Scalars distance speed mass time volume temperature work and energy

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1 Scalars and Vectors scalar is a quantit which has no direction associated with it, such as mass, volume, time, and temperature. We sa that scalars have onl magnitude, or size. mass ma have a magnitude of kilograms, a volume ma have a magnitude of 5 liters, and so on. But a vector is a quantit which has both magnitude (size) and direction (angle). For eample, if someone tells ou the are going to appl a 0 pound force on ou, ou would want to know the direction of the force, that is, whether it will be a push or a pull. So, force is a vector, since direction is important in specifing a force. The same is true of displacement, as we will see in the following sections. The table below lists some vectors and scalars ou will be using in our phsics course. Vectors displacement velocit acceleration force weight momentum Scalars distance speed mass time volume temperature work and energ Vector ddition and Subtraction We can graphicall add vectors to each other b placing the tail of one vector onto the tip of the previous vector: B - B = + B = + (- B), or = B In the diagram on the left above, we have added two vectors head to tail b placing the tail of vector B on the head (tip) of vector. When adding vectors graphicall, we ma move a vector anwhere we like, but we must not change its length or direction. The resultant is drawn from the tail of the first vector to the head of the last vector. The resultant is also called the vector sum of and B, and can replace the two vectors and ield the same result.

2 Eample 4 Displacement is also a vector. Consider a hiker who walks 8 kilometers due east, then 10 km due north, then 1 km due west. What is the hiker s displacement from the origin? vector can be represented b an arrow whose length gives an indication of its magnitude (size), with the arrow tip pointing in the direction of the vector. We represent a vector b a letter written in bold tpe. For this eample, we list the displacement vectors like this: = 8 km west B = 10 km north C = 1 km east We can graphicall add the second displacement vector to the first, and the third displacement vector to the second: 1 km N 10 km = 10.8 km at 68º from E W 8 km 68º E S The resultant vector is the displacement from the origin to the tip of the last vector. In other words, the resultant is the vector sum of the individual vectors, and can replace the individual vectors and end up with the same result. Of course, just adding the lengths of the vectors together will not achieve the same result. dding 8 km, 10 km, and 1 km gives 30 km, which is the total distance traveled, but not the straight-line displacement from the origin. We see in the diagram above that the resultant displacement is 10.8 km from the origin at an angle of 68º from the east ais.

3 We could have added the displacements in an order and achieved the same resultant. We sa that the addition of vectors is commutative. The equilibrant is the vector which can cancel or balance the resultant vector. In this case, the equilibrant displacement is the vector which can bring the hiker back to the origin. Thus, the equilibrant is alwas equal and opposite to the resultant vector. 1 km N 10 km E = - = 10.8 km at 68º + 180º from E W 8 km 68º E S

4 The Components of a Vector We ma also work with vectors mathematicall b breaking them into their components. vector component is the projection or shadow of a vector onto the - or -ais. For eample, let s sa we have two vectors and B shown below: B α B B We will call the projection of vector onto the -ais its -component,. Similarl, the projection of onto the -ais is.the vector sum of and is, and, since the magnitude of is the hpotenuse of the triangle formed b legs and, the Pthagorean theorem holds true: and from the figures above, = sin = cos tan We can write the same relationships for vector B b simpl replacing with B and the angles with in each of the equations above.

5 Eample 5 Find the - and -components of the resultant in Eample 4. The resultant is 10.8 km long at an angle of 68º from the east (+) ais: 68º Since the vector component is adjacent to the 68º angle, we have the relationship = cosθ = (10.8 km)cos 68º = 4 km Similarl, the vector component is opposite to the 68º angle, so we have = sinθ = (10.8 km)sin 68º = 10 km ddition of Vectors b Means of Components Earlier we added vectors together graphicall to find their resultant. Using the head-to-tail method of adding vectors, we can find the resultant of and B, which we called. We can also use components to find the resultant of an number of vectors. For eample, the -components of the resultant vector is the sum of the -components of, B, and C. Similarl, The - components of the resultant vector is the sum of the -components of and B. So, we have that = + B + C and = + B + C and b the Pthagorean theorem, z

6 B C Eample 6: Using the diagrams above, let = 4 meters at 30º from the -ais, B = 3 meters at 45º from the -ais, and C = 5 m at 5º from the -ais. Find the magnitude and direction of the resultant vector. (cos30 = 0.87, sin 30 = 0.50, cos45 = sin45 = 0.70, cos5 = 0.90, sin5 = 0.4) Solution: First, we need to find,, B, B, C, and C : = cos30 = (4 m)cos30 = (4 m)(0.87) = 3.5 m. = sin30 = (4 m)cos30 = (4 m)(0.50) =.0 m B = Bcos45 = (3 m)cos45 = (3 m)(0.70) =.1 m B = Bsin45 = (3 m)sin45 = (3 m)(0.70) =.1 m C = - Ccos5 = (5 m)cos5 = - (5 m)(0.90) = m C = - Csin5 = (5 m)sin5 = - (5 m)(0.43) = -. m

7 Note that C and C are both negative, since the point to the left and down, respectivel. Now we can find the - and -components of the resultant : = + B + C = 3.5 m +.1 m + (- 4.5 m) = 1.1 m = + B + C =.0 m +.1 m + (-. m) = 1.9 m The magnitude of the resultant C is 1.1m 1.9 m. m and its angle from the -ais can be found b tan tan m 1.1m 60 =. m at 60º from the +-ais The properties of vectors we ve discussed here can be applied to an vector, including velocit, acceleration, force, and momentum.

Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole

Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists