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Unit 2 Forces This unit will address the next logical question of why things move (or don’t move!), which is known as dynamics. The study of dynamics involves forces, which cause changes in motion (acceleration). You will also study how a force can apply pressure and some of the related applications.

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**Module 2.1 – Introduction to Forces**

This module serves as an introduction to how forces can affect the motion of objects, along with introducing some common forces. This basic understanding of forces will be required in module 3.2, which will involve more in-depth analysis of various situations.

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**Force Force A push or a pull**

A result of an interaction between objects Contact forces are forces that result when two objects are perceived to be physically in contact with one another Non-contact forces are forces which result when two objects are not in physical contact with each other

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Inertia Historically thought that forces provide speed (Aristotle) – rest was natural state Galileo proposed that friction caused objects to come to rest – being at rest not necessarily natural state Inertia - the natural tendency of an object to remain in its current state of motion (either moving or at rest)

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**Net Force is 4 N to the right**

Net Force – overall (total) force Net Force is zero Net Force is 4 N to the right

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Check Your Learning Why does a package on the seat of a bus slide backward when the bus accelerates quickly from rest? Why does it slide forward when the driver applies the brakes? The bus is initially at rest, as is the package. In the absence of any force, the natural state of the package is to remain at rest. When the bus pulls forward, the package remains at rest because of its inertia (until the back of the seat applies a forward force to make it move with the bus). From the point of view of someone on the bus, it appears that the package is moving backward; however, someone watching from outside the bus would see the bus move forward and the package trying to stay in its original position. Once the package is moving with the bus, its inertia has now changed. It now has a natural tendency to be moving forward with a constant speed. When the bus slows down, the package continues to move forward with the same constant speed that it had until some force stops it.

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Check Your Learning A marble is fired into a horizontal circular tube that is anchored onto a frictionless tabletop, as shown in the diagram below (as viewed from above). Which of the 3 paths will the ball take as it exits the tube? The marble will follow path #2. The ball is following a circular path while it is in the tube because the tube is exerting a force that is causing it to change its direction and travel in a circle; when that force is no longer applied, the ball continues to do the last thing that it was doing – it continues travelling in a straight line at a constant velocity.

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Mass Mass of an object refers to how much (the quantity) matter there is in an object. Mass is a scalar quantity and is measured in kilograms (kg). The mass of an object does not change depending on where the object is Mass can also be considered to be a measure of the inertia of an object

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**Force of Gravity and Weight**

Force of Gravity – non-contact force of attraction that is present between any two objects Directly proportional to the mass of each object and inversely proportional to the distance squared Weight is defined as the force of gravity acting on an object g is the acceleration due to gravity and can vary depending on where you are on Earth Weight can change depending on location

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**Values for g Location Acceleration due to gravity (m/s2) North Pole**

9.83 Equator 9.78 Mt. Everest 9.76 Marianas Ocean Trench International Space Station 9.08 Location Acceleration due to gravity (m/s2) Earth 9.80 Moon 1.64 Mars 3.72 Jupiter 25.9

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**Check Your Learning A person has a weight of 639 N at the North Pole.**

What is the person’s mass? Remember that weight is the same as the force of gravity. Since we are at the North Pole, the acceleration due to gravity can be obtained from Table Since the directions here are all toward the center of the Earth, we will use the scalar form of the equation and ignore directions.

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**Check Your Learning What is the person’s mass at the equator?**

Since mass is an indication of the amount of matter present in an object and is not dependent on location, their mass is still 65.0 kg. What is the person’s weight at the equator? Since we are looking for weight, we actually need to find the force of gravity on the person. Again, we can obtain the value of g from Table 1.

