Forces Equations Forces โ ๐๐๐ = ๐๐ โ ๐ญ โ ๐๐๐ = ๐ญ โ ๐+๐ญ โ ๐+๐ญ โ ๐+โฏ ๐ญ ๐ โ ๐ = ๐๐ โโ where ๐ โโ = ๐. ๐ ๐ [๐ ๐๐๐] ๐ญ ๐ โ๐ญ๐ฎ = ๐ฎ๐๐ ๐๐ ๐๐ ๐ญ๐ญ = ๐๐ญ๐ต โ๐ญ๐ = ๐โ๐ โ Trigonometry Right-Angled Triangles โ๐ฆ๐ ๐๐๐ ๐ Any Triangle ๐ ๐ต ๐ด ๐๐๐ ๐๐ฆ๐กโ๐๐๐๐๐๐๐ ๐โ๐๐๐๐๐ ๐ ๐ถ ๐ โ๐ฆ๐2 = ๐๐๐ 2 + ๐๐๐2 ๐๐๐๐ ๐ฟ๐๐ค sin ๐ด sin ๐ต sin ๐ถ = = ๐ ๐ ๐ ๐๐๐๐ ๐ผ๐๐๐๐ก๐๐ก๐๐๐ ๐๐๐ sin ๐ = โ๐ฆ๐ ๐ถ๐๐ ๐๐๐ ๐ฟ๐๐ค ๐ 2 = ๐2 + ๐ 2 − 2๐๐ cos ๐ถ ๐๐๐ โ๐ฆ๐ ๐๐๐ tan ๐ = ๐๐๐ cos ๐ = © Michelle Brosseau, Mrs. Brosseau’s Binder Forces Vocabulary Date: Name: Update this list throughout the unit with new Physics vocabulary. Term Definition © Michelle Brosseau, Mrs. Brosseau’s Binder Forces Preconceptions Date: Name: Answer these questions at the beginning of the Forces unit using the knowledge you already have. You will revisit these same questions at the end of the unit. What is a “force”? List all the forces you know. Is friction good or bad? What is “inertia”? Does an object moving with a constant velocity require a force to keep it moving? How can one lose weight without exercising? What are you excited to learn about in this unit? © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s First Law: The Law of Inertia Date: Name: Motion: Aristotle to Galileo Aristotle, a Greek philosopher and scientist (384-322 BC), uses common sense to explain physics. He noticed that some objects in motion maintained their motion without assistance. He called these “_________________________________”. These included rocks falling off ledges, liquids running downhill, air rising and flames going upward. His view was that any solid falls toward the Earth because it seeks to get closer to its natural resting place. Any motion that was not natural was “_________________________________,” that is, the object needed to be pushed or pulled. Aristotelian physics had a hard time explaining many things, though (sadly) most people still share his views on motion. Consider this: what happens when you drop two pieces of paper, one is flat and one is crumpled into a ball? What happens when you drop two objects of roughly the same shape but very different masses? Aristotelian physics could not explain either of these effects. His theory stated that the heavier object would seek the Earth’s center more strongly, thus falling faster. We can test the hypothesis that any two objects dropped from the same height will hit the ground at precisely the same time using a vacuum (a container that contains no air). This has been tested, and the objects fall precisely at the same rate. This leads to an important discovery by Galileo Galilei (1564-1642): Galileo’s Law of Falling © Michelle Brosseau, Mrs. Brosseau’s Binder This means that any two objects – a duck, a velociraptor, a notebook, a piece of chalk, even an individual atom – will fall together. Draw a picture here to help you remember Galileo’s Law of Falling. Astronauts confirmed this on the moon as well. A hammer and a feather were dropped together from the same height on the moon, and fell at the same rate, hitting the moon’s surface at the same time. Cool fact: The falling of an individual atom was tested in 1999, and in order to do so the researchers had to eliminate any thermal motion the atom had. This meant cooling the atoms within two-millionths of a degree of absolute zero. The atoms fell just like stones. Isn’t that cool? Galileo made many other contributions to the study of forces, including recognition of the effects of friction and being one of the first scientists to use the scientific method. His contributions helped future physicists develop the 3 laws of Newtonian Physics (to be improved upon by Quantum Physics in the 1900s). How Things Move You’ve heard “inertia is a property of matter” from the opening credits to Bill Nye the Science Guy. But what exactly is inertia? Inertia This leads us to Newton’s* First Law: The Law of Inertia. © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s First Law of Motion, The Law of Inertia *Though this law is attributed to Sir Isaac Newton, it was actually Rene Descartes who invented this law. Consider an object in space (far from the gravitational forces of other objects in space). If the object was stationary to begin with, then it will _____________________________________ until an ________________________ ____________ acts upon it. Similarly, if the object was moving with a constant velocity then it will ___________________________________________ until an ____________________________________________ acts upon it. Draw these situations to help remember Newton’s First Law, The Law of Inertia: Stationary object Object with constant velocity Example: Use Newton’s First Law to explain why it is important to wear your seatbelt in a moving vehicle. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Some people believe that more massive objects fall faster than less massive objects. Describe a demonstration you could perform in your everyday life to clarify this concept for others. Q2. Define “inertia” in your own words, then rank the following objects from lowest to highest inertia: plane, car, transport truck, motorcycle, toy car Q3. Suppose you are being chased by a massive bull. Use the concept of inertia to explain why you could escape by running in a zig-zag pattern. Q4. Use Newton’s First Law to explain how headrests help prevent whiplash in motor vehicle accidents. Well-labelled diagrams will aid your explanation. © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s Second Law: The Law of Motion Date: Name: The main idea to Newton’s theories is that ______________________________ _______________________________ (as opposed to velocities), and ___________________ is needed to keep an object _______________________ ________________________________. In fact, the acceleration, ๐, is proportional to total force, ๐น๐๐๐ก . This was counter-intuitive to most, but this can easily be demonstrated. Imagine a mass on a table connected by a string to a mass hanging off the edge of the table. What happens? What happens when the mass hanging off the edge of the table is doubled? _____________________________________ We write, ๐๐ผ๐น read, “acceleration is proportional to net force.” This means if one is increased; the other will increase by the same factor. Also, 1 ๐๐ผ ๐ read, “acceleration is inversely proportional to mass.” This means that if one is increased; the other