Introduction and Mathematical Concepts Chapter 1 Science is

Introduction and Mathematical Concepts Chapter 1

Science is the study of our universe • 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.

Physics • 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 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.

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

SI units System International (SI) units meter (m): unit of length kilogram (kg): unit of mass second (s): unit of time

Standards for the units kilogram: Block of platinum-iridium metal meter: Distance light travels in 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.

Converting units Example 1 The World’s Highest Waterfall The highest waterfall in the world is Angel Falls in Venezuela with a height of 979. 0 m. Express this height in feet. First find a relationship between meters and feet. Since 1 meter = 3. 281 feet, then

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.

Trigonometry review Rules for a right triangle

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.

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

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

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

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

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

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

What is the angle θ? 6. 32 m 2 m 6 m

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

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

Vector components

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

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 y The angle is in quadrant 1 because the vector has positive x and y components. 2 1 3 4 x

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 2 1 3 4 x

Mathematical vector addition

Add vectors using vector components

Example: Mathematically adding vectors vector magnitude direction x components y components A 6 cm 20° Ax = A cos θA 6 cm cos 20°= 5. 64 cm Ax = A sin θA 6 cm sin 20°= 2. 05 cm B 8 cm 65° Bx = B cos θB 8 cm cos 65° = 3. 38 cm Bx = B sin θB 8 cm sin 65° = 7. 25 cm C= A+B 12. 96 cm 45. 9° 9. 02 cm 9. 30 cm

Example: Mathematically subtracting vectors Subtract the B components from the A components. vector magnitude direction x components y components A 6 cm 20° Ax = A cos θA 6 cm cos 20°= 5. 64 cm Ax = A sin θA 6 cm sin 20°= 2. 05 cm B 8 cm 65° Bx = B cos θB 8 cm cos 65° = 3. 38 cm Bx = B sin θB 8 cm sin 65° = 7. 25 cm C= A-B 5. 67 cm -66. 5° 2. 26 cm -5. 20 cm The vector is in quadrant 4 because x is positive and y is negative.

The End