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Intro to Dynamics

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We all have an intuition about how objects move. Our beliefs are hard to change since they work well in our day to day lives. But they limit us in developing an understanding of how the world works - we must build on our intuition and move beyond it. Intuitive Physics

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Galileo vs. Aristotle In our experience, objects must be pushed in order to keep moving. So a force would be needed to have a constant velocity. This is what Aristotle claimed in his in his series of books entitled "Physics", written 2400 years ago. But 400 years ago, another scientist and astronomer, Galileo, proposed the following thought experiment which revealed another perspective.

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Imagine two perfectly smooth ramps connected together by a perfectly smooth surface. If a ball is let go at the top of the one ramp, what will happen? Thought Experiment

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Imagine two perfectly smooth ramps connected together by a perfectly smooth surface. If a ball is let go at the top of the one ramp, what will happen?

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Thought Experiment Imagine two perfectly smooth ramps connected together by a perfectly smooth surface. If a ball is let go at the top of the one ramp, what will happen?

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Thought Experiment Imagine two perfectly smooth ramps connected together by a perfectly smooth surface. If a ball is let go at the top of the one ramp, what will happen?

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If a ball rolls down one ramp, it keeps rolling up the other side until it reaches the same height. Thought Experiment

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Now repeat that experiment, but make the second ramp less steep. What Will Happen? Thought Experiment

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Now repeat that experiment, but make the second ramp less steep. What Will Happen?

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Thought Experiment Now repeat that experiment, but make the second ramp less steep. What Will Happen?

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Thought Experiment Now repeat that experiment, but make the second ramp less steep. What Will Happen?

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It will still keep rolling until it reaches the same height, but it has to roll farther! Thought Experiment

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Finally, make the ramp flat. Now what will happen?

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Thought Experiment Finally, make the ramp flat. Now what will happen?

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Thought Experiment Finally, make the ramp flat. Now what will happen?

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Thought Experiment Finally, make the ramp flat. Now what will happen?

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Thought Experiment Finally, make the ramp flat. Now what will happen?

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Thought Experiment Finally, make the ramp flat. Now what will happen?

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Thought Experiment It will keep rolling forever, no external force is necessary.

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Galileo vs. Aristotle It's not that Aristotle was wrong. In everyday life, objects do need to keep being pushed in order to keep moving. Push a book across the table. When you stop pushing, it stops moving. Aristotle is right in terms of what we see around us every day.

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Force and Motion It's just that Galileo, and later Newton, imagined a world where friction could be eliminated. F applied F friction In the absence of all external forces, an object's velocity remains constant. Two equal and opposite forces have the same effect, they cancel to create zero net force. Friction represents an external force acting on the object, just as your push is an external force.

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Newton's 1st Law of Motion

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Sir Isaac Newton Galileo's observations were more fully formed in 1687 by the 'father of physics, ' Sir Isaac Newton, who called this observation "The First Law of Motion".

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I. Newton's First Law of Motion In other words, an object maintains its velocity (both speed and direction) unless acted upon by a nonzero net force or an unbalanced fore. Having zero velocity, being at rest, is not special, it is just one possible velocity…a velocity which is no more special than any other. A n object at rest remains at rest, and an object in motion remains in motion, unless acted on by a net external force.

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Newton's First Law of Motion Mass as a Measure of the Amount of Inertia All objects resist changes in their state of motion. The tendency of an object to resist changes in its state of motion varies with mass. Mass is that quantity that is solely dependent upon the inertia of an object. The more inertia that an object has, the more mass that it has. A more massive object has a greater tendency to resist changes in its state of motion. A. Mass and Inertia

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Force is a vector quantity-a push or a pull 1. Balanced forces: equal and opposite forces Balanced forces give no motion No motion of hands Newton's First Law of Motion B. Combining Forces

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2. Unbalanced force – more force in one direction (net force)

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Bigger force to the right Smaller force to the left Net force to the right

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A.K.A. The Law of Inertia This law is often referred to as the "Law of Inertia." The word inertia comes from the latin word iners which means idle, or lazy. Inertia is the tendency of an object to resist any change in motion. Demo: hard boiled egg vs. raw egg. Click here.Click

