Forces in Two Dimensions

Forces in Two Dimensions
Chapter Forces in Two Dimensions 5 In this chapter you will: Add vectors graphically and mathematically. Use Newton’s laws to analyze motion when friction is involved. Analyze force vectors in equilibrium.

Section Warmup 5.1 Consider a person who walks 100 m due north and then loses all sense of direction. Without knowing the direction, the person walks another 100 m. What could the magnitude of displacement be relative to the original starting point? Hint: Draw a picture using two vectors.

Table of Contents 5 Chapter 5: Forces in Two Dimensions
Section 5.1: Vectors Section 5.2: Friction Section 5.3: Force and Motion in Two Dimensions (what does two- dimensions mean?) Homework for Chapter 5 Read Chapter 5 Study Guide 5, due before the Chapter Test HW 5.A, HW 5.B: Handouts. You will need a ruler / protractor.

Vectors 5.1 In this section you will:
Measure and report the magnitude of vectors. Evaluate the sum of two or more vectors in two dimensions. Determine the components of vectors.

Vectors 5.1 Vectors and Scalars
Section Vectors 5.1 Vectors and Scalars vector – A quantity with magnitude (size) and direction. ex: displacement, velocity, acceleration, and force Vectors are represented by arrows. scalar - A quantity with magnitude only. ex: distance, speed, mass, time, and volume tail tip or head

N NE NW W E SW SE S Vectors 5.1 Vectors in Two Dimensions
Section Vectors 5.1 Vectors in Two Dimensions The direction of a vector is where it points towards. You can specify the angle in degrees, as measured counter-clockwise from the x-axis. 0° 90° 180° 270° 360° θ = 225° N NW NE More generally, give the cardinal direction. Here, the vector’s direction is southwest, or SW. W E SW SE S

Vectors 5.1 Adding Vectors
Section Vectors 5.1 Adding Vectors Vectors can be added by placing arrows tip to tail. resultant - The arrow that extends fro the tail of the first vector to the tip of the last vector. It indicates both the magnitude and the direction of the vector sum. Vectors may be added in any order (commutative operation). A + B = B + A Vectors may be shifted around, but as long as the keep the same size and direction, they are the same vector.

Vectors 5.1 Adding Vectors Graphically
Section Vectors 5.1 Adding Vectors Graphically The figure below shows two force vectors on a free-body diagram. Vectors a and b are originally drawn tail-to-tail. They can not be added this way. Instead, shift vector b so it is tip-to-tail with vector a. Shift vector b so a and b are tip to tail. → vector b vector a vector b vector a

Vectors 5.1 Adding Vectors Graphically
Section Vectors 5.1 Adding Vectors Graphically You can draw the resultant vector pointing from the tail of the first vector to the tip of the last vector and measure it to obtain its magnitude. resultant vector b vector a Practice: Adding Vectors Graphically Worksheet.

Vectors 5.1 Adding Vectors Mathematically
Section Vectors 5.1 Adding Vectors Mathematically You may also use mathematical methods to determine the length or direction of resultant vectors. If you are adding together two vectors at right angles, vector A pointing north and vector B pointing west, you could use the Pythagorean theorem to find the magnitude of the resultant, R. R A B

Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors
Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors Find the magnitude of the sum of a 15-km displacement and a 25-km displacement when the angle between them is 90° and when the angle between them is 135°.

Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors Sketch the two displacement vectors, A and B, and the angle between them. B A θ1

Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors Identify the known and unknown variables. A B θ1 Known: A = 25 km B = 15 km θ1 = 90° θ2 = 135 ° Unknown: R = ? R

Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors When the angle is 90°, use the Pythagorean theorem to find the magnitude of the resultant vector. Substitute A = 25 km, B = 15 km

Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors When the angle does not equal 90°, use the graphical method to find the magnitude of the resultant vector: Draw the vectors to scale. Pick a scale that will fit on your paper. example: 15 km = 1.5 cm Measure the resultant. For our problem, the resultant measures 37 cm. Convert back to the original units. 3.7 cm = 37 km

Section Vectors 5.1 Finding the Magnitude of the Sum of Two Vectors Are the units correct? Each answer is a length measured in kilometers. Do the signs make sense? The sums are positive. Are the magnitudes realistic? The magnitudes are in the same range as the two combined vectors, but longer. The answers make sense.

