UNIT V: Multi-Dimensional Kinematics and Dynamics Page 1

UNIT V: Multi-Dimensional Kinematics and Dynamics

As we have already discussed, the study of the rules of nature (a.k.a. Physics) involves both scalar and vector quantities. Perhaps the most efficient way to remind ourselves of this is conducting a little thought experiment with Newton’s Second Law which, as we know, states that: The acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to the object’s mass. Often times when we write Newton’s Second Law in mathematical terms, we include small arrows over the net force and the acceleration to indicate that they are vectors (having both magnitude and direction). The mass does not have an arrow over it as it is a scalar (having only magnitude):

If we were to change the mass of an object … the magnitude of the acceleration would be affected. As the mass is increased/decreased, the acceleration will decrease/increase. However, the direction of the acceleration would remain unchanged. On the other hand, if we were to change the net force on the object by changing its magnitude and direction, then both the magnitude and direction of the acceleration would change.

The universe around us is multi-dimensional, with net forces and motion in many directions (three spatial dimensions and one temporal dimension in the world of Euclidean Geometry … but many more dimensions if the string theorists are correct regarding the structure of our universe!). Accordingly, we require a formal way to consider both the magnitude and direction of physics quantities in our analysis. This formal method is called vector analysis!

A. Vectors: a. Definitions i. Scalar – A quantity with only magnitude, like distance, speed, mass, charge, temperature, energy. Scalars indicate the amount of something: 2.0 miles, 44 mph, 3.5 kg, 21 coulombs, 273 degrees Kelvin, 42 joules

ii. Vector – A quantity with both magnitude and direction, like position, velocity, acceleration, force, displacement. A vector can often be most efficiently represented by an arrow:

Tail Head The tip of the arrow is known as the head of the vector. The arrow points in the appropriate direction for the quantity given. The other end of the arrow is known as the tail. The arrow is drawn to scale so that the length of the arrow corresponds to the magnitude of the vector. UNIT V: Multi-Dimensional Kinematics and Dynamics Page 2

b. Finding the components of a vector: Any 2-D vector can be split into two components that are perpendicular to one another. Typically, we chose one direction to be positive to the right (the positive x-axis) and another direction to be positive upward (the positive y-axis). This can either be done graphically or using trigonometric functions.

Example#b.1: Consider the vector illustrated, which is 5.00 units long at an angle of 36.9 degrees counterclockwise from the x-axis. Determine the x and y-components of this vector two ways  Graphically (measure the “run” and “rise” of the original vector … the run and rise will be the x and y-components, respectively)  Using Trigonometric Functions (think SOHCAHTOA … with the original vector, its x-component, and its y-component being the hypotenuse, adjacent side, and opposite side, respectively

Example#b.2: Repeat the steps in Example#b.1 for the vector illustrated below, which is 7.071 units long at an angle of 225 degrees counterclockwise from the x-axis.

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c. Finding the resultant of two perpendicular vectors: Any number of multi- dimensional vectors can be combined into an overall effect, which is represented by a single vector called the resultant. To begin, we will consider the “addition” of two perpendicular components to yield a single 2-D vector.

Example#c.1: Consider the scenario illustrated below, where the x- component is 3.00 units long and the y-component is 4.00 units long. Determine the resultant and direction (angle CCW from x-axis) two ways:  Graphically (you will need a straight edge)  Using the Pythagorean Theorem and Trigonometric Functions

Example#c.2: Repeat the steps in Example#c.1 for the situation illustrated below, where the x-component is -4.00 units and the y-component is -4.00 units.

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d. Applications – Finding the resultant of more than two perpendicular vectors

Example#d.1-Hiking: Suppose you walked 2.0 Km East, followed by 4.0 Km North, followed by 6.0 Km West, and finally 8.0 Km South. Sketch your path and determine the following quantities: your final position, the total distance you travelled, your displacement, and the direction you would need to travel in to head directly back toward your starting position.

Example#d.2-Swimming: Suppose a person swims Eastward at 2.5 m/sec, across a river that is moving at a constant velocity of 6.0 m/sec toward the South. Determine the swimmer’s resulting speed and direction (relative to due East).

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Example#d.3-FBD for Balanced Forces: Consider the FBD as illustrated below. Sketch the vector diagram on the axes provided, by placing the four vectors “head to tail” in the following order (Fgrav, Fapp, Fnorm, Ffrict). How can you tell that the net force is zero for this scenario?

Example#d.4-FBD for Unbalanced Forces: Consider the FBD as illustrated below. Sketch the vector diagram on the axes provided, by placing the four vectors “head to tail” in the following order (Fgrav, Fapp, Fnorm, Ffrict). How can you tell from the vector diagram that the net force is not zero for this scenario?

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Coordinate Planes for added practice:

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B. Generalized Vector Addition: Often times the vectors that need to be combined within a problem are not conveniently aligned with the x and y-axes. Accordingly, we need to generalize the methods from the last section to be applicable for any number of vectors.

a. Graphical Method – A Vector Diagram can be used to illustrate the vector sum of any number of vectors. Perhaps the best way to illustrate this is by way of an example. The top panel below shows three vectors. Our goal is to construct a Vector Diagram and determine the vector sum graphically on the bottom panel, using the steps indicated below

Begin by placing A on the lower panel with its original length and direction.