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**Check Your Learning What is the person’s mass on the moon?**

Since mass is an indication of the amount of matter present in an object and is not dependent on location, the mass is still 65.0 kg. What is the person’s weight on the moon? Since we are looking for weight, we actually need to find the force of gravity on the person. This time we can obtain the value of g from Table 2

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**Common Forces Name of Force Symbol Description Force of friction**

This is the force of resistance that a surface exerts on an object. It acts in a direction opposite that of the motion of the object. This will be examined more closely in the next section. Normal Force This is the force exerted by a surface on an object. It is always perpendicular to the surface and in a direction away from the surface. Air Resistance This is similar to the force of friction, but refers to the force of resistance exerted by air molecules on an object. It acts in a direction opposite that of the motion of the object. Tension This is the force exerted by a rope. The tension force is directed along the length of the rope and pulls equally on the objects on the opposite ends of the rope. Applied Force A force applied to an object by a person or another object.

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**Check Your Learning A book is at rest on a table.**

Identify the forces (with their direction) acting on the book in each of the following situations: A book is at rest on a table. force of gravity acting downward the normal force of the table pushing up A book is being pushed by a person horizontally to the right at a constant speed. the force of the person pushing to the right the force of friction acting on the box to the left

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Check Your Learning The book in the previous part is let go, allowing it to slow down and come to rest. Since there is no longer any interaction between the person and the book, there is no longer a acting on the book. The other forces remain the same. force of gravity acting downward the normal force of the table pushing up the force of friction acting on the box to the left A book is falling through the air, accelerating downward. force of air resistance acting upward

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Free Body Diagrams Free body diagram is essential when setting up a problem involving forces. Identifies all of the forces (including their directions) acting on a single object, and only the forces acting on that object. Each force should be represented by an arrow that is directed outward from the body in the direction of the force

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Example 1 Draw a free body diagram for a box sitting on a table. Solution:

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**Example 2 Consider the following free body diagram:**

In what direction is the net force? Solution: If we look at the vertical forces, we see that the force upward is equal in magnitude to the force downward (since the force vectors are the same length); therefore, the vertical net force is zero. If we look at the horizontal forces, it is obvious that the force to the right is bigger than the force to the left; therefore, there is a net force to the right and the object will have an acceleration to the right.

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Check Your Learning Draw a free body diagram for the book in each of the following situations: A book is at rest on a table. -force of gravity acting downward -the normal force of the table pushing up A book is being pushed by a person horizontally to the right at a constant speed. - force of gravity acting downward - the normal force of the table pushing up - the force of the person pushing to the right - the force of friction acting on the box to the left

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Check Your Learning The book in the previous part is let go, allowing it to slow down and come to rest. Since there is no longer any interaction between the person and the book, there is no longer a acting on the book. The other forces remain the same. - force of gravity acting downward the normal force of the table pushing up the force of friction acting on the box to the left A book is falling through the air, accelerating downward. - force of gravity acting downward - force of air resistance acting upward

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Friction Friction between two surfaces exists because of an interaction between these surfaces Opposes the motion Normal force tells us with how much force two bodies are being pushed together Must be calculated from the free body diagram!!

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Kinetic Friction When a body is already in motion, a force of kinetic friction (also sometimes called sliding friction) always acts to oppose the sliding of the two surfaces past each other. The magnitude of this kinetic friction depends on two things: The nature of the two sliding surfaces – different surfaces will have different amounts of friction. The normal force – the bigger the normal force, the more force there is pushing the two surfaces together.

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**Coefficient of Friction**

is the coefficient of kinetic friction Depends on the two surfaces involved Rearranging gives So coefficient of friction is dimensionless (no units)

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Static Friction When a body is at rest (relative to the surface that it is in contact with), a force of static friction always acts to resist any attempt to start a body moving. or μs is the coefficient of static friction. This coefficient of static friction generally has a larger value than the coefficient of kinetic friction

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**Sample Values of μ μk μs 0.2 0.4 0.03 0.1 0.8 1.0 0.5 0.7 1 1-4 0.07**