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1 In the absence of an external force, a moving object will Astop immediately. Bslow down and eventually come to a stop. Cgo faster and faster. Dmove with constant velocity. c c c c

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2 When the rocket engines on a spacecraft are suddenly turned off while traveling in empty space, the starship will Astop immediately. Bslowly slow down, and then stop. Cgo faster and faster. Dmove with constant velocity. c c c c

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3 A rocket moves through empty space in a straight line with constant speed. It is far from the gravitational effect of any star or planet. Under these conditions, the force that must be applied to the rocket in order to sustain its motion is Aequal to its weight. Bequal to its mass. Cdependent on how fast it is moving. D zero. c c c c

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4 You are standing in a moving bus, facing forward, and you suddenly fall forward. You can infer from this that the bus’s Avelocity decreased. Bvelocity increased. Cspeed remained the same, but it's turning to the right. Dspeed remained the same, but it's turning to the left. c c c c

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5 You are standing in a moving bus, facing forward, and you suddenly fall forward as the bus comes to an immediate stop. What force caused you to fall forward? Agravity Bnormal force due to your contact with the floor of the bus Cforce due to friction between you and the floor of the bus DThere is not a force leading to your fall. c c c c

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Newton's laws are only valid in inertial reference frames: An inertial reference frame is one which is not accelerating or rotating. It is an area in which every body remains in a state of rest unless acted on by an external unbalanced force. Inertial Reference Frames

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This is why a drink on the dashboard of a car can suddenly seem to accelerate backwards without any force acting on it. It's not accelerating, it's standing still. The reference frame, the car, is accelerating underneath it. Inertial Reference Frames When your car accelerates, it is not an inertial reference frame. Click here for a very famous video about frames of reference. watch the first 2:30 of the video

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II. Nature of Force

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A.Definition: A force is a push or pull resulting from the interaction between objects. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction stops, the two objects no longer experience the force. Forces only exist as a result of an interaction.

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II. Nature of Force B.Type of Forces A force is a push or pull acting upon an object as a result of its interaction with another object. There are two broad categories of forces based on whether the force resulted from the contact or non-contact of the two interacting objects. Contact ForcesNon Contact Normal ForceGravitational Force Frictional ForceMagnetic Force TensionElectrical Force Air Resistance Force Applied Force Spring Force

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There are many different types of forces that occur in nature, but perhaps none is more familiar to us than the force of friction (F fr ). 1. Friction - A Resistive Force Friction is a resistive force that opposes the motion of an object. What does sandpaper have to do with friction?

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Friction is the reason objects stop rolling or sliding along a surface. It is the reason it is difficult to start pushing a heavy box along the floor. Friction - A Resistive Force There are many different types of friction: Friction between solid objects and air is often called air resistance. Friction between two fluids is called viscosity, and so on.

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a. Factors affecting friction: Friction - A Resistive Force Type of surfaces How hard surfaces are pressed together

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b. Static v. Dynamic Friction Static Friction that acts on something that is not moving; No heat or wear is generated Dynamic Friction, aka Kinetic Friction, results from contact between moving objects (sliding, rolling, fluid)

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2. Gravity – force of attraction between two objects -- pulling force a. Factors Affecting Gravity i. Mass – more mass, more gravity ii. Distance – closer, more gravity

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b. Weight vs. Mass Define: weight: force of gravity on an object (F w ); dependent on location of the object (due to masses and distances) mass: amount of matter an object contains; does not change with location 1 kilogram mass has a weight of 9.8 N (on earth)

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Mass is the measure of the inertia of an object, the resistance of an object to accelerate. In the SI system, mass is measured in kilograms. Mass is not weight! Mass is a property of an object. It doesn't depend on where the object is located. Weight is the force exerted on that object by gravity. If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same. The case of mass versus weight Click on this link to see a Veritasium video about mass and weight!

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Weight is the force exerted on an object by gravity. Close to the surface of Earth, where the gravitational force is nearly constant, weight can be calculated with: Weight – the Force of Gravity Near the surface of Earth, g is 9.8 m/s 2 downwards. F g = mg

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18Determine the Force of Gravity (weight) of a 6.0 kg bowling ball.

