Barron's Physics Practice Plus: 400+ Online Questions and Quick Study Review

CHAPTER 1

Conventions and Graphing

Learning Objectives

In this chapter, you will learn how to:

∘Review the fundamental metric units (SI units) and some of the derived metric units (SI units) used in physics

∘Determine the dependent and independent variables of a graph

∘Explain the importance of slope and area to a graph

Fundamental and Derived Units

The fundamental metric units (SI units) in physics cover the basic quantities measured, such as length, mass, and time. The units measure a quantity and are given a unit name and symbol. Table 1.1 lists the fundamental quantities along with the unit names and symbols.

TABLE 1.1 Fundamental Quantities and Units

Derived units are combinations of one or more of the fundamental units. Table 1.2 lists common derived units used in physics.

TABLE 1.2 Derived Units

It is important to know which units correctly belong to a specific quantity. An easy way to do this is to write out the principal formula for the quantity and then replace each variable on the right side of the equation with its unit symbol. There may be more than one correct answer including the unit symbol, other derived units, and fundamental units. For example, all of the following are correct ways to express units of energy: J, N • m, and kg • m²/s².

EXAMPLE 1.1

Derived Units

The unit of force is the newton. What are the fundamental units that make up the newton?

WHAT’S THE TRICK?

Write down the foundational formula for force.

Replace the variable symbols with their matching units. Force is measured in newtons, N. Mass is measured in kilograms, kg. Acceleration is measured in meters per second squared, m/s².

N = kg • m/s²

Graphing Variables

The graphing techniques of mathematics are used in science to compare dependent and independent variables. In mathematics, you are familiar with the traditional x- and y-coordinate axes. In science, the x-axis represents the independent variable and the y-axis represents the dependent variable. The value of the dependent variable depends upon the independent variable.

Graphs are always titled so that the dependent variable is listed first, and the independent variable is listed second. As an example, a position versus time graph would have position (dependent variable) plotted on the y-axis and time (independent variable) plotted on the x-axis.

Slope and Area

Slope

Slopes are very important in physics. Slope is determined by dividing the rise (y-axis value) by the run (x-axis value). The trick is to look at the units written on the axes of the graph. If you divide these units, you can easily identify the significance of the slope.

EXAMPLE 1.2

Slope of a Graphed Function

(A) What is the value and significance of the slope in the time interval from 0 to 3 seconds?

WHAT’S THE TRICK?

Determining the slope is simply a matter of dividing the rise (y-axis values) by the run (x-axis values). The significance of the slope is determined by examining the resulting units.

The resulting units, meters per second (m/s), are the units of velocity. Therefore, the slope of the position versus time graph is equal to velocity. During the first 3 seconds, the object has a velocity of 5 m/s.

(B) What is the value and significance of the slope in the time interval from 3 to 5 seconds?

WHAT’S THE TRICK?

The slope in the interval between 3 and 5 seconds is zero.

During this time interval, the object has a velocity of zero and the y-axis value (position) is not changing. The object’s position remains constant at a location 15 m from the origin.

Area

The area formed by the boundary between the x-axis and the line of a graph is also very useful. Areas are calculated by multiplying the height (y-axis value) by the base (x-axis value). In problems where the area forms a triangle, the area is found with height × base. In cases where the line of the graph is below the x-axis, the area is negative. See Figure 1.1.

FIGURE 1.1 Calculating area

As with slope, you can easily determine the significance of the area. By multiplying the units written on the axes of the graph and then looking at the resulting units, you can quickly determine the significance of the area.

EXAMPLE 1.3

Area of a Graphed Function

What is the value and significance of the area of the graph during the time interval between 0 and 10 seconds?

WHAT’S THE TRICK?

Determine the area, and examine the resulting units.

area = height × base = (10 m/s)(10 s) = 100 m

Meters (m) are the units of displacement. The area under a speed versus time graph is therefore the displacement of the object during that time interval. The object graphed above traveled 100 meters in 10 seconds.

Interpreting Graphs

Consider the graph of velocity versus time in Figure 1.2.

FIGURE 1.2 Velocity versus time graph

The graph tells the story of an object, such as a car, as it moves over a 60-second period of time. At time zero, the object has a velocity of 0 meters per second and is therefore starting from rest. The y-intercept of a speed versus time graph is the initial velocity of the object, v0.

What the object is doing during the 60 seconds can be determined by analyzing the slope and area during the separate time intervals. Determine the significance of the slope by dividing the rise units (y-axis values) by the run units (x-axis values).

The slope units, meters per second squared (m/s²), are the units of acceleration. Thus, the slope of speed versus time is acceleration. Determine the significance of the area between the graphed function and the x-axis by multiplying the units of the y-axis by the units of the x-axis.

Meters (m) are the units of displacement. The area of a velocity versus time graph is displacement.

To analyze the motion mathematically, divide the graph into a series of line segments and evaluate each section. The following chart shows the acceleration and displacement for the time intervals corresponding to the graphed line segments.

