Vectors and Scalars

Scalars and Vectors

Physical quantities are divided into scalers, having only magnitude or size, and vectors, which have both magnitude and direction. Distance is an example of a scalar: 'The girl walked two kilometers." If a direction then is specified, it is called displacement, and is a vector. "The girl walked two kilometers toward the north."

A force is any influence that can produce a change in the velocity of an object. It may produce or prevent motion. It has both magnitude and direction, and is therefore a vector. If many forces act on an object at the same time, the resultant, or net force can be found by adding all of the individual forces. Of course, the direction must enter into the addition, and that makes vector addition different from the addition of scalars.

Example of scalar addition: A car travels 3 m and then 4 m more. The total distance is 3 m + 4 m= 7 m.

Example of vector addition: A car goes 3 m east then 4 m north. The total displacement is obtained by representing each vector as an arrow. One vector is oriented properly on an axis, starting at the origin. The tail of the next vector is oriented from the head of the first vector. If there are more than two vectors, the procedure continues. The order of vector selection does not matter, as long as they are all used. The resultant is obtained by drawing a vector from the origin to the head of the final vector. The magnitude and direction of the resultant are measured, or they are calculated using trigonometry. Here the answer is a magnitude of 5 m at a direction of [53 ° N of E]. (R = square root of (3² + 4²) = 5 m and angle theta = inverse tan (4/3))

Vectors and Scalars: Homework

1. Using the information given at the bottom of this page find the resultant vectors (give both magnitude and
direction for the F_R) given the following problems:

	Bob exerts a force of 12.5 N [due North] and Steve exerts a force of 4.5 N [due East]
	Ann exerts a force of 8.85 E 4 N [due West] and Carol exerts a force of 2.30 E 6 N [due South]

2. A huge beach ball lands on the playground at Smart Tech Undergraduate High School. In trying to move the huge beach ball 6 kids each push on the ball at the same time. Find the direction the ball would move and give the force (as though one kid did all the work, or in other words, give the F_R – another hint would be to draw the vectors graphically using cm for Newtons, etc.)

	kid 1: 14 N [35° North of East]
	kid 2: 9.5 N [due South]
	kid 3: 13.55 N [15.5° West of North]
	kid 4: 8.3 N [due West]
	kid 5: 12.4 N [due South]
	kid 6: 5.5 N [4.5° West of South]

3. On an uncharted desert island you must bury a treasure (science notes for the next test). You design a treasure map using vectors (as in the problem above). Give at least 5 different treasure map steps (leaving from a center location) that must be followed to reach the site of the treasure. Give your map to a classmate and see if he/she can find the treasure.

(hint: give your directions something like this: leave the center of the island and go 2.78 steps (cm on your map) [14.5° South of West]. From there go …