Vectors 5.1 Finding the Sum of Two Vectors p. 121 Practice Problems:
Section Vectors 5.1 Finding the Sum of Two Vectors p. 121 Practice Problems: 1 & 2: Use the mathematical method (Pythagorean theorem and calculator) to find the magnitude of the displacement (length of the resultant). Also, report the direction of displacement for #1. Possible answers are N, NE, E, SE, S, SW, W, NW. 3 & 4: Use the graphical method (ruler) to find the magnitude of the displacement. Also, report the direction of displacement for #4.

F Fy Fx Vectors 5.1 Vector Components
Section Vectors 5.1 Vector Components Because a vector has both magnitude and direction, you can separate it into horizontal (or x-) and vertical (or y-) components. To do this, draw a rectangle with horizontal and vertical sides and a diagonal equal to the vector. Draw arrow heads on one horizontal and one vertical side to make the original vector the resultant of the horizontal and vertical components. Label the vector and its x- and y- components. vector resolution - Breaking a vector into its x- and y- components. Fy Fx F

d v F a Vectors 5.1 Practice with Vector Resolution
Section Vectors 5.1 Practice with Vector Resolution Use a ruler to draw and label the x- and y- components of these vectors. 2. 1. d v 3. F 4. a HW 5.A Handout

Section Check 5.1 Question 1
Jeff moved 3 m due north, and then 4 m due west to his friends house. What is the magnitude of Jeff’s displacement? 3 + 4 m 4 – 3 m m 5 m

Section Check 5.1 Answer 1 Answer: D
Reason: When two vectors are at right angles to each other as in this case, we can use the Pythagorean theorem of vector addition to find the magnitude of resultant, R.

Section Check 5.1 Answer 1 Answer: D
Reason: Pythagorean theorem of vector addition states If vector A is at right angle to vector B then the sum of squares of magnitudes is equal to square of magnitude of resultant vector. That is, R2 = A2 + B2  R2 = (3 m)2 + (4 m)2 = 5 m

Section Check 5.1 Question 2 Answer 2
Is the distance that you walk equal to the magnitude of your displacement? Give an example that supports your conclusion. Answer 2 Not necessarily. For example, you could walk around the block (one km per side). Your displacement would be zero, but the distance that you walk would be 4 kilometers.

Could a vector ever be shorter than one of its components? Equal in length to one of its components? Explain. Answer 3 It could never be shorter than one of its components, but if it lies along either the x- or y- axis, then one of its components equals its length.

The order in which vectors are added does not matter. Mathematicians say that vector addition is commutative. Which ordinary arithmetic operations are commutative? Answer 4 Addition and multiplication are commutative. Subtraction and division are not.

Which of the following actions is permissible when you graphically add one vector to another: moving the vector, rotating the vector, or changing the vector’s length? Answer 5 Allowed: moving the vector without changing length or direction.

In your own words, write a clear definition of the resultant of two or more vectors. Do not explain how to find it; explain what it represents. Answer 6 The resultant is the vector sum of two or more vectors. It represents the quantity that results from adding the vectors.

Section Vectors 5.1 Please Do Now Write the three most important things you’ve learned and want to remember about adding vectors.

Friction 5.2 In this section you will: Define the friction force.
Distinguish between static and kinetic friction.

Friction 5.2 Static and Kinetic Friction
Section Friction 5.2 Static and Kinetic Friction Push your hand across your desktop and feel the force called friction opposing the motion. A frictional force acts when two surfaces touch. When you push a book across the desk, it experiences a type of friction that acts on moving objects. This force is known as kinetic friction, and it is exerted on one surface by another when the two surfaces rub against each other because one or both of them are moving.