CHAPTER 2

Vectors

Learning Objectives

In this chapter, you will learn how to:

∘Identify a mathematical coordinate system that will provide a common frame of reference to orient direction in physics problems

∘Understand the differences and similarities among scalar and vector quantities

∘Resolve vectors into components and add vector quantities

Coordinate System

Problems in physics often involve the motion of objects. Position, displacement, velocity, and acceleration are key numerical quantities needed to describe the motion of an object. Position involves a specific location, while velocity and acceleration act in specific directions. Using the mathematical coordinate system is ideal to visualize both position and direction. The coordinate system provides a common frame of reference in which the quantities describing motion can be easily and consistently compared with one another.

We can place an axis anywhere, and we can orient the axis in any direction of our choosing. If a problem does not specify a starting location or direction, then position the origin at the object’s starting location. In Figure 2.1, a problem involving the motion of a car can be visualized as starting at the origin and moving horizontally along the positive x-axis.

FIGURE 2.1 Horizontal motion

In more complex problems, some quantities cannot be oriented along a common axis. In these problems, direction must be specified in degrees measured counterclockwise (ccw) from the positive x-axis, as shown in Figure 2.2.

FIGURE 2.2 Coordinate system

A coordinate system is a valuable tool that provides a frame of reference when position and direction are critical factors.

Scalars

A scalar is a quantity having only a numerical value. No direction is associated with a scalar. The numerical value describing a scalar is known as its magnitude. Some examples of commonly used scalars are listed in Table 2.1.

TABLE 2.1 Commonly Used Scalars

The symbols representing scalars are printed in italics. For example, a mass of 2.0 kilograms will be written as m = 2.0 kg. Scalars can have magnitudes that are positive, negative, or zero. For example, time = 60 seconds, speed = 0 meters per second, and temperature = −10°C.

Vectors

Although scalars possess only magnitude, vectors possess both magnitude and a specific direction. Examples of commonly encountered vectors are listed in Table 2.2.

TABLE 2.2 Commonly Used Vectors

Formal vector variables are usually written in italics with a small arrow drawn over the letter, as shown in the middle column in Table 2.2. You may encounter vector quantities, such as force, in any one of these forms: , F, Fx, Fy, and F. The first, , is the most accepted and distinctly indicates a vector quantity. It is the format used in this book. The second, F, is an alternate way to indicate a vector quantity. The next two, Fx and Fy, signify vector components that lie along the specific axis indicated by their subscripts. The last, F, appears to be the convention to indicate a scalar quantity. It is typically used when only the magnitude of the vector is needed and the direction is understood.

Distinguishing between vectors and scalars by simply looking at an equation can be confusing. How, then, do you tell scalars and vectors apart? Physics problems may contain clues in the text of the problem to help distinguish vectors from scalars. The mention of a specific direction definitely indicates a vector quantity. However, it is up to you to learn which quantities are vectors and when the use of vector components is necessary. Counting on the use of a specific set of symbol conventions may not be wise.

Vectors do follow certain mathematical conventions that are worth noting. Vector magnitudes can be only positive or zero. However, vectors can have negative direction. Consider the acceleration of gravity, a vector quantity acting in the negative y-direction. The gravity vector includes both magnitude and direction . Substituting this exact expression, including the negative y-direction, into an equation is not really workable. Instead the value −10 m/s² may be substituted into equations. The negative sign in front of the magnitude indicates the negative y-direction. This can be done only if all the vector quantities used in an equation lie along the same axis and it is understood that the signs on all vector quantities represent direction along that axis. This essentially transforms the vector quantities into scalar quantities, allowing normal mathematical operations. As a result, the variable may be shown as a scalar in italics (g = −10 m/s²) rather than in bold print. When a negative sign is associated with a vector quantity, it technically specifies the vector’s direction and assists with proper vector addition.

Vectors are represented graphically as arrows. For displacement vectors, the tail of the arrow is the initial position of the object, xi, and the tip of the arrow is the final position of the object, xf. The length of the arrow represents the vector’s magnitude, and its orientation on the coordinate axis indicates direction. This may give some insight into the reason that some vector quantities are displayed in italics.

Figure 2.3 shows a car moving 200 meters and its associated vector.

FIGURE 2.3 Horizontal displacement

The magnitude of the displacement vector, Δx, is the absolute value of the difference between the final position, xf, and the initial position, xi. Direction can be seen in the diagram.

Δx = xf − xi = 200 − 0 = 200 m, to the right (+x)

For other vectors, such as velocity and force, the quantity described by the vector occurs at the tail of the arrow. The tail of the arrow shows the actual location of the object being acted upon by the vector quantity. The tip of the arrow points in the direction the vector is acting. The length of the arrow represents the magnitude of the vector quantity. The magnitude and direction described by these types of vectors may be instantaneous values capable of changing as the object moves. In addition, the object may not reach the location specified by the tip of the arrow.