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19 Determine the weight of a small car with a mass of 900 kg.

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20Using a spring scale, you find that the weight of a friction block in the lab is around 24 N. What is the mass of the block, in kilograms?

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21A 120 lb woman has a mass of about 54.5 kg. What is her weight?

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22What is the weight of a 25 kg object located near the surface of Earth?

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23Which of the following properties of an object is likely to change on another planet? AMass BWeight C Color DVolume (size and shape) c c c c

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24The acceleration due to gravity is lower on the Moon than on Earth. Which of the following is true about the mass and weight of an astronaut on the Moon's surface, compared to Earth? AMass is less, weight is same BMass is same, weight is less CBoth mass and weight are less DBoth mass and weight are the same c c c c

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FGFG An object at rest must have no net force on it. If it is sitting on a table, the force of gravity is still there; what other force is there? Weight – the Force of Gravity; and the Normal Force

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FGFG FNFN An object at rest must have no net force on it. If it is sitting on a table, the force of gravity is still there; what other force is there? The force exerted perpendicular to a surface is called the normal force. It is exactly as large as needed to balance the force from the object (if the required force gets too big, something breaks!) The words "normal" and "perpendicular" are synonyms. Weight – the Force of Gravity; and the Normal Force

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25A 14 N brick is sitting on a table. What is the normal force supplied by the table? A14 N upwards B28 N upwards C14 N downwards D28 N downwards c c c c

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26What normal force is supplied by a desk to a 2.0 kg box sitting on it?

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3. ‘Other’ In this unit, we will be concerned with weight (force of gravity on an object) and friction (force that opposes motion). We will also conside other contact forces: Normal Force Applied Force Tension Air Resistance

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Return to Table of Contents C. Free Body Diagrams

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Free Body Diagrams A free body diagram is a drawing physicists use in order to show all the forces acting on an object. Drawing free body diagrams can help when trying to solve for unknown forces or showing the motion of the object. Click here for a Veritasium video on free body diagrams and reviewing Normal Force!

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Free Body Diagrams 1. Draw and label a dot to represent the first object. 2. Draw an arrow from the dot pointing in the direction of one of the forces that is acting on that object. Label that arrow with the name of the force. 3. Repeat for every force that is acting on the object. Try to draw each of the arrows to roughly the same scale, bigger forces getting bigger arrows. mg FTFT

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Free Body Diagrams 4. Once you have finished your free body diagram, recheck it to make sure that you have drawn and labeled an arrow for every force. This is no time to forget a force. 5. Draw a separate arrow next to your free body diagram indicating the likely direction of the acceleration of the object. This will help you use your free body diagram effectively. 6. Repeat this process for every object in your sketch. mg FTFT FTFT a

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A hot air balloon hovers in the sky. Draw it's free body diagram: FBFB mg a = 0

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A boy pulls a toy along (with constant velocity) on a string. Draw the toy's free body diagram: mg FNFN FTFT a = 0 (const. velocity)

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An object on a crane descends with constant velocity. Draw the object's free body diagram: FTFT mg a = 0

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Kinetic Friction Force Friction that acts on an object that is already in motion is called kinetic friction. For kinetic – sliding friction, we write: Kinetic friction is the product of two things: μ k is called the coefficient of kinetic friction, and is different for every pair of surfaces. F N is simply the Normal Force, which, on flat surfaces, is equal to the weight of the object.

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Kinetic Friction Force A larger coefficient of friction means a greater frictional force. Notice the friction that occurs between different materials in the table below: Surface Coefficient of Kinetic Friction Wood on Wood0.2 Ice on Ice0.03 metal on metal (lubricated)0.07 Steel on steel (unlubricated)0.6 Rubber on dry concrete0.8 Rubber on wet concrete0.5 Rubber on other solid surface0.04 Teflon on Teflon1.0 Human Joints in limbs0.01

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Σ F y = 0 F N = mg A man accelerates a crate along a rough surface. Draw the crate's free body diagram: Determine ΣF in the x and y directions FNFN mg F app F fr a The object has a net force to the right: ΣF x = ma x F app - F fr = ma x

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34A 4.0kg brick is sliding on a surface. The coefficient of kinetic friction between the surfaces is What it the size of the force of friction?