will decrease by the same factor. ๐๐ผ ๐น ๐ If we put these two proportionalities together we find an object’s acceleration is proportional to the force exerted upon it divided by its mass. If the mass is given in kg and the acceleration in m/s2, the proportionality can be written as an equality: The unit of force is appropriately called a newton, abbreviated N, where N = kg · m/s2. ๐น๐๐๐ก , the net force on an object, is the vector sum of all forces acting upon the object. (Yes, we still use vectors!) © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s Second Law, The Law of Motion An object’s acceleration is determined by the net force exerted on it by its environment and by the object’s mass. The direction of acceleration is the same as the direction of the net force. Quantitatively, the acceleration is proportional to the net force divided by the mass: Draw diagrams to help you remember this law. Example 1: A 5.7 kg wooden crate experiences a net force of 6.1 N [West]. Determine the crate’s acceleration. This example is simple. To tackle more complex problems, we will need to use a diagram to stay organized. © Michelle Brosseau, Mrs. Brosseau’s Binder Free body diagrams One of our initial steps to solving any problem is to draw a picture. A free-body diagram, abbreviated FBD, is a picture used to solve problems involving forces. Example 2: A raft is free to move on the surface of the water. There is an upward buoyant force equal in magnitude to the force of gravity, 1225 N [down]. There is a force of 150 N [E] from the strong winds and 215 N [E] from the current. Draw the FBD for the raft, then find ๐น๐๐๐ก and ๐. We begin by drawing a rectangle (or square, or circle) and the object inside the shape. We also include the mass (in kg) of the object within the shape. We then draw all forces acting upon the object using vectors pointing in the appropriate directions (for most questions this will be up, down, left and right). We generally choose to use the standard reference system, where up/North and right/East are positive while left/West and down/South are negative. Notes: Only draw the forces that are ______________________ the object. We do not include any forces the object is exerting on other objects (like the raft pushing the water). Multiple forces in the same direction are drawn next to each other. If a force is acting __________________ on an object we draw this force _______________ the shape. Force vectors always point __________________ the object. The length of force vectors are relative to their strength, that is _________________ forces have _______________ vectors. © Michelle Brosseau, Mrs. Brosseau’s Binder To find ๐น๐๐๐ก we use the equation: ๐น๐๐๐ก = ๐น1 + ๐น2 + ๐น3 + ๐น4 + โฏ That is, the net force is the (vector) sum of all the forces acting on the object. Consider the horizontal and vertical components separately first, then add them using vector addition if needed. © Michelle Brosseau, Mrs. Brosseau’s Binder More complicated examples require vector addition to find the net force and acceleration. Example 3: A 19-kg sled is being pulled on ice by two siblings, each holding a rope. The younger sibling pulls with a force of 17 N [E15°N] and the elder sibling pulls with a force of 21 N [E12°S]. Find the net force and acceleration of the sled, assuming no friction on the ice. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Summarize Newton’s second law in your own words. Q2. What is a “newton”? What does it measure? Show 1 N as its base SI units. Q3. A trio of students push a 65 kg crate. The first student pushes 31 N [E], the second student pushes 28 N [S] and the third student pushes 39 N [W]. Draw the FBD for the crate. © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. For each FBD, find the net force and acceleration. a) b) c) © Michelle Brosseau, Mrs. Brosseau’s Binder Q5. Two people push a 2250 kg car, each with a force of 275 N [forward]. There is a frictional force of 310 N [backwards]. a) Draw the FBD for the car. b) Find the net force acting on the car. c) Find the acceleration of the car. d) Find how far the car has travelled after 10.0 seconds of pushing if the car starts from rest. © Michelle Brosseau, Mrs. Brosseau’s Binder Q6. Two mechanics use pulleys to lift a 105 kg engine out of a car. The first mechanic pulls with a force of 675 N [U21°R] and the second mechanic pulls with a force of 712 N [U33°L]. The force of gravity acting on the engine is 1029 N [D]. a) Draw the FBD for the engine. b) Determine the net force acting on the engine. c) Find the acceleration of the engine. d) Determine how long it will take to lift the engine 1.25 m from rest. © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s Third Law: The Law of Force Pairs Date: Name: Forces always come in pairs. Always. How do we know? -slap the tabletop/countertop with your hand -grab your table, now pull hard on it -now push hard on the table When you slap the table, it slaps back – that’s why your hand is stinging. You exert a force on the table. The table exerts a force back on your hand because it decelerates your hand (in that it stops your hand) When you pull on the table, the table pulls you toward it as well. When you push on the table, the table also pushes you away. Forces always come in pairs. Always. Every force is an interaction between two objects – rather than one object acting on another. Think of slapping the table as an interaction between your hand and the table – then you should see that each exerts a force on the other. If you tap the table lightly, then slap the table hard, can you feel the difference in the reaction force? This tells us something quantitative about the force pairs – if one increases the other does too. In fact, through experimentations we find that the two forces in any force pair have the same strength. That is – the forces have the same magnitude (but clearly opposite directions). Newton saw this too, and summarized it in the Law of Force Pairs: Newton’s Third Law: The Law of Force Pairs © Michelle Brosseau, Mrs. Brosseau’s Binder We can see the interactions between two objects by drawing FBDs for both objects. Example: Draw the FBDs for each object in all of the following situations. A 100.0 kg hockey player checks a You lean against a wall with a 85.0 kg hockey player with a force force of 25 N. of 65 N. You push against the ground as you walk. The Earth pulls on the Moon. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. An apple is pulled towards the ground by the Earth. We can see the apple accelerate, but we can’t see the Earth accelerate. Does this break Newton’s third law? Explain. Q2. Explain these situations using Newton’s third law. a) A person can move forward by pressing backwards against the ground. b) Two skaters, close together, are stationary on the ice. When one pushes the other they both move away from one another. c) In space, astronauts can propel themselves in a direction by throwing an object in the opposite direction. © Michelle Brosseau, Mrs. Brosseau’s Binder Q3. A 3500 kg van hits a 2500 kg car with a force of 1480 N [E]. a) What force does the van experience? b) Calculate the acceleration of both vehicles after the collision. Q4. If the 3500 kg van and the 2500 kg car each hit each other with a force of magnitude 1480 N in a head-on collision. a) Does the net force acting on each vehicle change? b) Calculate the acceleration of both vehicles after the collision. © Michelle Brosseau, Mrs. Brosseau’s Binder Newton’s Laws of Motion Summary Date: Name: Can you fill in the ______________________? ______________________’s Law of Falling If air ___________________________ is negligible, then any two _______________________ that are ______________________ together will fall together, regardless of their ___________________________ and their ___________________________, and regardless of the substances of which they are made. _____________________’s First Law: The Law of ________________________ A body that is subject to no ________________________ forces will stay at ___________________________ if it was at rest to begin with and keep ___________________________ if it was moving to begin with (in which case its motion will be in a straight ___________________________ at an unchanging ___________________________). Or, more simply: An object that is subject to no external ___________________________ must ___________________________ an unchanging __________________________. ____________________’s Second Law: The Law of ______________________ An object’s ___________________________ is determined by the ___________________________ (2 words) exerted on it by its environment and by the object’s ___________________________. The direction of acceleration is the ___________________________ as the direction of the net force. Quantitatively, the acceleration is proportional to the net force divided by the mass: ___________________________ ____________________’s Third Law: The Law of ___________________________ (2 words) Every ___________________________ is an interaction between two objects. Thus forces must come in ___________________________ – if one object exerts a force on a second object then the second exerts a force on the first. In fact, the two forces have the same ___________________________ (magnitude) but opposite ___________________________. © Michelle Brosseau, Mrs. Brosseau’s Binder Gravity Date: Name: There are four fundamental forces that govern the universe. ๏ท ๏ท ๏ท ๏ท Have you ever wondered why the nucleus of an atom stays together, with all those positively charged protons that repel each other? The ________________________________________ is responsible for holding the nuclei of atoms together. It is by far the strongest force but only acts over short ranges of order 10-15 meters. The ________________________________________ is the second strongest force. It causes electric and magnetic effects such as the repulsion between like electrically charged particles and the interaction of magnets. It is long-ranged, can be attractive or repulsive, and acts only between pieces of matter carrying electrical charge. The ________________________________________ is responsible for radioactive decay and neutrino interactions. It has a very short range and, as its name indicates, it is very weak. The ________________________________________ is by far the weakest force, but very long ranged. Furthermore, it is always attractive, and acts between any two pieces of matter in the Universe. These four fundamental forces determine the structure of the universe. If the strength of any of these forces changed the universe would change dramatically. All four forces are very interesting, but since we’re focusing on Newtonian Mechanics (as opposed to Quantum Mechanics) let’s focus on the gravitational force. Recall the story of Newton and the apple. What caused the apple to fall on his head? What causes the moon to revolve around the Earth, and the Earth around the Sun? The answer is the force of ___________________. Gravity is an ___________________________ force, meaning that the force always pulls objects closer rather than pushing them away. Also, gravity is an ________________________________________ force – that is, the apple didn’t need to be touching the Earth to experience its pull. Did you know that everything that has mass has a gravitational field? That means that everything with mass – the Earth, the Sun, an apple, you and even an electron is attracted to everything else with mass. Have a look at the person next to you. Did you know that you are attracted to that person? Gravitationally at least! © Michelle Brosseau, Mrs. Brosseau’s Binder In fact, this is how our Sun and planets and satellites were created – bits of cosmic dust attract each other and combine to create larger cosmic dust bunnies, which attract even more cosmic dust eventually creating stars, planets and moons. Thought Experiment: Imagine that you and your beloved are the only two things left in the universe, billions of kilometers apart. Would the two of you ever meet? What if you both had a velocity that was sending you away from each other? On Earth we know that the acceleration due to gravity is about ๐ = ๐ 9.8 2 [๐๐๐ค๐]. Using Newton’s second law, we see that the force of ๐ gravity on an object must be: We use this relationship when we are dealing with objects on or near the surface of the Earth. The force of gravity acting on an object is more colloquially called the object’s weight. Physicists distinguish mass (which is the _____________________________________ an object has, and is measured in ______) with weight (which is a _____________ and is measured in ______). If you were to travel to a different planet, one with a different acceleration