These types of vectors are readily seen in projectile motion. In Figure 2.4, a projectile is launched with a speed of 50 meters per second at an angle of 37° above the horizontal.

FIGURE 2.4 Projectile motion

Although only three key velocity vectors are shown in the diagram, they clearly demonstrate how the magnitude and direction of velocity change throughout the flight. During the motion depicted in the diagram, no two instantaneous velocity vectors are completely alike.

Knowing how to recognize vectors quantities like displacement, velocity, acceleration, and force will improve your problem-solving skills. The importance of vector direction cannot be overstated. Including the correct sign representing a vector’s direction is often the key to arriving at the correct solution. The next sections will demonstrate the importance of vector direction as we review basic vector mathematics.

Vector Mathematics

Components

Vectors aligned to the x- and y-axes are mathematically advantageous. However, some problems involve diagonal vector quantities. Diagonal vectors act simultaneously in both the x- and y-directions, and they are difficult to manipulate mathematically. Fortunately, diagonal vectors can be resolved into x- and y-component vectors. The x- and y-component vectors form the adjacent and opposite sides of a right triangle where the diagonal vector is its hypotenuse. Aligning the component vectors along the x- and y-axes simplifies vector addition.

The magnitudes of component vectors are determined using right-triangle trigonometry. In Figure 2.5, vector A is a diagonal vector. It has a magnitude of A and a direction of θ.

FIGURE 2.5 Magnitudes of vectors

Vector Ax is the x-component of and is adjacent to angle θ. Vector Ay is the y-component of and is opposite angle θ. Normally, the magnitude of the components of vector would be determined using the following right-triangle trigonometry.

Ax = A cos θ, +x-direction

Ay = A sin θ, +y-direction

EXAMPLE 2.1

Determining Component Vectors

A projectile is launched with an initial velocity of 50 meters per second at an angle of 37° above the horizontal. Determine the x- and y-component vectors of the velocity.

WHAT’S THE TRICK?

Draw the component vectors and identify the adjacent and opposite sides.

Determining the magnitudes of each component requires multiplying the hypotenuse by the correct fraction. The direction of each component can be determined by looking at the diagram.

vx = 4/5 hyp = 4/5 (50) = 40 m/s, +x-direction

vx = 3/5 hyp = 3/5 (50) = 30 m/s, +y-direction

In some problems, the component vectors are known or given and you must determine the vector they describe. Pythagorean theorem and inverse tangent are used to calculate the magnitude and direction of the diagonal vector described by the component vectors.

Adding Vectors

One important aspect of working with vectors is the ability to add two or more vectors together. Only vectors with the same units for magnitude can be added to each other. The result of adding vectors together is known as the vector sum, or resultant.

You can use two visual methods to add vectors. The first is the tip-to-tail method, and the second is the parallelogram method. In some problems, the resultant is known or given and you must determine the magnitude and direction of one of the vectors contributing to the vector sum. The sections below detail examples of each of these scenarios.

Tip-to-Tail Method

Adding vectors tip to tail is advantageous when a vector diagram is not given. Begin by sketching a coordinate axis. Vectors can be added in any order. However, drawing x-direction vectors first, followed by y-direction vectors, is best. Choose the first vector and draw it starting from the origin and pointing in the correct direction. Start drawing the tail of the next vector at the tip of the previous vector. Keep the orientation of the second vector the same as it was given in the problem. Continue this process, adding any remaining vectors to the tip of each subsequent vector. Finally, draw the resultant vector from the origin (tail of the first vector) pointing to the tip of the last vector. You will encounter three common cases of vector addition.

•Vectors pointing in the same direction

•Vectors pointing in opposite directions

•Vectors that are 90° apart

EXAMPLE 2.2

Adding Vectors Pointing in the Same Direction

A person walks 40 meters in the positive x-direction, pauses, and then walks an additional 30 meters in the positive x-direction. Determine the magnitude and direction of the person’s displacement.

WHAT’S THE TRICK?

When vectors point in the same direction, simply add them together. Sketch or visualize the vectors tip to tail. The resultant is equal to the total length of both vectors added together.

Resultant = 40 m + 30 m = 70 m

EXAMPLE 2.3

Adding Vectors Pointing in Opposite Directions

A person walks 40 meters in the positive x-direction, pauses, and then walks an additional 30 meters in the negative x-direction. Determine the magnitude and direction of the person’s displacement.

WHAT’S THE TRICK?

When a vector points in the opposite (negative) direction, you can insert a minus sign in front of the magnitude. Technically, vectors cannot have negative magnitudes. The minus sign actually indicates the vector’s direction, and it represents a vector turned around 180°. Again, sketching or visualizing the vectors tip to tail will help you arrive at the correct resultant. The resultant is drawn from the origin to the tip of the last vector added.

Resultant = 40 m + (−30 m) = 10 m

EXAMPLE