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35A brick is sliding to the right on a horizontal surface. What are the directions of the two surface forces: the friction force and the normal force? Aright, down Bright, up Cleft, down Dleft, up c c c c

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Static friction is the frictional force between two surfaces that are not moving along each other. Static friction keeps objects from moving when a force is first applied. Static Friction Force F applied v = 0 **

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μ s is the coefficient of static friction, and is different for every pair of surfaces. Static Friction Force F applie d v = 0 **

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Static Friction Force ** Note the ≤ symbol in this equation. Imagine pushing on a box until it moves. You can apply a small force... nothing happens. You apply more and more force until the box finally starts moving - this is the maximum amount of static friction. The friction can be LESS than the maximum amount or EQUAL to the maximum amount, but never greater. The force of friction is equal to μ s F N at the instant when the object starts to move. Then what happens?

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Friction Force The static frictional force increases as the applied force increases, always equal to the net applied force. Until it reaches its maximum, μ s F N. Then the object starts to move, and the kinetic frictional force takes over, μ K F N Friction force, f Applied force, F A f = μ S F N μ k F N 0 no motion sliding Friction force, f **

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Friction Force Surface Coefficient of Static Friction Coefficient of Kinetic Friction Wood on wood Ice on ice Metal on metal (lubricated) Steel on steel (unlubricated) Rubber on dry concrete Rubber on wet concrete Rubber on other solid surfaces1-41 Teflon on Teflon in air0.04 Joints in human limbs0.01 ** The table below shows values for both static and kinetic coefficients of friction. Notice that static friction is greater than kinetic friction. Once an object is in motion, it is easier to keep it in motion.

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36A 4.0 kg brick is sitting on a table. The coefficient of static friction between the surfaces is What is the largest force that can be applied horizontally to the brick before it begins to slide? **

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37 A 4.0kg brick is sitting on a table. The coefficient of static friction between the surfaces is If a 10 N horizontal force is applied to the brick, what will be the force of friction? **

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Return to Table of Contents Tension

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FTFT mg a When a cord or rope pulls on an object, it is said to be under tension, and the force it exerts is called a tension force, F T. Any object that is hanging or suspended is considered to have tension acting upward. In some cases, tension can be applied in other directions as well. Tension Force

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FTFT mg a There is no special formula to find the force of tension. We need to use force diagrams and net force equations to solve for it! Tension Force

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38 A 25 kg lamp is hanging from a rope. What is the tension force being supplied by the rope?

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39 A crane is lifting a 60 kg load at a constant velocity. Determine the tension force in the cable.

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D. Net Force Sum off all forces acting on an object

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II. Newton's 2nd Law of Motion

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An object doesn't change its velocity unless a force acts on it. How does an object respond to a force when it is applied? Newton’s Second Law of Motion

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Newton’s second law identifies the relationship between acceleration and force. Namely, as a net force is applied, an object accelerates. A. Calculating with Newton’s Second Law of Motion F net = ma *the word 'net' means overall, or total. We will discuss this in further detail later, but for now just think of the sum of all forces, sometimes written as ΣF as any force on an object

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Units of Force Mass is measured in kilograms (kg). And we already know acceleration is measured in meters/second 2 (m/s 2 ). Therefore, the unit of force, the Newton, can be found from the second law F net = ma N = kg*m/s 2 The unit of force in the SI system is the newton (N). F net = ma

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6 A 3.5 kg object experiences an acceleration of 0.5 m/s 2. What net force does the object feel?

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7A 12 N net force acts on a 36 kg object? How much does it accelerate?

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8How much net force is required to accelerate a 0.5 kg toy car, initially at rest to a velocity of 2.4 m/s in 6 s?