Now, add B to A in the lower panel: C This is accomplished by “sliding” B B along the page, maintaining the same orientation as in the top panel, until its “tail” lies at the “head” of A. A

Next, add C to the sum of A and B on the lower panel: This is accomplished by “sliding” C along the page, maintaining the same orientation as in the top panel, until its “tail” lies at the “head” of B.

Sketch the resultant (the vector sum D = A + B + C): This is accomplished by drawing a vector from the origin to the tip of C on the bottom panel.

Finally, use a ruler and protractor to determine the magnitude and direction of the resultant.

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Mathematical Method – The vector sum of a number of vectors can also be determined mathematically. As in the previous example, let’s illustrate this is by way of an example. The top panel below shows three vectors. Our goal is to determine the resultant, using the steps indicated below:

Determine the horizontal and vertical components for each of the vectors, using appropriate trigonometric functions:

C B

A

Add the horizontal components to get the horizontal component of the resultant:

Add the vertical components to get the vertical component of the resultant:

Finally, use the Pythagorean Theorem and an appropriate trigonometric function to determine the magnitude and direction of the resultant.

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C. Physics Examples: Equilibrium

EQUIPMENT NEEDED: Ruler, Protractor

PURPOSE: To graphically illustrate how a Force Diagram can be F3 η constructed from a Free-Body Diagram and make connections with prior knowledge regarding geometry. F2 λ THEORY: An object is said to be in equilibrium if it is at rest and is δ acted on by forces that balance in all directions. Figure 1 illustrates a Free Body Diagram for the scenario where three forces act on the F1 object. From Newton’s First Law, we know that the net force must equal zero. This can be illustrated graphically through the creation of Figure 1: the corresponding Force Diagram. FBD We begin by drawing F1 to scale on the page with its original magnitude and direction (see Figure 2). We then add F2 to F1. This is accomplished by “sliding” F2 along the page, maintaining the same orientation as in Figure 1, until its “tail” lies at the “head” of F1 (see Figure 3). Finally, we follow the same procedure to add F3 to the sum F1 of F2 and F1 (see Figure 4). The result is called a Force Diagram of the scenario. The angles between the forces in the Free-Body Diagram (Figure 1) can be related to the interior angles of the triangle between the same Figure 2 forces in the Force Diagram (Figure 4) using supplements:     180o    180o     180o If the interior angle between the shortest two sides in the Force F Diagram (Figure 4) is a right angle, then the triangle is a right triangle 1 and we can use the Pythagorean Theorem and trigonometric functions (remember SOHCAHTOA) to relate the lengths of the sides and the F2 interior angles1: 2 2 2 Figure 3 F1  F2  F3 F F F cos()  3 ; sin()  2 ; tan()  2 F F F 1 1 3 F2 F3 F3 F3 cos()  ; sin()  ; tan()  θ F1 F1 F2 F 1 β α

F2 Figure 4: Vector Diagram 1 If none of the angles is a right angle then the Law of Sines and Law of Cosines will be used instead. UNIT V: Multi-Dimensional Kinematics and Dynamics Page 10

PROCEDURE/ANALYSIS CASE 1 - Equal Angles: Suppose three 2.0 Newton forces are acting on an object and that the angles shown on the Free-Body Diagram below: a. Confirm that the Force Diagram was created using procedure outlined earlier. b. Measure the angles on the Free-Body Diagram and the Force Diagram and show that corresponding angles between forces on Free-Body Diagram and Force Diagram are supplements.

F1 = 2.0N

F3 = 2.0 N

F2 = 2.0N

F1 = 2.0N

F2 = 2.0N

F3 = 2.0 N

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CASE 2 - A Common Pythagorean Triple: Suppose the forces as illustrated below are acting on an object at the angles shown on the Free-Body Diagram: a. Create the associated Force Diagram using the procedure outlined earlier. b. Measure the angles on the Free-Body Diagram and the Force Diagram and show that corresponding angles between forces on Free-Body Diagram and Force Diagram are supplements. c. Show that the Pythagorean Theorem and the trigonometric functions are valid for the Force Diagram.

F3 = 2.0N

F2 = 1.5 N

F1 = 2.5N

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CASE 3 – Finding Unknown Forces: Suppose an object is under the action of three forces, with the magnitude and direction of one force being known (solid arrow) and the direction of the other two forces also being known (dashed lines, one in the –y direction and the other in the –x direction), as shown on the Free-Body Diagram: a. Create the associated Force Diagram using a modified procedure to that outlined earlier. b. Measure the angles on the Free-Body Diagram and the Force Diagram and show that corresponding angles between forces on Free-Body Diagram and Force Diagram are supplements. c. Determine the magnitudes for the unknown forces by measuring them. d. Show that the Pythagorean Theorem and the trigonometric functions are valid for the Force Diagram.

F1 = 2.0 N

F3 = ______N

F2 = ______N