due to gravity, your ___________ would remain the same but your ___________________ would change depending on the characteristics of that planet. © Michelle Brosseau, Mrs. Brosseau’s Binder Example: A 150-gram apple falls experiences gravity and falls towards the Earth. Determine the force of gravity acting on the apple. When an object is in free fall, it does not necessarily accelerate towards the ground for its entire trip. If air resistance is large enough, the _________________ force of gravity will _________________ with the _________________ force of air resistance. When this happens the object no longer accelerates towards the ground, instead it moves at a constant velocity called the _________________________________. The terminal velocity of an object depends on many factors, including the object’s mass and surface area, and the density of the air. Example: Draw the FBD, then find the acceleration of a 65-kg skydiver who a) jumps out of a plane and experiences 115 N of air resistance. b) achieves terminal velocity. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. What does it mean to be an “at-a-distance” force? Q2. Explain the difference between mass and weight. Q3. “Force of gravity” and “acceleration due to gravity” are often confused. Use the example of a small rock and a more massive cannonball falling from the same height to compare the objects’ force of gravity and their acceleration due to gravity. Include a diagram. © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. Find the force of gravity acting on the following objects: a) A 1.2 kg rabbit b) A 55 kg wolf c) An 800 kg giraffe d) A 1500 kg hippopotamus Q5. In 2012, Felix Baumgartner, an Austrian skydiver, ascend into the stratosphere using a helium balloon. He jumped from a height of 39 km above Earth’s surface. a) Do you think that 9.8 m/s2 can be used to determine his acceleration as he jumped out of the capsule? Why or why not? b) Draw the FBD for Baumgartner before he pulls his parachute. c) Baumgartner’s maximum speed was 1357.64 km/h. Draw the FBD for Baumgartner after he pulls open his parachute. © Michelle Brosseau, Mrs. Brosseau’s Binder Universal Law of Gravity Date: Name: The value ๐ = 9.8 ๐⁄๐ 2 [๐๐๐ค๐] is special. It is a measured value that applies to objects on or near Earth’s surface. This means, that the value for ๐ could not be used for objects located significantly above or below the average radius of Earth, or for objects located on other astronomical objects like planets, dwarf planets and asteroids. How do we find the force of gravitational attraction between the Earth and the Moon, between the Earth and the Sun, or between you and your Physics textbook? Newton to the rescue once again! Newton proved using Calculus that objects of finite size (objects that can be measured, even if they are very large) can be considered as particles. That is, even the largest of objects can be considered as a point, with all its mass at one point (usually the center of the object). Using this fact - and the facts that the force of gravitational attraction is proportional to the masses of both objects in question and inversely proportional to the distance between them squared – we can develop a proportionality statement: But this is not an equality, for the two sides to be equal we need to factor in the universal gravitational constant, G = 6.67 x 10-11 Nโm2/kg2. (Notice the universal gravitational constant is capital G, whereas the Earth’s gravitational constant is lowercase g – think the universe is bigger so it has the bigger G). Now we have a formula for the force of gravitational attraction between any two objects: Where G = 6.67 x 10-11 Nโm2/kg2, m1 is the mass of the first object, m2 is the mass of the second object and r is the distance between the centers of the two objects. © Michelle Brosseau, Mrs. Brosseau’s Binder Example: Let’s see how attracted two members of our class are to each other. This equation works for any two objects in the universe. Example: Determine the force of gravity between the Earth and the Sun. mE = 5.972 x 1024 kg, mS = 1.989 x 1030 kg, r = 1.496 x 1011 m © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Determine the force of gravitational attraction between a 92 kg student and a 550 g slice of pizza that are 25 cm apart. Q2. Neptune’s largest moon, Triton, has an orbital radius of 354 800 km. Find the force of gravitational attraction between Neptune (mN = 1.024 x 1026 kg) and Triton (mT = 2.14 x 1022 kg). © Michelle Brosseau, Mrs. Brosseau’s Binder Q3. Find the weight of a 65 kg person… a) When they are on the surface of Earth (2 different ways). b) When they are 36576 m above the surface of Earth (Felix Baumgartner’s jump altitude). c) When they are at an altitude 408 km above Earth’s surface (the same altitude maintained by the International Space Station). rE = 6371 km, mE = 5.972 x 1024 kg © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. Find the of acceleration due to gravity on the surface of Triton. mT = 2.14 x 1022 kg, rT = 1353 km Q5. Find the gravitational field strength of Neptune. mN = 1.024 x 1026 kg, rN = 24622 km Q6. Explain why different values for the distances are used in Q2, Q4 and Q5. © Michelle Brosseau, Mrs. Brosseau’s Binder Physicists’ Weight Loss Plan Date: Name: Choose a person (or object, or pet) and determine its mass in kilograms. Person/Object Mass (kg) Q1. Research the acceleration due to gravity (also called gravitational field strength) in 3 different cities around the world. Calculate the person’s weight (in Newtons) for each of the cities. Acceleration due City, Country to gravity (m/s2 or Person’s Weight (N) N/kg) Q2. Research and list at least 2 reasons why the acceleration due to gravity differs in cities around the world. Q3. A person wants to lose weight, which area of the world should they move to and why? © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. Research the radius and mass of 3 astronomical objects like planets, dwarf planets or stars (you need not restrict yourself to our solar system, or even real life Pandora and Tatooine are acceptable!). Then find the person’s weight in Newtons for each of the three. Astronomical Object: Radius: Mass: Find the person’s weight on the surface of this astronomical object. Astronomical Object: Radius: Mass: Find the person’s weight on the surface of this astronomical