Surfaces μk μs Wood on wood 0.2 0.4 Ice on ice 0.03 0.1 Rubber on dry concrete 0.8 1.0 Rubber on wet concrete 0.5 0.7 Rubber on other surfaces 1 1-4 Metal on metal (lubricated) 0.07 0.15

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Example 1 A 7.0 kg box is being pushed horizontally at a constant speed. If the coefficient of friction is 0.30, how much force is being used to push the box? Solution

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Example 1 Since the speed is constant, the net force must be zero. The force that is pushing the box must balance the force of friction (the net force must be zero since there is no change in motion)

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Example 2 You are pushing horizontally on a book against a wall so that it does not slide down the wall. Draw a free body diagram for this situation. What condition is necessary for the box to not slide down the wall? If you lessen the horizontal push that you are exerting, the box will start to slide down the wall. Explain why this happens.

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**Example 2 Assuming that you are pushing to the right**

In order for the box to not slide down the wall, the force of gravity must be balanced by the force of friction According to the free body diagram, the normal force and the pushing force must be equal. If you push with less force, then the normal force will be less. A smaller normal force provides a smaller force of friction, which mean the force of gravity is now larger than friction and the box will slide down the wall.

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Check Your Learning A friend pushes a 625 g textbook horizontally along a table at a constant velocity with 3.50 N of force. Determine the normal force supporting the textbook. As can be seen in the free body diagram, the normal force must equal the force of gravity (since there is no vertical acceleration).

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Check Your learning Calculate the force of friction between the book and the bench. Since the textbook has a constant velocity, the net force must be zero; therefore, the force pushing to the right must equal the force of friction to the left: Calculate the coefficient of friction between the book and the bench. Since we already know the force of friction and the normal force, we can find the coefficient of friction

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Check Your Learning Which coefficient of friction have you found, static or kinetic? Since the textbook was moving at a constant velocity, this is the coefficient of kinetic friction.

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**Module Summary In this module you have learned that**

Inertia is the tendency of an object to maintain its motion and that a net force is requires to change the motion of an object. The force of gravity can be found using the equation and that the weight of an object is the force of gravity acting on that object. Free body diagrams can be used to identify and analyze the forces in a problem. The force of friction can be found using the equation

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**Newton’s First Law Law of Inertia**

Inertia is the tendency of an object to maintain its motion unless acted on by an outside net force. Newton’s First Law: An object in uniform motion (or at rest) will remain in uniform motion (or at rest) unless acted on by an outside net force.

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Frames of Reference Inertial frame of reference can be defined as one in which Newton’s First Law applies Any frame of reference that moves at a constant velocity relative to an inertial frame of reference is also an inertial frame of reference Any frame of reference that accelerates relative to an inertial frame of reference is not an inertial frame of reference (it can be referred to as a noninertial frame of reference)

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Check Your Learning A physics book is motionless on the top of a table. If you give it a hard push, it slides across the table and slowly comes to a stop. Use Newton’s first law of motion to answer the following questions. Why does the book remain motionless before the force is applied? The book is rest; it remains at rest since there is no outside force acting on it. Why does the book move when the hand pushes on it? The hand provides an outside force, so the book no longer remains at rest. Why does the book eventually come to a stop? When you remove your hand, the only horizontal force acting on the book is friction. Because of this outside force, the book does not maintain its uniform motion and begins slowing down. Under what conditions would the book remain in motion at a constant speed? If there were no friction or if you push the book with a force exactly equal to friction (so that the net force were zero) then Newton’s First Law applies and the book remains in uniform motion.

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Check Your Learning What determines if a frame of reference is inertial or not? Give an example of each type of frame of reference. A frame of reference is inertial if Newton’s First Law is true. This will occur if the frame of reference is at rest or moving at a constant velocity. A frame of reference is not inertial if Newton’s First Law does not hold; this will occur if the frame of reference is accelerating. An example of an inertial frame of reference would be an elevator moving at a constant speed. If the elevator starts accelerating upward, it is no longer an inertial frame of reference.

