Physics Chapter 1

The following basic steps are commonly used in problem solving:

1. Read the problem carefully at least twice. Be sure you understand the nature of the problem before proceeding further. 2. Draw a labeled diagram (with coordinate axes if needed). 3. Visualize the problem in your head. 4. As you examine what is being asked in the problem, identify the basic physical principle or principles that are involved, listing the known and unknown quantities. • 5. Select a basic relationship or derive an equation that can be used to find the unknown• 6. Substitute known values, with the appropriate units, into the equation.

In order for two or more vectors to be added, they must have the same units; their sum is independent of the order of addition. The triangle method and the parallelogram method are graphical methods for determining the resultant or sum of two or more vectors.

Addition of Vectors

The magnitude and direction of vector A

Can be determined from the values of the x and y components of A.

The location of a point P in a plane can be specified by either

Cartesian coordinates, x and y, or polar coordinates, r and θ.

Any vector can be completely described by its components. A unit vector is a dimensionless vector one unit in length used to specify a given direction.

Components of a Vector and Unit Vectors

In order to add vector A to vector B using the graphical method,

Construct vector A, and then draw vector B such that the tail of vector B starts at the head of vector A . The sum of A + B is the vector that completes the triangle by connecting the tail of A to the head of B.

The location of any point in space can be specified by the use of a coordinate system. Two frequently used coordinate systems are the Cartesian (or rectangular) coordinate system and the plane polar coordinate system. In general, any coordinate system consists of: • A fixed reference point O, called the origin • A set of specified axes or directions with an appropriate scale and axis labels • A set of instructions that tell us how to label a point in space relative to the origin and axes.

Coordinate Systems

Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities. That is, quantities can be added or subtracted only if they have the same dimensions. Furthermore, the terms on both sides of an equation must have the same dimensions. By following these simple rules, you can use dimensional analysis to help determine whether or not an expression has the correct form, because the relationship can be correct only if the dimensions on the two sides of the equation are the same.

Dimensional Analysis

Test the result by asking these questions:

Do the units match? Is the magnitude of the answer reasonable? Is the positive or negative sign proper or meaningful based on the description of the problem?

Two vectors are equal vectors if they have the same magnitude and the same direction. It is not necessary that they act along the same line.

Equality of Two Vectors

Mechanical quantities can be expressed in terms of three fundamental quantities, mass, length, and time, which in the SI system have the units kilograms (kg), meters (m), and seconds (s), respectively.

Fundamental Quantities

When a vector is multiplied or divided by a positive (negative) scalar, the result is a vector in the same (opposite) direction. The magnitude of the resulting vector is equal to the product of the absolute value of the scalar and the magnitude of the original vector.

Multiplication by a Scalar

The sum of a vector and its negative is zero. A vector and its negative have the same magnitude, but have opposite directions.

Negative of a Vector

When one performs measurements on certain quantities, the accuracy of the measured values can vary; that is, the true values are known only to be within the limits of the experimental uncertainty. The value of the uncertainty can depend on various factors such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed. When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the least accurate of the quantities being multiplied. In this context "least accurate" means "having the lowest number of significant figures." The same rule applies to division. When numbers are added (or subtracted), the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum (or difference).

Significant Figures

Review carefully the discussion of forming models and alternative representations presented in Section 1.10 of your textbook.

Student Response

When two or more vectors are to be added, all of them must represent the same physical quantity

That is, have the same units.

A + B = B + A

The commutative law of addition states that when two or more vectors are added, the sum is independent of the order of addition.

The operation of vector subtraction utilizes the definition of the negative of a vector.

The vector (− A) has a magnitude equal to the magnitude of A, but acts or points along a direction opposite the direction of A. The negative of vector A is defined as the vector that when added to A gives zero for the vector sum.

In the graphical or geometric method of vector addition

The vectors to be added (or subtracted) are represented by arrows connected head-to-tail in any order. The resultant or sum is the vector that joins the tail of the first vector to the head of the last vector. The length of each arrow must be proportional to the magnitude of the corresponding vector and must be along the direction that makes the proper angle relative to the others.

A vector is a physical quantity that must be specified by both magnitude and direction. A scalar quantity has only magnitude.

Vectors and Scalars

≈

approximately equal to

The resultant vector (vector sum or difference) of two vectors,

can be expressed in terms of the components of the two vectors.

Δ

change in a quantity

≡

defined as

Unit vectors are

dimensionless and have a magnitude of exactly 1. Vector A lying in the x-y plane (see figure at right), having rectangular components Ax and A y, can be expressed in unit vector notation. Unit vectors specify the directions of the vector components: ˆ i along the x axis, ˆ j along the y axis and ˆ k along the z axis.

The component vectors of a vector are

directed along the coordinate axes and have magnitudes which equal the projections onto the respective coordinate axes. They should not be confused with the scalar quantities Ax and Ay which are the rectangular components of a vector.

=

equality of two quantities

< , >

less than, greater than

IxI

magnitude of the quantity x

~

on the order of

The density of any substance is defined as the ratio of mass to volume.

p= m/v

∝

proportional to < less than > greater than Δ change in a quantity x magnitude of the quantity x x ∑ 1 sum of the set x

During your progress through this course you will use,

simplification, analysis, and structural models in solving physics problems.

If one set of coordinates is known, values for the other set can be calculated using Equations,

x = r cosθ, y = r sinθ, tanθ = y/x and r = square root of x^2 + y^2