23m

• Σ_y = 0 + 7.5m = 7.5m

• d_R = √((Σ_x)² + (Σ_y)²)

• d_R = √((23m)² + (7.5m)²)

• d_R = 24.18m

• θ = tan^-1[7.5m / 23m]

• θ = 18.06^o N of E

• d_R = 24.18m 18.07^o N of E

Component Method Example 2:

• d₁ = 5 m, S

• d₂ = 12 m, E

• d₃ = 4 m, SW

• d₄ = 6 m, W

• d_1x = 0

• d_1y = -5 m

• d_2x = 12 m

• d_2y = 0

• d_3x = (cos 45^o)(4 m)

• d_3y = (sin 45^o)(4 m)

• d_4x = -6m

• d_4y = 0

• Σ_x = 3.17m

• Σ_y = -7.83m

• d_R = √((3.17m)² + (-7.83m)²)

• = 8.45m

• θ = tan^-1[7.83m / 3.17m]

• = 67.96^o S of E

• d_R = 8.45m, 67.96^o S of E

Kinematic Equation of Motion

Motion - change in position in respect to the observer

Position - refers to the location of an object with respect to a reference point or origin.

Study of motion is divided into two:

1. Kinematics - describes motion in terms of displacement, velocity, and acceleration.

2. Dynamics - the study of force in relation to motion

Distance - refers to total length of path by any object in moving from it’s initial position

Displacement - is defined as the change in position of an object in relation to it’s starting position.

Ex. Andee jogged 12 times around a circular track of radius 5m. Find the total distance & displacement

c = 2πr

• c = 2π(5) = 31.4 m

• distance = 31.4 m × 12 = 377

• displacement = 0

Speed = rate of change of distance

s = d / t

Velocity = rate of change of displacement

v = s / t

Average Speed - total distance traveled by the total time elapsed

s_avg = d_t / t_T

Instantaneous Speed - The recorded speed at a particular moment in time.

Average Velocity - displacement divided by the total time elapsed.

v_avg = Δx / Δt = x - x₀ / t - t₀

• Δx = change in position

• Δt = change in time.

Instantaneous Velocity - Velocity at a specific point in time.

v_ins = lim(Δt → 0) (Δx/Δt)

Accelerations exists when there is a change in velocity.

a = v - u / t

• u = initial velocity

• v = find velocity

• t = time

Instant Acceleration

a_ins = lim(Δt → 0) (Δv/Δt)

Example 1:

There is a 360 m long path, V starts walking at a rate of 3 m/s, A starts walking 10 s after at a rate of 4 m/s. How long will it take for A to overtake V?

• V_V = 3 m/s

• V_A = 4 m/s

• d - 360 m

• 10 s

• d_v = d_a

• 3(t + 10) = 4t

• 3t + 30 = 4t

• 30 = 4t - 3t

• t = 30 s

How far is V, when overtaken by A?

• d_A = 4t = 4(30) = 120

• 360 - 120 = 240 m

Example 2:

s(t) = -2t³ + 13t², where s in meters and t in seconds.

a. find the average velocity from t = 4 s to t = 6 s.

• s(4) = 80 m

• s(6) = 36 m

• v_avg = 36 m/s - 80 m/s / 6 s - 4 s

• v_avg = 22 m/s

b. Find the instantaneous velocity at t = 4

• v_ins = -6t² + 26t

• = -6(4)² + 26(4)

• = -6(16) + 104

• = -96 + 104

• v_ins = 8 m/s

Example 3:

Lean and MIckey are 150 m apart. They start walking at 3 m/s and 5 m/s, respectively.

Lean represents x / 3, Mickey represents 150 - x / 5

a. How long will it takes for them to meet?

• t_i = t_m = x / 3 = 56.25 m / 3 m/s = 18.75 s

b. Find the corresponding distance traveled by each?

• t_i = t_m

• x / 3 = 150 - x / 5

• 5x = 3(150 - x)

• 5x = 450 - 3x

• x = 450 / 8

• x = 56.25 m

Example 4:

A particle moves along the x-axis so that it’s position at any time t greater than or equal to 0 is given by the function x(t) = t³ - 12t + 1, where x is measured in terms of feet, and t in seconds.

a. Find the displacement during t = 4

• x(4) = 4³ - 12(4) + 1

• x(4) = 17 ft

b. Find the average velocity of the particle for 4 less than or equal to t less than or equal to 6.

• t(4) = 17 ft

• t(6) = 145 ft

• v_avg = 145 ft - 17 ft / 6 ft - 4 ft

• v_avg = 64 fps

c. Find Instantaneous Velocity at 3 s

• d’(t) = 3t² - 12

• x(3) = 3(3)² - 12

• = 27 - 12

• = 15 fps

Example 5:

The position of an object as a function of time t is given by x(t) = 3.5t³ - 2t + 5, where x is measured in meters and t in seconds.

a. Give the velocity and acceleration as a function of time.

• V_ins = 10.5(2)² - 2

• = 40 m/s

• a_ins = 21(2)

• = 42 m/s

Example 6:

The position, in meters, of a body at time t seconds is x(t) = t³ - 6t² + 9t. Find the body’s acceleration each time the velocity is zero.