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Newton’s Second Law A net force must provide an acceleration if uniform motion is no longer being maintained. How much will an object accelerate if there is a net force? Larger force provides a larger acceleration Acceleration can be said to be directly proportional to the net force

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**Newton’s Second Law Larger mass reduces the acceleration**

Acceleration can be said to be inversely proportional to the mass Proportionality constant of 1

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Newton’s Second Law Newton’s Second Law: The net force needed to accelerate an object is a product of the object’s mass and acceleration. Notice in this equation that both net force and acceleration are vectors and must have the same direction Important to remember that net force is an overall force requirement – other forces must act to provide this net force

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Example A 1300 kg car is moving at a constant speed when the brakes are applied, providing a frictional force of 6500N. What is the acceleration? Solution

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Example The vertical forces of gravity and the normal force balance one another, resulting in no vertical acceleration The acceleration is horizontal the net force can only be provided by horizontal forces the only horizontal force is friction. Using Newton’s Second Law (and using the right as the positive direction)

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Check Your Learning A race car has a mass of 710 kg. It starts from rest and travels 40.0 m in 3.0 s. What net force is applied to it?

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Newton’s Third Law What happens if you are standing on a skateboard and you push a person standing on a different skateboard? You both move! A force must have been applied to each of you. Newton’s Third Law: For every action force, there is an equal and opposite reaction force. These forces always act on different objects!

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Newton’s Third Law With equal and opposite forces, how does anything ever move? Picking up a ball: Ball exerts an equal force on your hand, but this is not on the ball and does not appear in the free body diagram

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Example Suppose you are floating around in space (many km from any planet so that you feel no gravity) outside of your spaceship. You get frustrated and decide to kick your spaceship. Does your foot hurt? Solution Yes, your foot will hurt. Even though there is no gravity, Newton’s Third law still applies. If you kick the spaceship, it applies an equal and opposite force on your foot.

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Check Your Learning A 60.0 kg boy and a 40.0 kg girl use an elastic rope while engaged in a tug of war on a frictionless icy surface. If the acceleration of the girl toward the boy is 3.0 m/s2, what is the acceleration of the boy toward the girl? Looking at the girl first,

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Check Your Learning According to Newton’s Third Law, an equal and opposite force will be applied to the boy:

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Module 3.2 Objective 2 Upon completion of this module, the participant will be able to: Analyze complex dynamics situations using Newton’s Second Law.

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**Multiple Force Problems**

Most real world situations involve multiple forces acting on an object that contribute to the net force Must draw a free body diagram!! Free body diagram helps you to do a number of things: Identify the forces involved, with their directions Separate the vertical from horizontal forces Determine what forces must be equal to one another (in order to determine the normal force, for example) Determine what forces contribute to the acceleration (if there is one) Assign a positive or negative sign to the forces to indicate directions

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Vector Notation Refers to the force vector, both magnitude and direction Refers to the force magnitude only – direction must be specified in another way

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Example A 43.0 kg chair is being pushed across a floor with an acceleration of 3.1 m/s2 to the right. If the coefficient of friction between the chair and the floor is 0.44, with how much force is the chair being pushed? Solution

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**Example Vertical Forces – only normal force and force of gravity**

Since there is no vertical acceleration Horizontal Forces

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Check Your Learning A force of 40. N accelerates a 5.0 kg block at 6.0 m/s2 to the right along a horizontal surface. What is the coefficient of friction?

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Check Your Learning Vertical Forces Horizontal Forces

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**Apparent Weight Weight = force of gravity**

Apparent Weight – how heavy you feel in certain situations Consider a person standing on a bathroom scale in an elevator at rest: Scale exerts a force equal to the force of gravity

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Apparent Weight Consider what happens when the elevator accelerates upward: Scale exerts a force larger than the force of gravity since net force is upwards According to Newton’s Third Law, you exert a force on the scale equal to the force that the scale exerts on you You feel heavier!

