IB Physics: Topic 1.

Physics and

Measurement.

 

1. The scientific method could be the most efficient problem-solving tool

ever devised. There are 6 steps in the scientific method: (I) Define the

problem, (II) Gather information, (III) State your hypothesis, (IV) Test

the hypothesis, (V) Form your conclusion, (VI) Publish the results. The

scientific method is the process by which scientists, collectively and over

time, endeavor to construct an accurate (that is, reliable, consistent and

non-arbitrary) representation of the natural world.

2. The Metric System (Systeme International, SI) was first introduced by

the French Academy of Science in 1795 as an attempt to unify existing

systems. The SI contains the basic units for length (meter, m), mass

(kilogram, kg), and time (second, s), with speed (m/s), volume (m3),

and density (kg/m3) as some derived units.

3. All calculations must be done observing significant digits and scientific

notation. When a number is expressed in scientific notation, the number

of significant figures is the number of digits needed to express the number

to within the uncertainty of measurement.  We always round to the least

precise measurement. Here is an example problem, if you have to multiply

 2.5 cm x 1.23 cm the result is 3.1 cm2.

4. The number of significant figures of a product or quotient of two or more

quantities is equal to the smallest number of significant figures for the

quantities involved. For example, if you multiply 5.2 x 3.751 x 6.43, your

answer must be written using only two significant digits as 130.

For addition or subtraction, the number of significant figures is determined

with the smallest significant figure of all the quantities involved. For example,

the sum 10.234 + 5.2 + 100.3234 is 115.7574, but should be written 115.8

(with rounding), since the quantity 5.2 is significant only to 0.1.

 5. Order of magnitude (power of 10) calculations provide quick estimates

for answers to certain questions. An order of magnitude calculation is an

estimate to determine if a more precise calculation is necessary. We round

off or guess at various inputs to obtain a result that is reliable to within a

factor of 10. Specifically, to get the order of magnitude of a given quantity,

we round off to the closest power of 10 (ex: 75 kg is expressed as 102 kg).

Another ex., the average distance from the Earth to the Sun is 93,000,000

miles. In scientific notation it is 9.3x107 miles. But since 9.3 is closest to 101,

we would express the order of magnitude as 108 miles.

6. A frame of reference is a coordinate system for specifying the precise

location of objects in space. You make a convenient choice for the origin of

the system. Maybe you have heard the expression, "It depends on your

frame of reference."

7. Accuracy refers to the agreement of a measured value with an accepted

value. Percent error measures accuracy. Precision is the agreement of a set

of measured values with each other. It can be measured by average deviation.

8. Each instrument has an inherent amount of uncertainty in its measurement.

Even the most precise measuring device cannot give the actual value because

to do so would require an infinitely precise instrument.

9. A measure of the precision of an instrument is given by its uncertainty.

As a good rule of thumb, the uncertainty of a measuring device is 20%

of the least count. Recall that the least count is the smallest subdivision given

on the measuring device. The uncertainty of the measurement should be given

with the actual measurement, for example, 41.64 0.02cm.

10. Here are some typical uncertainties of various laboratory instruments:

  • Meter stick: 0.02cm
  • Vernier caliper: 0.01cm
  • Triple-beam balance: 0.02g
  • Graduated cylinder: 20% of the least count

Here's an example. The uncertainty of all measurements made with a meter

stick whose smallest division (least count) is one millimeter is 20% of 1mm

or 0.02 cm. Say you use the meter stick to measure a metal rod and find that

the rod is between 10.2 cm and 10.3 cm. You may think that the rod is closer

to 10.2 cm than it is to 10.3 cm, so you make your best guess that the rod

is 10.23 cm in length. Since the uncertainty in the measurement is 0.02cm,

you report the length of the rod as 10.23 0.02 cm (0.1023 0.0002 m).

When a quantity is graphed, it is common for the uncertainty of that quantity

to be represented by error bars.

11. Graphs are plotted with the independent (control) variable on the x-axis,

and the dependent (measured) variable on the y-axis.

12. Graphs can show direct (linear), inverse (hyperbolic), periodic (sinusoidal),

quadratic (parabolic), or chaotic relationships.

13. All equations must be dimensionally correct. We use dimensional analysis

(factor labeling) to determine if equations are correct.

14. The number of atoms in one mole of any element or compound is 6.02x1023

which we know as Avogadro's number.

15. The density of a substance is defined as its mass per unit volume. We use

the Greek letter rho, ρ, for density and the equation is ρ = m / V.

16. Some equations that you may remember from Mathematics are A = πr2,

C = 2πr, A = 4πr2, V = πr2h, V = 4/3 πr3, and d = vt. Notice that variables in

Physics are case sensitive. For example, A is area, but a is acceleration.

Another example, T is temperature, but t is time.

17. We also need to remember SOHCAHTOA to compute the value of unknown

 sides and angles of right triangles.

18. We need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct formula to use

(iv) isolate the unknown

(v) substitute-in the given information and simplify.

View the PowerShow: HERE.

Very Important: If you have any questions or were absent from

class, see me before school (7:30 - 8:00 AM), during Lunch, or

after school (4:00 5:00 PM).

Best to send an email to rpersin@fau.edu.