object. Astronomical Object: Radius: Mass: Find the person’s weight on the surface of this astronomical object. © Michelle Brosseau, Mrs. Brosseau’s Binder Q5. Choose the celestial body from Q4 that gave the person the lowest weight. Using the radius and mass that you’ve researched, calculate the acceleration due to gravity (also called gravitational field strength) at the surface of that celestial body (this is the value that is equivalent to Earth’s 9.8 m/s2). Determine how many “g”s this is equivalent to by dividing this number by 9.8 m/s2. Q6. Create a poster, brochure, travel advertisement etc. to show the benefits of the Physicist’s Weight Loss program. -Include either the area/city on Earth or the celestial body (or both) that a person looking to lose the most weight should travel to. -Include at least 3 benefits/facts about what they will feel with a lower weight. -Make it very attractive (aesthetically or gravitationally) © Michelle Brosseau, Mrs. Brosseau’s Binder The Normal Force Date: Name: Draw the free body diagrams for the following situations: A 0.225 kg apple is free falling A 0.225 kg apple is sitting on a towards Earth. desk. Of course, gravity is acting in both situations, but there must be a second force acting on the apple that is sitting on a desk. How do we know? Recall Newton’s 2nd Law equation, ๐น๐๐๐ก = ๐๐. The apple that is sitting on the desk is _________________________________, so the net force must be _____________. For the net force to be zero there would have to be an _______________________________________ acting on the apple that is keeping it from accelerating. This force is called the ________________________________. Remember our saying, “all forces come in pairs”, so the force of the apple pushing down on the tabletop must be met with an equal and opposite force. The normal force is the __________________________ from Newton’s Third Law “every action force has an equal and opposite reaction force.” In this context “normal” does not mean regular or ordinary – in Math and Physics “normal” means ________________________. The normal force acts ___________________________________________ to the surface the object is on. Draw the normal force, ๐น๐ , acting on the apple in the following situations: © Michelle Brosseau, Mrs. Brosseau’s Binder We’ve discussed how the force of gravity can change depending on the masses and the distance between two objects. Can the normal force change too? The Great Compensator The normal force is the great compensator of forces. Provided the surface is strong enough to withstand the force applied to it, the normal force will ________________________ as the force applied increases. Try This 1. Place your binder/book on the desk. 2. Place your hand on the binder. 3. Apply a large downward force on the binder from your hand. Did your binder accelerate through the desk? That must mean that the forces are all _______________________________. Draw the free body diagrams for both situations. Binder with your hand resting lightly Binder with your hand pressing on the binder. down hard on the binder. โ๐ต To solve problems using ๐ญ 1. Draw a free body diagram and label all forces – expect to label ๐น๐ if there is a surface involved. 2. Create an ๐น๐๐๐ก statement and solve for any unknown variables. Note: If the surface is on an angle, you will usually need to find the xand y-components of ๐น๐ using Trigonometric Ratios (sine, cosine and tangent). © Michelle Brosseau, Mrs. Brosseau’s Binder Example 1: Normal isn’t always up! a) An elephant of mass 4700 kg steps onto a platform. What is the normal force acting on the elephant? Draw the FBD for the elephant. b) What is the normal force of a magnet being held to the chalkboard with a force of 1.5 N? Draw the FBD for the magnet. c) You are helping to install an 11 kg decorative beam to a ceiling. You apply a force of 170 N upwards to hold the beam in place while you wait for your co-worker to install the beam. What is the normal force acting on the beam? Draw the FBD for the beam. © Michelle Brosseau, Mrs. Brosseau’s Binder Example 2: Bathroom Scales Bathroom scale questions are very common when investigating the normal force. The scale’s reading (called the apparent weight) is always the same as the normal force acting on the person (force-pairs). Consider a 75 kg person standing on a scale in an elevator. Hypothesize the strength of the normal force relative to the force of gravity (greater than, or less than), then find the normal force if… a) the elevator is stationary. b) the elevator accelerates upwards at 0.25 m/s2. c) the elevator accelerates downwards at 0.6 m/s2. d) the elevator is moving upwards at a constant velocity. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Determine the normal force acting on a 250-gram apple, at rest on a table. Include an FBD. Q2. A 1.2 kg textbook rests on top of another 1.2 kg textbook. Draw the FBDs for each textbook including all values. Q3. Describe a situation in which the normal force is not directed up. © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. An 89-kg person stands on a scale (calibrated in newtons) in an elevator. Determine the reading on the scale if a) the elevator is stationary. b) the elevator accelerates upwards at 0.18 m/s2. c) the elevator accelerates downwards at 0.75 m/s2. d) the elevator is moving upwards at a constant velocity. © Michelle Brosseau, Mrs. Brosseau’s Binder The Force of Friction Date: Name: The Force of Friction Consider the apple on the slanted surface. If the apple is not accelerating, what force is most likely to prevent the apple from slipping off the surface? Friction is the Jekyll and Hyde of forces. We need friction to move, but it also hinders our movement by causing kinetic (moving) energy to turn into heat and causes wear and tear on objects. Friction is believed to be caused by microscopic welds created by intermolecular forces. Imagine the molecules of both substances “welding” together where they are in contact to form an attractive force. To overcome this force (friction) these welds must be broken. This belief, along with inertia, also explains why it is harder to get an object moving than it is to keep it move. Friction occurs when two surfaces are touching and is split into two categories: 1. ____________________________ occurs when an object is not moving Examples: 2. ____________________________ occurs when an object is moving Examples: © Michelle Brosseau, Mrs. Brosseau’s Binder Criteria for a Frictional Force 1) There exists a “coefficient of friction” – that is, we do not choose to ignore friction 2) There exists a normal force between two surfaces 3) There is an applied force trying to move that object If any of these are not met, there is no friction. Coefficient of Friction, μ What is a coefficient of friction? Imagine you have two coins, one on smooth ice and one on rough plywood. You flick each with an equal force, which is going to move further before stopping? This is the idea of the coefficient of friction – different surfaces resist motion differently. The coefficient of friction is determined by using the ratio between two forces: It is the ratio of how much frictional force there is to the normal force acting on the object. Note how the amount of friction does not depend on the push or pull applied to the object, instead it is based on the force of the surface acting on the object – the normal force! What are the units of the coefficient of friction? © Michelle Brosseau, Mrs. Brosseau’s Binder We represent the coefficient of friction by the Greek letter μ, “mu”. There are two types of coefficients of friction – the coefficient of static friction (μs) used when the object is stationary, and the coefficient of kinetic friction (μk) used when an object is moving. To find the force of friction we re-arrange the previous equation to find: FF = μsFN for a stationary (static) object FF = μkFN for a moving (kinetic) object Coefficients of Friction for Common Material Pairs Surface A Surface B Coefficient of Static Friction μs Coefficient of Kinetic Friction μk Steel Steel Steel Rubber Teflon Wood Wood Wood Human synovial fluid Steel Leather Copper Brass Steel Concrete Teflon Dry snow Wet snow Wood 0.53 0.51 0.74 1.10 0.04 0.22 0.14 0.40 0.36 0.44 0.57 1.0 0.04 0.18 0.10 0.20 Cartilage 0.01 0.003 Ice Rock 0.1 1.0 0.01 0.8 Examine the table. What range of values does μ fall in? When would you expect a small value for μ? When would you expect a large value for μ? © Michelle Brosseau, Mrs. Brosseau’s Binder For a stationary object to move the force applied must be greater than the force of static friction. Example 1: Overcoming Static Friction A pack of dogs are pulling a wooden sled and rider of combined mass of 238 kg across dry snow. How much force do the dogs need to apply to cause the sled to begin to move? Example 2: Kinetic Friction You push your 27 kg wooden desk across the wood floor with a constant velocity. Find the force of friction and applied force. © Michelle Brosseau, Mrs. Brosseau’s Binder Sometimes when we perform calculations, the force of static friction is greater than that of the applied force. Be cautious with the free body diagrams in these situations. The object will not move until the applied force is greater than the force of static friction. Example 3: Will it move? Two sisters are fighting over the contents of a 48 kg crate. One sister pulls with a force of 79 N [right], while the other pulls with a force of 51 N [left]. The coefficient of static friction between the crate and the floor is 0.32. Find the acceleration of the crate. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Explain the difference between static and kinetic friction. Which one has the higher coefficients of friction? In which situations would you use μs versus μk? Q2. Draw an FBD for a 3 kg steel block being pulled at a constant speed on ice. Show all values for the four forces. (Hint: use the coefficient of friction table in your notes.) Q3. Explain the concept of coefficient of friction using words that a 5year old would understand. © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. Determine the force needed to get a 9 kg piece of rubber moving on concrete. Q5. A 2250 kg car has rubber tires and drives on concrete. How much force must the engine apply to keep the car moving at a constant speed? © Michelle Brosseau, Mrs. Brosseau’s Binder Q6. A child pulls a 5 kg wooden sled on dry snow at a constant velocity forward. Determine the frictional force if a) The child pulls the rope at angle 35° above the horizontal. b) The child pulls the rope at angle 35° below the horizontal. © Michelle Brosseau, Mrs. Brosseau’s Binder Mandatory Winter Tires Assignment Date: Name: If you’ve driven down the highway during a harsh winter, surely you have seen the staggering number of vehicles damaged and abandoned in the ditches. The number of collisions dramatically increases when the weather turns cold: drivers don’t change their driving habits and the vehicle is more likely to slide or get stuck in the presence of ice or snow. The end result is that emergency responders and tow-trucks are overworked, roads need to be closed down, there is damage to vehicles and property and passengers and pedestrians are harmed. The local government is now looking to enforce a law requiring all drivers to put winter tires on their vehicles. In this assignment you will analyze data on all season tires vs. winter tires, create a campaign for the mandatory winter tires law and suggest an appropriate fine for drivers who violate this law. Create an informative piece that would be distributed to the local citizens explaining the reasoning behind this new law. This should be a high-quality, professionally designed piece that will inform and persuade the citizens to start using winter tires. The format is up to you; brochure, slideshow, commercial, etc. What to include: a) A detailed, written explanation into why mandatory winter tires are now becoming law. This will include a physics-based justification of why winter tires are superior to all-season tires for use in the cold months of the year. It may also include statistics that you research into the number of accidents in Ontario based on weather/road conditions and type of tires, relevant images (informative or persuasive), cost of repair for damaged vehicles, loss of work time for common car collision related injuries, etc. Convince the reader they must invest in winter tires. b) An analysis of the different stopping distances for one of the vehicle classes provided (which one might most citizens drive?), across the 8 different tire/road conditions at 50 km/h, 80 km/h and 100 km/h. That is, calculate how far each type of vehicle will travel before