• d(x) / t = 3t² - 12t + 9 —> 3t² / 3 - 12t / 3 + 9 /3 —> t² - 4t + 3 = 0

• (t - 1)(t - 3) = 0

• t = 1, t = 3

• d(x) / t = 6t - 12

• t = 1 —> 6(1) - 12 = - 6 m/s²

• t = 3 —> 6(3) - 12 = 6 m/s²

Free Fall

When the only force acting on an object is gravity, the object is said to be in free fall thus, acceleration is constant.

In a vacuum, all objects in free fall accelerate at the same rate, regardless of mass.

Newton’s Law of Gravitation

Characteristics of Free Fall

• Free-falling objects do not encounter air resistance.

• All free-falling (on Earth) accelerate downwards at a rate of 9.8 m/s²

Forward and Downward Motion

• At the object’s maximum height, upward velocity and downward velocity are both equal to zero.

Terminal Velocity

• When an object’s weight due to gravity is balanced with air resistance, it stops it’s acceleration and starts moving at a constant speed.

Equations for Uniformly Accelerated Motion

V_F = V_i + gt

d = V_i t+ (1/2)gt²

V_F² = V_i² + 2gd

• V_F = Final Velocity

• V_i = Initial Velocity

• g = acceleration due to gravity

• d = displacement

• t = time elapsed

Example 1:

Ralph dropped a stone from his bedroom window.

a. What is it’s velocity after 2 seconds

• V_i = 0 V_f = ? (0) + (9.8m/s)(2 s)

• t = 2 s V_f = 19.6 m/s²

• g = 9.8 m/s²

b. How far from the window has it fallen after 2 seconds?

• d = (0)(2 s) + (1/2)(9.8 m/s²)(2 s)²

• d = 19.6 m from the window.

Example 2:

Paul threw a ball vertically upward from the ground. The ball returned to the ground in 4 s.

a. How high did the ball rise?

• d = (0)(2 s) + 1/2(9.8 m/s²)(2 s)²

b. With what velocity was it thrown?

• 0 = V_i + (-9.8 m/s²)(2 s)

• 0 = V_i - 19.6 m/s²

• V_i = 19.6 m/s²

Example 3:

If a body falls freely from rest, how far does it move during the twelfth second?

• d = (0)(12) + (1/2)(9.8 m/s²)(12 s)²

• d = 705.6 m

Example 4:

A person throws an object into the air with an initial velocity 4.9 m/s off an 52 m tall building.

a. max. height relative to the ground.

• V_f = V_i² + 2gd

• 0 = (4.9 m/s²) + 2(-9.8 m/s²)d

• 0 = 24.01 m/s² - (19.6 m/s²)d

• (19.6 m/s²)d / 19.6 m/s² = 24.01 m/s² / 19.6 m/s²

• d = 1.225 m

• d_T = 52 m + 1.225 m

• d_T = 53.225 m

b. It’s time of flight or the total time in the air.

• V_f = V_i + gt

• 0 = 4.9 m/s + (-9.8 m/s²)t

• (9.8 m/s²)t / 9.8 m/s² = 4.9 m/s / 9.8 m/s²

• t = 0.5 s

• d = V_it + (1/2)gt²

• 53.225 = 1/2(9.8 m/s²)t²

• √((53.225 m)/4.9 m/s²) = (4.9 m/s²)√t² / 4.9 m/s²

• t₂ = 3.3 s

• t_T = 3.3 s + 0.5 s =3.8 s

c. Velocity before it reaches the ground.

• V_F = 0 + (9.8 m/s²)(3.3 s)

• V_F = 32.34 m/s

• V_t = 42.14 m/s

Projectile Motion

Projectile - any object upon which the only force acting is gravity.

A horizontally launched will reach the ground at the same as an object as an object dropped vertically from rest. The initial horizontal velocity is irrelevant.

All projectiles follow a parabolic path called a trajectory.

Horizontal Distance (d_x) = V_xt

• V_x = Horizontal Velocity

• t = time

Vertical Distance (d_y) = (1/2)gt²

• g = acceleration due to gravity

• t = time

Horizontal Velocity (v_x) = constant

Vertical Velocity (v_y) = gt

• g = acceleration due to gravity

• t = time

Time (t) = √(2d_y / g)

• d_y = Vertical Distance

• g = acceleration due to gravity

Actual Velocity (v) = √((v_x)² + (v_y)²)

• v_x = Horizontal Velocity

• v_y = Vertical Velocity

Projectile Motion with an Angle

Initial Velocity X (v_xi) = v_i cos θ

• v_i = Initial Velocity

• θ = Angle

Initial Velocity Y (v_yi) = v_i sin θ

• v_i = Initial Velocity

• θ = Angle

Actual Final Velocity (v) = √((v_xf)² + (v_yf)²)

• v_xf = Final Horizontal Velocity

• v_yf = Final Vertical Velocity

Distance X (d_x) = v_xit

• v_xi = Initial Horizontal Velocity

• t = time

Maximum Height (h_max) = (2v_isin θ)² / 2g

• v_i = Initial Velocity

• θ = Angle

• g = gravity

Time (t) = 2(v_i sin θ) / g

• v_i = Initial Velocity

• θ = Angle

• g = gravity