Section Friction 5.2 Static and Kinetic Friction To understand the other kind of friction, imagine trying to push a heavy couch across the floor. You give it a push, but it does not move. Because it does not move, Newton’s laws tell you that there must be a second horizontal force acting on the couch, one that opposes your force and is equal in size. This force is static friction, which is the force exerted on one surface by another when there is no motion between the two surfaces.

Section Friction 5.2 Static and Kinetic Friction You might push harder and harder, as shown in the figure below, but if the couch still does not move, the force of friction must be getting larger. This is because the static friction force acts in response to other forces.

Section Friction 5.2 Static and Kinetic Friction Finally, when you push hard enough, as shown in the figure below, the couch will begin to move. Evidently, there is a limit to how large the static friction force can be. Once your force is greater than this maximum static friction, the couch begins moving and kinetic friction begins to act on it instead of static friction.

Section Friction 5.2 Static and Kinetic Friction Friction depends on the materials in contact. For example, there is more friction between skis and concrete than there is between skis and snow. The normal force between the two objects also matters. The harder one object is pushed against the other, the greater the force of friction that results. Since the normal force depends on the mass of the object, greater mass means greater friction. Surface area does NOT contribute to friction. Give an example of low friction: Give and example of high friction:

Section Friction 5.2 Static and Kinetic Friction If you pull a block along a surface at a constant velocity, according to Newton’s laws, the frictional force must be equal and opposite to the force with which you pull. You can pull a block of known mass along a table at a constant velocity and use a spring scale, as shown in the figure, to measure the force that you exert. You can then stack additional blocks on the block to increase the normal force and repeat the measurement.

Section Friction 5.2 Static and Kinetic Friction Plotting the data will yield a graph like the one shown here. There is a direct proportion between the kinetic friction force and the normal force. The different lines correspond to dragging the block along different surfaces. Note that the line corresponding to the sandpaper surface has a steeper slope than the line for the highly polished table.

Friction 5.2 Static and Kinetic Friction Kinetic friction force
Section Friction 5.2 Static and Kinetic Friction You would expect it to be much harder to pull the block along sandpaper than along a polished table, so the slope must be related to the magnitude of the resulting frictional force. The slope of this line, designated μk, is called the coefficient of kinetic friction between the two surfaces and relates the frictional force to the normal force, as shown below. Kinetic friction force The kinetic friction force is equal to the product of the coefficient of the kinetic friction and the normal force.

Section Friction 5.2 Static and Kinetic Friction The maximum static friction force is related to the normal force in a similar way as the kinetic friction force. The static friction force acts in response to a force trying to cause a stationary object to start moving. If there is no such force acting on an object, the static friction force is zero. If there is a force trying to cause motion, the static friction force will increase up to a maximum value before it is overcome and motion starts.

Friction 5.2 Static and Kinetic Friction Static Friction Force
Section Friction 5.2 Static and Kinetic Friction Static Friction Force The static friction force is less than or equal to the product of the coefficient of the static friction and the normal force. In the equation for the maximum static friction force, μs is the coefficient of static friction between the two surfaces, and μsFN is the maximum static friction force that must be overcome before motion can begin.

Static and Kinetic Friction
Section 5.2 Static and Kinetic Friction Note that the equations for the kinetic and maximum static friction forces involve only the magnitudes of the forces. The forces themselves, Ff and FN, are at right angles to each other. The table here shows coefficients of friction between various surfaces. Although all the listed coefficients are less than 1.0, this does not mean that they must always be less than 1.0. p.129

Friction 5.2 Balanced Friction Forces
Section Friction 5.2 Balanced Friction Forces You push a 25.0 kg wooden box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box? Identify the forces and establish a coordinate system.

Balanced Friction Forces
Section Balanced Friction Forces 5.2 Draw a motion diagram indicating constant v and a = 0. Draw the free-body diagram.

Balanced Friction Forces 5.2
Section Balanced Friction Forces 5.2 Identify the known and unknown variables. Known: m = 25.0 kg v = 1.0 m/s a = 0.0 m/s2 μk = 0.20 Unknown: Fp = ?