it stops with different coefficients of friction between the tires and the road surface. The average citizen requires a visually impactful message; graph this in an appropriate fashion, or include an infographic to communicate this data. This involves a number of calculations. If you are efficient, you will find a way to automate this calculation (Google Sheets, excel, clever calculator tricks). Average citizens don’t care about your calculations, but your teacher does – you’ll submit them separately from the informative piece. c) Information on the fine for not using winter tires. The fine for not using winter tires should be large enough to scare strongly encourage drivers to change their tires for the winter while balancing the fact that, if charged, the driver still has to be able to afford the winter tires. Use the following information to © Michelle Brosseau, Mrs. Brosseau’s Binder determine and justify how much the fine should be for drivers not using winter tires during the winter months: Snow tires vary in price from $50 - $450 per tire. Many drivers use inexpensive rims at about $50 per wheel for their winter tires and pay $20 for the tires to be changed at the shop. If the winter tires are not already on their own rims, the cost to change the tires increases to about $90 twice a year (winter tires put on and taken off). Winter tires generally last 5 years before needing to be replaced. The citizens may appreciate a breakdown of how you determined the appropriate fine. You may include the calculations in your informative piece, or separately with your calculations to part b. In addition to the fine amount, you can decide where the funds from this fine would go (for example, road improvements, police enforcement, etc.), consequences for not paying the fine (car impounded, jail time, etc.) and justify that with an explanation. Data: Calculate the stopping distances for one of the different classes of vehicles under the different tire and road conditions at 50 km/h, 80 km/h and 100 km/h. (1 vehicle x 4 road conditions x 2 types of tires x 3 initial speeds = 24 values) Vehicle Class Compact car Midsize car Large car Compact Truck or SUV Midsize Truck or SUV Large Truck or SUV School Bus Transport Truck Average Mass in kilograms 1350 1590 1985 1575 1940 2460 11000 36000 Road Condition Coefficient of friction of dry tires on dry roads, (7°C) Coefficient of friction of dry tires on dry roads, cold day (-10°C) Coefficient of friction of tires on snow Coefficient of friction of tires on ice All Season Tires Winter Tires 1.0 1.0 0.72 0.89 0.24 0.57 0.05 0.38 Due Date: © Michelle Brosseau, Mrs. Brosseau’s Binder Hooke’s Law Date: Name: When you step on a scale or weigh the fish you caught, you are typically using a device that involves a spring. Whether you _____________________ a spring (bathroom scale) or _____________________ a spring (Newton spring scale) the natural ___________________________ of the spring is to bring the spring back to its natural length (its equilibrium position). Each spring is different; its _____________________, _____________________ and _____________________ of the springs affects how much force it takes to extend or compress the spring. When a spring is being extended by means of an additional mass, the net force on the mass will be zero when the spring settles to its new ____________________________________. The amount of restoring force, ๐น๐ , of a spring is determined by the extension or compression of the spring (โ๐ฅ) and its spring constant, ๐. Mathematically, Hooke’s Law This equation works for any spring, provided the force applied to the spring is not too great to cause _____________________ – unravelling of the spring that is non-repairable. The Spring Constant, ๐ This value is unique to each spring. As discussed before, the material, thickness and tightness of a spring affects how much it can be stretched (or compressed). The units of ๐ are __________________________________. What this unit is telling us is that the spring constant is the relationship between force and stretch (or compression). The value of ๐ tells us ______________________________________________________________________. Therefore, spring with larger values for ๐ require ________________________ to cause a stretch (or compression). Artwork © Glitter Meets Glue Designs © Michelle Brosseau, Mrs. Brosseau’s Binder Try These Complete the table to determine the missing piece of information. Try it without a calculator. Spring constant, ๐ Stretch/Compression, โ๐ฅ Force applied, โโโ ๐น 200 N/m 200 N 600 N/m 300 N 25 N/m 100 N 400 N 1m 500 N 0.5 m 700 N/m 0.5 m 250 N/m 2m One of the many advantages to using Hooke’s Law and that is need not be restricted to springs. Elastic materials, such as bungee cords and elastic bands, and flexible materials such as plastics and wood also obey Hooke’s Law to some extent. Example 1: Finding the new length A fish of mass 12.9 kg is attached to a Newton spring scale that has a spring constant of 665 N/m. Determine the stretch of the spring. © Michelle Brosseau, Mrs. Brosseau’s Binder Example 2: Finding the force A composite hockey stick (k = 9481 N/m) bends 5.0 cm during a slap shot. How much force did the stick apply to the puck? Example 3: Finding the spring constant A 61.2 kg bungee jumper jumps from a bridge. She is tied to a 11.2 m long (unstretched) bungee cord and falls a total of 32.4 m once her motion settles. Calculate the spring constant of the bungee cord. © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. A gemstone of mass 1.8 kg compresses a scale’s spring by 2.6 cm. Determine the spring constant. Q2. How much would the spring in the previous question compress if a 5.2 kg mass was placed on the scale? Q3. A Newton spring scale is being calibrated such that 10 N of force results in an extension of 0.50 cm. a) Determine the spring constant. b) Determine the extension of the spring when a mass of 1.36 kg is suspended from the spring scale. c) Determine the mass of a suspended object that stretches the spring by 0.97 cm. © Michelle Brosseau, Mrs. Brosseau’s Binder Summary Define and summarize the characteristics and equations involved with the following four forces. Include relevant diagrams. The Force of Gravity The Normal Force The Force of Friction The Spring Force © Michelle Brosseau, Mrs. Brosseau’s Binder Systems of Objects Date: Name: An object is in equilibrium when all forces acting upon it are _______________________. This results in no acceleration for the object (remember that an object can have all forces balanced and still maintain its velocity). When two or more objects are involved in a force problem we call this a ____________________. If an object is not in equilibrium the forces are _________________________. This means that the object will accelerate in the direction of the net force (Newton’s second law). Systems involving more than one object give you an opportunity to flex your problem-solving muscles. Several strategies arise in the process of solving these problems. We will to employ Newton’s second and third laws frequently. Most problems can be solved by using the following steps: 1. Read the problem and record known values and unknown variables. 