B. Vector and Scalar Quantities
Vector quantities consist of both magnitude and direction. The magnitude is made up of a number and a unit.
Scalar quantities consist of only a magnitude.
Position, displacement, velocity, acceleration, force, weight, and momentum are examples of vector quanti- ties.
The direction of a vector is stated using square brackets behind its magnitude.
(i.e., 12 km [N])
Scalar quantities include such things as mass, time, distance, speed, work, and energy, to name but a few.
Vector quantities are represented on a diagram by a directed line segment, drawn to scale, with reference coordinates to show direction.
The tail of a vector is called the origin and the tip is called the terminal point.
Equivalent vectors have the same magnitude and direction.
Collinear vectors can be added algebraically or graphically. The resultant vector is obtained from vector addition. Graphical addition of vectors is performed on a neat, accurate, scale diagram.
Non-collinear vectors exist in more than one dimension.
Many important applications in kinematics and dynamics require an understanding of vectors.
The sum of any two or more vectors can be determined graphically or mathematically.
Vectors are added by aligning the tail of one vector (origin) with the tip of another vector (terminal point) on a neat, accurate, properly scaled diagram.
The vector sum of two or more vectors is called the resultant vector.
The resultant vector points from the tail of the first vector to the tip of the last vector being added.
To make a vector negative, change its direction. A negative vector has the opposite direction of a positive vector.
The magnitude and direction must be stated for the resultant vector, as for any other vector quantity.
To subtract one vector from another, the same rule applies as in one dimension; change the direction of the vector being subtracted and then proceed as with vector addition.
Vector and scalar operations yield totally different results. The two must not be confused.
The following mathematical concepts might need reinforcement.
For two perpendicular vectors:
Trigonometric relationships:
Vectors act independently of one another.
The vector sum of two perpendicular vectors can be determined mathematically using the Pythagorean theorem and trigonometric relationships.
A single vector can be regarded as the resultant of two components usually, but not necessarily, acting perpendicularly. (i.e., horizontally and vertically, or along the x and y axis.)
The process of determining effective values of component vectors is called vector resolution.
Vector components can be determined using graphical and mathematical methods.
Graphical methods for solving vector problems help to conceptualize abstract situations and provide a good first approximation for mathematical methods.
Mathematical methods for determining vector components involve the use of the Pythagorean Theorem, trigonometry, and other important mathematical concepts.
A good understanding of mathematics is essential in the study of physics. Many important concepts in physics are based on mathematical relationships developed experimentally and theoretically.
The vector sum of two non-perpendicular vectors can be found by using the vector component method. Each vector is resolved into perpendicular components. The components are added separately using methods for one dimensional vector addition. The resulting component sums are then added using the rules for adding perpendicular vectors to obtain the resultant vector. This method is preferable when more that two vectors are involved.
For a vector F making an angle Ø (where Ø is measured counterclockwise from the positive x axis), the component of F in the x direction is given by Fx = FcosØ, and the component of in the y direction is given by Fy = FsinØ.
To get the resultant components for several vectors being added, decompose into x and y components.
The net x and y components are, respectively
The magnitude of the resultant vector is
The direction of the resultant vector is
(The terms inverse tan or tan-1 are
acceptable, as long as
.)
Students will increase their abilities to:
.
The following mathematical concepts might need reinforcement.
For two non - perpendicular vectors:
Law of sines:
Law of cosines:
(Use this method for verification. Stress the vector component
method.)
Given the magnitude and direction of two of the forces acting on the test object, predict the magnitude and direction of a third force needed to maintain static equilibrium. Test the prediction experimentally. Account for any slight deviations from predicted values.
A similar set of experiments can be performed on a non-concurrent force apparatus.
The object of the activity is to learn about projectile motion -- in an enjoyable way. Number the slots in the egg carton from 1 to 6, with the largest numbers in the centre of the carton. Students score "points" for getting a ball to fall into the holes in the egg cartons.
The positions of the cartons can be adjusted back and forth, in the line along the floor where the balls will strike. Depending on the size and mass of the balls being used, students will have to determine where on the ramp the balls should be placed in order for them to land in the holes scoring the highest point values.
Teams can compete against one another, placing the cardboard carton in different positions and giving the other team a particular ball to use. The cardboard carton can be placed anywhere from underneath the ramp, to the maximum distance that the balls could travel along the ramp. Prior to having the teams compete with one another, students can observe the way the balls fall and make measurements to determine where a particular ball will land if it is released from a certain starting position on the ramp.
The bubbles will form shapes which have the smallest possible surface area of fundamental shape (Plateau's bubbles). Interference fringes can be seen in the soap film. A qualitative analysis of the forces acting on the soap bubbles can be done.