Section Balanced Friction Forces 5.2 Solve for the Unknown The normal force is in the y-direction, and there is no acceleration. FN = Fg = mg Substitute m = 25.0 kg, g = 9.80 m/s2 FN = 25.0 kg(9.80 m/s2) = 245 N

Section Balanced Friction Forces 5.2 The pushing force is in the x-direction; v is constant, thus there is no acceleration. Fp = μkmg Substitute μk = 0.20, m = 25.0 kg, g = 9.80 m/s2 Fp = (0.20)(25.0 kg)(9.80 m/s2) = 49 N

Section Balanced Friction Forces 5.2 Are the units correct? Performing dimensional analysis on the units verifies that force is measured in kg·m/s2 or N. Does the sign make sense? The positive sign agrees with the sketch. Is the magnitude realistic? The force is reasonable for moving a 25.0 kg box.

Section Section Check 5.2 Question 1 Define friction force.

Section Section Check 5.2 Answer 1 A force that opposes motion is called friction force. There are two types of friction force: Kinetic friction—exerted on one surface by another when the surfaces rub against each other because one or both of them are moving. Static friction—exerted on one surface by another when there is no motion between the two surfaces.

Juan tried to push a huge refrigerator from one corner of his home to another, but was unable to move it at all. When Jason accompanied him, they where able to move it a few centimeter before the refrigerator came to rest. Which force was opposing the motion of the refrigerator? Static friction Kinetic friction Before the refrigerator moved, static friction opposed the motion. After the motion, kinetic friction opposed the motion. Before the refrigerator moved, kinetic friction opposed the motion. After the motion, static friction opposed the motion.

Section Check 5.2 Answer 2 Answer: C
Reason: Before the refrigerator started moving, the static friction, which acts when there is no motion between the two surfaces, was opposing the motion. But static friction has a limit. Once the force is greater than this maximum static friction, the refrigerator begins moving. Then, kinetic friction, the force acting between the surfaces in relative motion, begins to act instead of static friction.

On what does the magnitude of a friction force depend? The material that the surface are made of The surface area Speed of the motion The direction of the motion

Section Check 5.2 Answer 3 Answer: A
Reason: The materials that the surfaces are made of play a role. For example, there is more friction between skis and concrete than there is between skis and snow.

A player drags three blocks in a drag race, a 50-kg block, a 100-kg block, and a 120-kg block with the same velocity. Which of the following statement is true about the kinetic friction force acting in each case? Kinetic friction force is greater while dragging 50-kg block. Kinetic friction force is greater while dragging 100-kg block. Kinetic friction force is greater while dragging 120-kg block. Kinetic friction force is same in all the three cases.

Section Check 5.2 Answer 4 Answer: C
Reason: Kinetic friction force is directly proportional to the normal force, and as the mass increases the normal force also increases. Hence, the kinetic friction force will hit its limit while dragging the maximum weight.

Force in Two Dimensions
Section Force in Two Dimensions 5.3 Equilibrants and the Parallelogram Method In this section you will: Determine the force that produces equilibrium when multiple forces act on an object. Learn the parallelogram method of vector addition.

Section Force in Two Dimensions 5.3 Equilibrium Revisited Now you will use your skills in adding vectors to analyze situations in which the forces acting on an object are at angles other than 90°. Recall that when the net force on an object is zero, the object is in equilibrium. According to Newton’s laws, the object will not accelerate because there is no net force acting on it; an object in equilibrium is motionless or moves with constant velocity.

Section Force in Two Dimensions 5.3 Equilibrium Revisited It is important to realize that equilibrium can occur no matter how many forces act on an object. As long as the resultant is zero, the net force is zero and the object is in equilibrium. The figure here shows three forces exerted on a point object. What is the net force acting on the object? Remember that vectors may be moved if you do not change their direction (angle) or length.

Section Force in Two Dimensions 5.3 Equilibrium Revisited The figure here shows the addition of the three forces, A, B, and C. Note that the three vectors form a closed triangle. There is no net force; thus, the sum is zero and the object is in equilibrium.