2. Draw a system diagram that includes all objects. 3. Define the positive and negative directions. It is best practice for the acceleration to be in the positive direction. 4. Draw an FBD for each object. Use Newton’s third law to identify any action-reaction force pairs that apply to the FBDs of two objects. 5. Calculate the values of forces, use horizontal and vertical components if needed. 6. Use Newton’s second law to solve for missing values. Sometimes, trigonometry can speed up the solving process. 7. Check that your answers are reasonable. General Tips: © Michelle Brosseau, Mrs. Brosseau’s Binder Example 1: Unbalanced system of two objects The diagram shows two masses connected via a string across two pulleys. The masses are initially held in place, then allowed to move freely. If m1 = 4.7 kg and m2 = 1.8 kg: a) Which direction will the masses accelerate? b) Draw the FBD for each mass. c) Determine the magnitude of acceleration of the masses. d) Determine the force of tension in the string. © Michelle Brosseau, Mrs. Brosseau’s Binder Example 2: Three objects in equilibrium As shown in the diagram, three masses are suspended by strings. The system is in equilibrium, no masses are accelerating. If m2 = 5.0 kg, and the angles are θ = 45º and β = 30º, determine the masses m1 and m3. © Michelle Brosseau, Mrs. Brosseau’s Binder Example 3: Perpendicularly accelerating objects Examine the diagram of two masses connected by a strong, taut string over a frictionless pulley where m1 = 3.2 kg, m2 = 4.1 kg, μs = 0.35 and μk = 0.28. a) Show that the two masses will accelerate. b) Determine the acceleration of the two objects. c) Determine the force of tension in the string. © Michelle Brosseau, Mrs. Brosseau’s Binder The previous problem involved perpendicular forces. How do the problem-solving strategies change when the forces are not perpendicular? Examine the system involving two masses connected by a string. One mass is on an inclined plane, the other mass is suspended over a frictionless pulley. Draw the free body diagrams for both masses. If these two masses are allowed to move, we can assume that m 1 would accelerate up the inclined plane as m2 accelerates downward. Of course, their motion will depend on the values of ๐1 , ๐2 , ๐, ๐๐ , and ๐๐ , but let’s examine how we can simplify this problem. © Michelle Brosseau, Mrs. Brosseau’s Binder Example 4: Objects accelerating on an inclined plane Examine the diagram of two masses connected by a strong, taut string over a frictionless pulley where m1 = 7.4 kg is on an inclined plane with θ = 24°, m2 = 5.9 kg, μs = 0.35 and μk = 0.28. a) Show that the two masses will accelerate. b) Determine the acceleration of the two objects. c) Determine the force of tension in the string. © Michelle Brosseau, Mrs. Brosseau’s Binder Let’s look at this system of accelerating object in general, that is, without any values associated with each of the variables. We will derive a general equation for the acceleration of the masses in this system that only involves ๐1 , ๐2 , ๐, ๐, and ๐๐ . © Michelle Brosseau, Mrs. Brosseau’s Binder Practice Q1. Three masses are suspended by strings. If m2 = 12.7 kg, and the angles are θ = 42º and β = 28º, determine the masses m1 and m3 that cause this system to be in equilibrium. © Michelle Brosseau, Mrs. Brosseau’s Binder Q2. Three masses are connected by ropes over frictionless pulleys. The masses are known, m1 = 6.0 kg, m2 = 2.0 kg, and m3 = 1.0 kg. What is the minimum coefficient of static friction between m2 and the surface to causes all masses to be stationary? © Michelle Brosseau, Mrs. Brosseau’s Binder Q3. Examine the diagram of two masses connected by a strong, taut string over a frictionless pulley where m1 = 13.5 kg is on an inclined plane with θ = 27°, m2 = 12.2 kg, μs = 0.41 and μk = 0.31. a) Show that the two masses will accelerate. b) Determine the acceleration of the two objects. c) Determine the force of tension in the string. © Michelle Brosseau, Mrs. Brosseau’s Binder Q4. Derive a general equation for the acceleration of m2 given that m2 accelerates upwards. © Michelle Brosseau, Mrs. Brosseau’s Binder Q5. Determine the minimum coefficient of static friction such that the masses remain stationary (assuming that m2 would accelerate downward if not for the force of friction), written in terms of m1, m2, g and θ. © Michelle Brosseau, Mrs. Brosseau’s Binder Q6. Two masses connected via a strong rope across two pulleys. The masses are initially held in place, then allowed to move freely. If m1 = 23.7 kg and m2 = 25 kg: a) Which direction will the masses accelerate? b) Draw the FBD for each mass. c) Determine the magnitude of acceleration of the masses. d) Determine the force of tension in the string. © Michelle Brosseau, Mrs. Brosseau’s Binder Q7. Three masses are connected by ropes over frictionless pulleys. The masses are known, m1 = 10.2 kg, m2 = 3.1 kg, and m3 = 4.8 kg. Determine the acceleration of m1 if μs = 0.38 and μk = 0.26. © Michelle Brosseau, Mrs. Brosseau’s Binder Forces Preconceptions Revisited Date: Name: After completing the Forces unit, revisit the questions from the beginning of the unit. What have you learned? Have your responses changed? What is a “force”? List all the forces you know. Is friction good or bad? What is “inertia”? Does an object moving with a constant velocity require a force to keep it moving? How can one lose weight without exercising? What was the most useful piece of information or skill you learned in this unit? © Michelle Brosseau, Mrs. Brosseau’s Binder