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Example A person has a mass of 82 kg. He is standing on a scale (which measures force in newtons) in an elevator. If the elevator is accelerating upward at 2.3 m/s2, what is the person’s apparent weight? Solution

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**Example Using up as positive**

The person’s apparent weight is 990 N, which is heavier than their actual weight of 800 N.

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Check Your Learning You are in an elevator at rest and you are standing on a scale. The scale reads 715 N. When the elevator begins accelerating, the scale reads 820 N. What is the acceleration of the elevator? When the elevator is at rest, the upward force that the scale is exerting will equal the downward force of gravity.

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**Check Your Learning When the elevator is accelerating upward,**

Using up as positive

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Check Your Learning If the scale were calibrated in kg (as many typical bathroom scales are), what would the two readings be? At rest, the scale would read 73.0 kg (which corresponds to a weight of 715 N on Earth). While accelerating upward, the scale would read 84 kg (which corresponds to the apparent weight of 820 N). Has your mass actually changed? No, your mass has not actually changed. The scale measures force, not mass. It is simply calibrated to provide the mass that corresponds to the measured weight on Earth. Since the scale is measuring your apparent weight (not your real weight which is simply the force of gravity that is acting on you), the scale gives your “apparent mass” which is what you feel like. Your mass is still 73.0 kg, although you feel as heavy as you would if you had a mass of 84 kg.

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**Systems of Masses Systems involve multiple masses and ropes/pulleys**

Tension in a rope can be considered to be the same everywhere in the rope and exerts a force on whatever it is connected to that is equal to its tension Elevator can use a counterweight to exert an upward force on it to help overcome gravity:

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**Systems of Masses Consider following system:**

Assign the heavier mass m2 to be the positive end of the rope and the lighter mass m1 to be the negative end We can now redraw the system to linearize it

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**Systems of Masses If we keep the two masses separate**

If we combine the two masses (eliminating tension)

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**Individual Objects When treating the system as individual objects:**

Use the individual mass of the object that you are studying when applying Newton’s Second Law. Draw a free body diagram for each object individually. Tension forces exerted along a cable or rope do contribute to the motion of an individual object and should be included as part of the net force. Set up Newton’s Second Law for each object. It may then be necessary to solve a system of equations using the equation for each object.

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**One Big Object When treating the system as one big object:**

The mass of the system is the total mass of all of the objects. This is the mass that should be used when applying Newton’s Second Law. Tension forces exerted along a rope or a cable between any two objects in the system are called internal forces and should not be included as part of the net force. These internal forces do not affect the motion of the system as a whole; only external forces such as friction and gravity affect the motion of the system and should be included in Newton’s Second Law.

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Example An Atwood machine is a simple machine consisting of two objects connected by a rope hanging over a pulley as shown in the picture below. What is the acceleration of the system?

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Method 1

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Method 1 Solving this system of equations,

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Method 2

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Other Systems Systems can also involve forces that are at angles to one another, such as a falling weight that exerts a horizontal force on a lab cart, Analyze the forces on each individual object. Assign a direction to the motion along the rope. Pretend the motion is linear, including only forces that are parallel to the motion in the calculation of net force. Draw both individual free body diagrams and a combination free body diagram of the system.

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Check Your Learning Ignoring friction, calculate the acceleration of the system below and the tension in the rope.

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Check Your Learning Individual Free Body Diagrams

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**Check Your Learning Linearized Free Body Diagram**

Using clockwise (or toward the 7.0 kg mass) as positive

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**Module Summary In this module, you learned **

Newton’s First Law – An object in uniform motion (or at rest) will remain in uniform motion (or at rest) unless acted on by an outside net force. Newton’s Second Law - The net force needed to accelerate an object is a product of the object’s mass and acceleration. Newton’s Third Law – For every action force, there is an equal and opposite reaction force. How to solve complex single body problems using free body diagrams and Newton’s Laws How to calculate apparent weight How to solve systems of masses involving ropes and pulleys

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