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We can use this equation to understand how force, mass, and acceleration are related. When two variables are 'directly proportional', it means that if we increase one, the other will increase as well. Newton’s Second Law of Motion F net = m a In Newton's 2nd Law, force and mass are directly proportional. More mass requires more force to create the same acceleration. Similarly, force and acceleration are directly proportional. To increase the acceleration of an object, more force must be applied.

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We can rearrange this equation to see how mass is related to acceleration: Newton’s Second Law of Motion a = F net m Acceleration is directly proportional to force and inversely proportional to mass.

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Newton’s Second Law of Motion a = F net m When two variables are 'inversely proportional', it means that if we increase one, the other will decrease! So if we increase the mass of an object under a constant force, the acceleration will decrease. Think about what happens when you move a filled wheelbarrow...

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9A net force F accelerates a mass m with an acceleration a. If the same net force is applied to mass 2m, then the acceleration will be A4a B2a Ca/2 Da/4 c c c c

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10A constant net force acts on an object. The object moves with: Aconstant acceleration Bconstant speed Cconstant velocity Dincreasing acceleration c c c c

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11A net force F acts on a mass m and produces an acceleration a. What acceleration results if a net force 2F acts on mass 4m? Aa/2 B8a C4a D2a c c c c

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12The acceleration of an object is inversely proportional to: Athe net force acting on it. Bits position. Cits velocity. Dits mass. c c c c

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Return to Table of Contents Net Force F net (ΣF)

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The greek letter sigma "Σ" means "the sum of". Sometimes ΣF is written as F net, it means the same thing. It means you have to add up all the forces acting on an object. Net Force ΣFΣF ΣF = ma Let's look at the left side of this equation first.

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Net Force The arrow above "F" reminds you that force is a vector. We won't always write the arrow but remember it's there. It means that when you add forces, you have to add them like vectors: forces have direction, and they can cancel out. ΣFΣF

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Example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object? Net Force ΣFΣF First we'll draw a free body diagram. We will discuss these in more detail later on but for now, follow these simple directions. FBDs consists of a dot, representing the object, and arrows representing the forces. The direction of the arrows represents the direction of the forces...their length is roughly proportional to their size.

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Newton’s Second Law of Motion ΣFΣF The first force (F 1 ) acts to the right with a magnitude of 20 N F1F1 Example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object?

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Newton’s Second Law of Motion ΣFΣF The second force, F 2, acts to the right also, with a greater magnitude of 30N. This is drawn slightly larger than F 1. F1F1 F2F2 Example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object?

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Newton’s Second Law of Motion ΣFΣF F1F1 F2F2 To add vectors, move the second vector so it starts where the first one ends. Example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object? The sum is a vector which starts where the first vector started, and ends where the last one ends.

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Newton’s Second Law of Motion ΣFΣF F1F1 F2F2 ΣFΣF These free body diagrams are critically important to our work. Once done, the problem can be translated into an algebra problem. Example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object?

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For example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). What is the net force on the object? Newton’s Second Law of Motion First we will define "to the right" as positive. Then we can interpret our diagram to read: ΣF = F 1 + F 2 ΣF = 20N + 30N ΣF = 50N to the right we get the direction from our diagram and from our positive answer, which we defined as meaning "to the right" F1F1 F2F2 ΣFΣF

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13Two forces act on an object. One force is 40N to the west and the other force is 40N to the east. What is the net force acting on the object?

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14Two forces act on an object. One force is 8.0 N to the north and the other force is 6.0N to the south. What is the net force acting on the object? A14 N to the north B14 N to the south C2 N to the north D2 N to the south c c c c

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Now let's look at the right side of our equation. Mass is a scalar...it does not have a direction. But acceleration does have a direction...it is a vector. The direction of the acceleration vector is always the same as the direction of the net force, ΣF, vector. Newton’s Second Law of Motion ma

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For example: A 5.0 kg object is being acted on by a 20N force to the right (F 1 ), and a 30N force, also to the right (F 2 ). We found the net force on the object to be 50N to the right. Now let's find its acceleration. Newton’s Second Law of Motion F1F1 F2F2 ΣFΣF ΣF = ma a = ΣF / m a = 50N / 5.0 kg a = 10 m/s 2 to the right s 2 to the right Answer

