Scalars and Vectors
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The document provides an introduction to scalars and vectors in physics, highlighting the mathematical basis for concepts like distance, speed, and acceleration. It distinguishes between scalars, which have only magnitude, and vectors, which have both magnitude and direction, and illustrates methods to find resultant vectors using head-to-tail and parallelogram methods. Additionally, it includes exercises to apply these concepts, such as calculating resultant vectors and utilizing the law of sines and cosines.
Scalars and Vectors
- 1. By Prof. Liwayway Memije-Cruz Scalars and Vectors
- 2. Introduction Physics is a mathematical science. Its concepts and principles have a mathematical basis. Mo tio n o f o bje cts in physics are e xpre sse d by distance , displace m e nt, spe e d, ve lo city, and acce le ratio n which are asso ciate d with mathematical quantities. Mathematical quantities used to describe the motion of objects are vector and a scalar.
- 4. Mathematical Quantities Scalars are quantities that are fully described by a magnitude (or numerical value) alone. Vectors are quantities that are fully described by both a magnitude and a direction.
- 5. Consider the following quantities listed below. Be able to categorize each quantity as being either a vector or a scalar. Quantity Category 20 s 80 km 4000 cal 70 m South 30 m/sec, East 25 degrees Celsius 5 hrs
- 6. Resultant and Equilibrant Resultant is the vector sum of two or more vectors. Equilibrant is a force capable of balancing another force and producing equilibrium.
- 7. Methods of Finding the Resultant Head to Tail Method 1. Place the two vectors next to each other such that the head of the one vector is touching the tail of the other vector. 2. Draw the resultant vector by starting where
- 8. Answer To find the resultant vector's magnitude, use the Pythagorean theorem.
- 9. Exercise 1 1. You left your house to visit your sister. You got in your car drove 40 miles east, then got on a highway and went 50 miles north. 2. Draw a vector from the beginning of your journey, your home, and the end, your friend’s house.
- 10. Answer How long is the vector that you drew?
- 11. Exercise 2 What is the sum of the two vectors? Use the head to tail method to calculate the resultant vector in the picture on the right
- 12. Answer
- 13. Parallelogram Method Basic sine, cosine and tangent (SOHCAHTOA)
- 14. To which triangle(s) below does SOHCAHTOA apply?
- 15. Answer
- 16. Law of Cosine
- 17. Law of Sines The law of sines provides a formula that relates the sides with the angles of a triangle. This formula allows you to relatively easily find the side length or the angle of any triangle.
- 19. Exercise 3 To best understand how the parallelogram method works, examine the two vectors below. The vectors have magnitudes of 17 and 28 and the angle between them is 66°. Use the parallelogram method to determine the magnitude of the resultant.
- 20. Step 1) Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them) Step 2) We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360.
- 21. Step 3 Draw the parallelograms diagonal. This diagonal is the resultant vector. Step 4 Use the law of cosines to determine the length of the resultant
- 22. Use the law of cosines to calculate the resultant.
- 23. References: http://www.mathwarehouse.com/vectors/result ant-vector.php#headToTailMethod http://www.s-cool.co.uk/a- level/physics/vectors-and-scalars-and-linear- motion/revise-it/resolving http://www.physicsclassroom.com/class/vector s/Lesson-1/Vector-Addition http://www.physicsclassroom.com/class/vector s/Lesson-1/Resultants