Section Force in Two Dimensions 5.3 Equilibrium Revisited Suppose that two forces are exerted on an object and the sum is not zero. How could you find a third force that, when added to the other two, would add up to zero, and therefore cause the object to be in equilibrium? To find this force, first find the sum of the two forces already being exerted on the object. This single force that produces the same effect as the two individual forces added together, is called the resultant force.

Section Force in Two Dimensions 5.3 Equilibrium Revisited The force that you need to find is one with the same magnitude as the resultant force, but in the opposite direction. A force that puts an object in equilibrium is called an equilibrant.

Section Force in Two Dimensions 5.3 Equilibrium Revisited The figure below illustrates the procedure for finding the equilibrant for two vectors. This general procedure works for any number of vectors.

Section Force in Two Dimensions 5.3 Example: Forces of 6.0 N east and 8.0 N south are applied to the top of a pole. What is the equilibrant? R =  (6.0 N)2 + ( 8.0 N)2 = 10 N Equilibrant = 10 N, NW

Section Force in Two Dimensions 5.3 The Parallelogram Method for Vector Addition This is a graphical method that can only be used for adding two vectors. Place the vectors tail-to-tail The vectors are two sides of the parallelogram. Draw in the other two sides. Draw the resultant from the tails to the tips. Worksheet: The Parallelogram Method

If three forces A, B, and C are exerted on an object as shown in the following figure, what is the net force acting on the object? Is the object in equilibrium?

Section Section Check 5.3 Answer 1 We know that vectors can be moved if we do not change their direction and length. The three vectors A, B, and C can be moved (rearranged) to form a closed triangle. Since the three vectors form a closed triangle, there is no net force. Thus, the sum is zero and the object is in equilibrium. An object is in equilibrium when all the forces add up to zero.

What is the difference between a resultant and an equilibrant? A resultant is the sum of two or more vectors. An equilibrant is the same magnitude as the resultant, but opposite in direction.

Compare and contrast the tip-to-tail method for vector addition with the parallelogram method. The tip-to-tail method can be used for adding any number of vectors. The parallelogram method can only be used to add two vectors. In the tip-to-tail method, vectors are lined up head-to-tail. In the parallelogram method, vectors are lined up tail-to-tail. They are both graphical methods.

Chapter 5 Review 5 Graphical Method of Vector Addition
add vectors tip-to-tail; the sum of the vectors is the resultant the resultant is from the tail of the first to the tip of the last vector addition is commutative vectors have magnitude and direction (ex: force, velocity, displacement) scalars only have magnitude (ex: mass, speed, total distance) displacement ≠ total distance

Chapter Chapter 5 Review 5 Mathematical Method for Adding Vectors (use when you have a right triangle) magnitude: R = √ A2 + B2 direction: Give the general cardinal direction ( north, west, southwest, etc.) For vectors, you must specify both magnitude and direction. Finding the Component of Vectors The x- component is the part of the vector parallel to the x-axis. The y- component is the part of the vector parallel to the y-axis.  is labeled so the x- component is adjacent and the y-component is opposite.

Chapter 5 Review 5 Friction Ff = μ FN = μ mg
Friction only depends on μ and m (not surface area). Ff static > Ff kinetic

Inclined planes An object placed on a tilted surface will often slide down the surface. The rate at which the object slides down the surface is dependent upon how tilted the surface is; the greater the tilt of the surface, the faster the rate at which the object will slide down it. In physics, a tilted surface is called an _______________________________.

Objects are known to accelerate down inclined planes because of an unbalanced force.
There are always at least __________forces acting upon any object that is positioned on an inclined plane - the force of gravity and the normal force. The _____________________ (also known as weight) acts in a downward direction; yet the ____________________________acts in a direction perpendicular to the surface (in fact, normal means "perpendicular").

The first peculiarity of inclined plane problems is that the normal force is____________directed in the direction that we are accustomed to. Up to this point in the course, we have always seen normal forces acting in an upward direction, opposite the direction of the force of gravity. But this is only because the objects were always on horizontal surfaces and never upon inclined planes. The truth about normal forces is not that they are always upwards, but rather that they are always directed ________________________________ to the surface that the object is on.