Website Homework Assignment #1. Show all work.

1. Write a paragraph that illustrates how a student would use the scientific

method to determine what college to attend.

2. Find an English-to-Metric System of Measurement conversion table.

Then use it to make the following conversions:

(a) 6.75 inches = ______ centimeters      (b) 3.0 miles = _____ kilometers

(c) 400 cubic inches = ______ Liters        (d) 100 lbs = ______ kilograms

3. How many significant digits are in each of the following measurements:

(a) 12.375 cm   ____       (b) 3.000 m   _____       (c) .00075 s  _____  

(d) 6.0075 in.  _____       (e) 93,000,000 mi. ____ (f) 25,000 km  ______

4. Express all the measurements given in #3 in scientific notation.

5. Perform the following calculations and express your answers in scientific

notation with the correct amount of significant digits:

(a) 2500 m x 2.75 m = ___________   (b) 2.4x107 cm / 1.45 cm = __________

(c) 4.3 cm + 1.75 cm + .041 cm = ___________

(d) (6.3x108 m x 4x109 m) / 7.11x10-5 m = _____________

6. Express each answer in #5 as an order of magnitude.

7. Solve for the variable in red. Show all of your steps.

(a)  F = mv2/r  ________  (b) E = mc2 ________

(c) T = 2π√(L ⁄ g) _______  (d) d = at2 _______

(e) 1/p + 1/q = 1/f _______  (f) F = (G m1m2)/r2 _______

(g) Combine these equations E = mc2 , c = fλ , E = hf  to get

an equation for  λ  that does not involve E or f .

 Website Notes for Scalars and Vectors:

I. Vectors.

Vectors are used to describe multi-dimensional quantities. Multi-dimensional

quantities are those which require more than one number to completely

describe them. Vectors, unlike scalars, have two characteristics, magnitude

and direction. A vector is indicated by an uppercase letter either in boldface

or with an arrow over the top.  For example,  A or  . Examples of vector

quantities are: position in a plane, position in space, velocity, acceleration,

and force.

II. Scalars.

Scalars are used to describe one-dimensional quantities, that is, quantities

which require only one number to completely describe them. They have

magnitude only. Direction does not apply. There are cases where scalars can

be combined mathematically, but we will save that for later. Some examples

of scalar quantities are: temperature, mass, time, volume, density, length,

area, and energy.

III. Vector Diagram.

Any vector can be resolved into perpendicular component vectors using sine

and cosine functions. Actually, for all vector problems just remember

SOHCAHTOA. This allows us to find the x and y components of any vector.

For the horizontal, or x-component of a vector A, we have, A(cos F ).

Then to find the y-component we use  A(sin F). This can easily be proven

with the Pythagorean Theorem.

 

IV. Some Properties of Vectors.

Two vectors are equal only if they have the same magnitude and direction.

To find the opposite of a given vector just keep the same magnitude but

point it in the opposite direction.  ex. A - B = A + (-B). Vectors can also

be expressed using polar coordinates (r , θ) specifying the length of the

radius vector r , and the angle of rotation, F ("Phi"), from the positive

x-axis.

Additionally, in a two-dimensional coordinate system, vectors can be

denoted using the unit vectors and ĵ. Each unit vector has magnitude

equal to 1, and they point in the x and y directions, respectively. We can

easily add the third dimension, or z direction using unit vector k.

 

 V. Addition of Vectors

Vectors can be added graphically using the head-to-tail method.

 

You begin by drawing the first vector in a coordinate system,

and then drawing the second vector from the endpoint of the

first, and so on. Then you can draw a single vector from the

origin to the head of the last vector.

 VI. Vector Subtraction.

The vector difference works the same as vector addition except

that we multiply the vector we are subtracting by -1. In other

words, to subtract a vector, just add the opposite.

 

It is much like subtracting two numbers: A - B = A + (-B).

This diagram illustrates vector subtraction in the tip-to-tail style.

The original B vector is shown as a dashed line.

VII. Multiplication of a Vector by a Scalar.

A vector may be multiplied by a scalar by multiplying each of its

components by that number.

 

Notice above, that the vector does not change direction,

only length. If A = (1,2) then 3A = (3,6).

 

 Website Homework Assignment #2: Solve These Vector

Problems and Show All Work.

1. Find the x and y components of the following vectors:

a. 240 km at 330                  b. 34 m/s at 210
c. 15 m at 12                        d. 20 m/s2 at 90

2. From the x and y components given, find the direction and magnitude

of the resultant.

a. Fy = 120 N, Fx = 345 N                 b. vy = 31 m/s, vx = 8 m/s
c. ax = -15 m/s2, ay = 12 m/s2

3. Add the three vectors below. Use the graphical method to show a picture

of the addition of the vectors. Use the mathematical method to obtain the

magnitude and direction of the resultant vector.

A = 450 N at 20, B = 250 N at 270, C = 100 N at 70

 

4. A soccerball is kicked with a horizontal velocity of 11.3 m/s and a vertical

 velocity of 3.5 m/s. What is the magnitude and direction of the resultant

velocity of the ball?

 

5. A pole-vaulter applies a force of 415 N to the pole at an angle of 37.

What are the horizontal and vertical components of this force?