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Force is a vector, so ΣF = ma is true along each coordinate axis. That means we can add up all the forces in the vertical direction and those will equal "ma" in the vertical direction. And then can do the same thing in the horizontal direction. Newton’s Second Law of Motion F1F1 F2F2 F3F3 a = 1 m/s 2 F 1 +(-F 2 ) = m*a F 1 - F 2 = 0 F 3 = m*a F 3 = (2kg)(1 m/s 2 ) F 3 = 2 N F1F1 F2F2 F3F3 a = 1 m/s 2

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15A force F 1 = 50N acts to the right on a 5.0 kg object. Another force, F 2 = 30N, acts to the left. Find the acceleration of the object:

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16A force F 1 = 350N pushes upward on 20.0 kg object. Another force, F 2 = 450N pulls downward. Find the acceleration of the object:

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17 An object accelerates downward at a rate of 4.9 m/s 2. If the downward force on the object is 500N and the upward force is 250N, what is the mass of the object?

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40 A 90 kg climber rappels from the top of a cliff with an acceleration of 1 m/s 2. Determine the Tension in the climber's rope.

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41 A crane lifts a 400 kg crate upward with an acceleration of 3 m/s 2. Determine the Tension in the crane.

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Return to Table of Contents Newton's 3rd Law of Motion

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Any time a force is exerted on an object, that force is caused by another object. You need two objects to have a force. Newton’s Third Law of Motion Newton’s third law: Whenever one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first object. Force exerted on cat by table Force exerted on table by cat

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Newton’s Third Law of Motion Whenever one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first object. For every action, there is an equal, opposite reaction. This is another way to state Newton's 3rd Law. It is important to remember that the forces (or actions) are always applied to two different objects.

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Newton’s Third Law of Motion A key to the correct application of the third law is that the forces are exerted on different objects. Make sure you don’t use them as if they were acting on the same object. Then they would add to zero! Force on hands Force on floor

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Rocket propulsion can also be explained using Newton’s third law. Hot gases from combustion spew out of the tail of the rocket at high speeds. The reaction force is what propels the rocket. Newton’s Third Law of Motion Note that the rocket does not need anything to “push” against.

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Helpful notation: the first subscript is the object that the force is being exerted on; the second is the source. Subscripts help keep your ideas and equations clear. Newton’s Third Law of Motion Horizontal force exerted on the ground by person's foot F GP Horizontal force exerted on the person's foot by ground F PG F GP = -F PG

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27 When you sit on a chair, the net external force on you is Azero. Bdependant on your weight. Cup. Ddown. c c c c

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28 An object of mass m sits on a flat table. The Earth pulls on this object with force mg, which we will call the action force. What is the reaction force? AThe table pushing up on the object with force mg B The object pushing down on the table with force mg C The table pushing down on the floor with force mg D The object pulling upward on the Earth with force mg c c c c

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29 A 20-ton truck collides with a 1500-lb car and causes a lot of damage to the car. Since a lot of damage is done on the car Athe force on the truck is greater then the force on the car B the force on the truck is equal to the force on the car C the force on the truck is smaller than the force on the car D the truck did not slow down during the collision c c c c

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30As you are sitting in a chair, you feel the chair pushing up on you. The reaction force in this situation is: AThe chair pushing down on the ground BGravity pulling down on you CYou pushing down on the chair DThe ground pushing up on the chair c c c c

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31 A student is doing push-ups in gym class. A reaction pair of forces is best described as: A The student pushes down on the ground - The ground pushes up on the student B Gravity is pulling the student down - The ground is pushing the student up C Gravity is pulling the student down - The student's arms push the student up D The student's hands push down on the ground - The students arms push the student up c c c c

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32Which of Newton's laws best explains why motorists should wear seat belts? Athe first law Bthe second law Cthe third law Dthe law of gravitation c c c c

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33If you blow up a balloon, and then release it, the balloon will fly away. This is an illustration of: (Note: there may be more than one answer. Be prepared to explain WHY!) A Newton's first law B Newton's second law C Newton's third law D Galileo's law of inertia c c c c

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