The task of determining the net force acting upon an object on an inclined plane is a difficult manner since the two (or more) forces are not directed in opposite directions. Thus, one (or more) of the forces will have to be resolved into perpendicular components so that they can be easily added to the other forces acting upon the object. Usually, any force directed at an angle to the horizontal is resolved into horizontal and vertical components. However inclined planes are a special case.

However, this is not the process that we will pursue with inclined planes. Instead, the process of analyzing the forces acting upon objects on inclined planes will involve resolving the ________________vector (Fgrav) into two perpendicular components. This is the second peculiarity of inclined plane problems. The force of gravity will be resolved into two components of force - one directed parallel to the inclined surface and the other directed perpendicular to the inclined surface.

The perpendicular component of the force of gravity is directed opposite the normal force and as such balances the normal force. The parallel component of the force of gravity is not balanced by any other force. This object will subsequently ________________ down the inclined plane due to the presence of an unbalanced force. It is the parallel component of the force of gravity that causes this acceleration. The parallel component of the force of gravity is the net force.

The task of determining the magnitude of the two components of the force of gravity is a mere manner of using the equations. The equations for the parallel and perpendicular components are:

In the absence of friction and other forces (tension, applied, etc
In the absence of friction and other forces (tension, applied, etc.), the acceleration of an object on an incline is the value of the parallel component (m*g*sine of angle) divided by the mass (m). This yields the equation (in the absence of friction and other forces)

In the presence of friction or other forces (applied force, tensional forces, etc.), the situation is slightly more complicated. Consider the diagram shown at the right. The perpendicular component of force still balances the normal force since objects do not accelerate perpendicular to the incline. Yet the frictional force must also be considered when determining the net force. As in all net force problems, the __________________is the vector sum of all the forces. That is, all the individual forces are added together as vectors. The perpendicular component and the normal force add to 0 N. The parallel component and the friction force add together to yield 5 N. The net force is 5 N, directed along the incline towards the floor.

All inclined plane problems can be simplified through a useful trick known as "tilting the head." An inclined plane problem is in every way like any other net force problem with the sole exception that the surface has been tilted. Thus, to transform the problem back into the form with which you are more comfortable, merely tilt your head in the same direction that the incline was tilted. Or better yet, merely tilt the page of paper.

Once the force of gravity has been resolved into its two components and the inclined plane has been tilted, the problem should look very familiar. Merely ignore the force of gravity (since it has been replaced by its two components) and solve for the net force and acceleration.

As an example consider the situation depicted in the diagram at the right. The free-body diagram shows the forces acting upon a 100-kg crate that is sliding down an inclined plane. The plane is inclined at an angle of 30 degrees. The coefficient of friction between the crate and the incline is 0.3. Determine the net force and acceleration of the crate.

Finding the force of gravity acting upon the crate and the components of this force parallel and perpendicular to the incline. The force of gravity is 980 N and the components of this force are Fparallel = 490 N (980 N • sin 30 degrees) and Fperpendicular = 849 N (980 N • cos30 degrees). Now the normal force can be determined to be 849 N (it must balance the perpendicular component of the weight vector). The force of friction can be determined from the value of the normal force and the coefficient of friction; Ffrict is 255 N (Ffrict = "mu"*Fnorm= 0.3 • 849 N). The net force is the vector sum of all the forces. The forces directed perpendicular to the incline balance; the forces directed parallel to the incline do not balance. The net force is 235 N (490 N N). The acceleration is 2.35 m/s/s (Fnet/m = 235 N/100 kg).

Problems Involving Friction, Inclines
An object sliding down an incline has three forces acting on it: the normal force, gravity, and the frictional force. The normal force is always perpendicular to the surface. The friction force is parallel to it. The gravitational force points down. If the object is at rest, the forces are the same except that we use the static frictional force, and the sum of the forces is zero. © 2014 Pearson Education, Inc.

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