Chapter 1 Introduction and Mathematical Concepts

Knowledge about our universe is improved by accurately adding to previous knowledge. –Each generation verifies and tries to improve on the knowledge discovered by previous generations. Scientists improve knowledge based on evidence. –To improve, you must be willing to change. Scientists are free to change their ideas because they are free to doubt and question. –Evidence helps evaluate which answers may be correct. New evidence may lead to new ideas about nature. Science is the study of our universe

Physics developed from the efforts of men and women to explain the behavior of our physical universe (including ourselves) Physics helps to explain... –Hitting of a baseball –Planetary orbits –Gymnastics –Electricity –Cell phone communication –Origin of the atoms in your body Physics knowledge is shared with everyone. Physics

Physics principles help us to predict how nature will behave in future events using rules developed by observing previous events. –Observations lead to generalized principles –Generalized principles predict future behavior Examples: Newton’s Laws → Spacecraft to Mars Maxwell’s Equations → Cell phones.

Physical quantities must be accurately communicated to other people. Correct use of units is essential for accurate communication of physical quantities. Units

SI units meter (m): unit of length kilogram (kg): unit of mass second (s): unit of time S ystem I nternational (SI) units

kilogram: Block of platinum-iridium metal Standards for the units meter: Distance light travels in second second: Time for a cesium-133 atom to vibrate 9,192,631,770 times.

Metric prefixes This chart is available inside the front cover of your textbook.

Example 1 The World’s Highest Waterfall The highest waterfall in the world is Angel Falls in Venezuela with a height of m. Express this height in feet. First find a relationship between meters and feet. Since 1 meter = feet, then Converting units

Success Strategy: 1. Write down units with every physical quantity. 2. Treat all units as algebraic quantities. Cancel the top and bottom units. 3. Gather the remaining units together. 4. Check that the remaining units are correct. For example: lengths should be in meters, velocities should be in meters/second, etc.

Rules for a right triangle Trigonometry review

How high is the building? 50° was measured at ground level

Finding angles

Pythagorean theorem: Relates the lengths of the three sides of a right triangle.

Scalars and Vectors Scalar quantity Described using just a value and units. Magnitude is the positive size of the value. Examples: 7 seconds, 106°F, 12 dollars Vector quantity Described using a magnitude and a direction. Examples: 20 meters/second north, 34 newtons east Important: Vectors are added and subtracted using different rules than the rules used for scalars.

The length of a vector arrow is proportional to the magnitude of the vector. 8 m 4 m Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector.

Displacement of a car is a vector quantity. displacement is 2 km at 30° north of east North East

Vectors are added using a head-to-tail method. Graphical vector addition rule:

5 m 3 m 8 m Vectors are added head-to-tail. Vector A is 5 m long and vector B is 3 m long. What is the sum of the two vectors? Adding vector with the same direction

Adding vector with the different directions Vectors are added head-to-tail.

What is the magnitude of the resultant vector R? 2 m 6 m R

2 m 6 m 6.32 m What is the angle θ?

Vector subtraction uses vector addition rules but uses a negative vector. Making a vector negative just reverses its direction. Graphical vector subtraction

A vector can be represented using x and y components. Vector components

It is easier to work with scalar components of vectors than with their vector components. Scalar components of vectors

Example: A vector has a magnitude of 175 m and direction angle of 50° relative to the x axis. Find the vector’s x and y components. (112 m) in the x direction + (134 m) in the y direction.

Using the x and y components with positive x Finding the vector magnitude Finding the vector direction The angle is in quadrant 1 because the vector has positive x and y components. y 3 21 x 4

Using the x and y components with negative x Finding the vector magnitude Finding the vector direction The vector is in quadrant 2 because the vector has a negative x and a positive y component so -36.9° is not correct. When the x component is negative, you need to add or subtract 180° to the tan -1 result to get an angle in the correct quadrant. y 3 21 x 4

Mathematical vector addition

Add vectors using vector components

Example: Mathematically adding vectors vectormagnitudedirectionx componentsy components A6 cm20° A x = A cos θ A 6 cm cos 20°= 5.64 cm A x = A sin θ A 6 cm sin 20°= 2.05 cm B8 cm65° B x = B cos θ B 8 cm cos 65° = 3.38 cm B x = B sin θ B 8 cm sin 65° = 7.25 cm C = A+B cm45.9° 9.02 cm9.30 cm

Example: Mathematically subtracting vectors vectormagnitudedirectionx componentsy components A6 cm20° A x = A cos θ A 6 cm cos 20°= 5.64 cm A x = A sin θ A 6 cm sin 20°= 2.05 cm B8 cm65° B x = B cos θ B 8 cm cos 65° = 3.38 cm B x = B sin θ B 8 cm sin 65° = 7.25 cm C = A-B 5.67 cm-66.5° 2.26 cm-5.20 cm Subtract the B components from the A components. The vector is in quadrant 4 because x is positive and y is negative.

The End