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SPH4U: Lecture 1 Dynamics How and why do objects move

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Describing motion – so far… (from last Year) Linear motion with const acceleration: Linear motion with const acceleration:

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What about higher order rates of change? If linear motion and circular motion are uniquely determined by acceleration, do we ever need higher derivatives? If linear motion and circular motion are uniquely determined by acceleration, do we ever need higher derivatives? Certainly acceleration changes, so does that mean we need to find some “action” that controls the third or higher time derivatives of position? Certainly acceleration changes, so does that mean we need to find some “action” that controls the third or higher time derivatives of position? NO. NO. Known as the “Jerk”

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Dynamics Isaac Newton ( ) published Principia Mathematica in In this work, he proposed three “laws” of motion: Isaac Newton ( ) published Principia Mathematica in In this work, he proposed three “laws” of motion: principia principia Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. Law 2: For any object, F NET = F = ma (not mv!) Law 3: Forces occur in pairs: F A,B = - F B,A (For every action there is an equal and opposite reaction.) (For every action there is an equal and opposite reaction.) These are the postulates of mechanics They are experimentally, not mathematically, justified. They work, and DEFINE what we mean by “forces”.

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Newton’s First Law An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. If no forces act, there is no acceleration. If no forces act, there is no acceleration. The following statements can be thought of as the definition of inertial reference frames. The following statements can be thought of as the definition of inertial reference frames. An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”. An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”. If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector! If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector! If you can eliminate all forces, then an IRF is a reference frame in which a mass moves with a constant velocity. (alternative definition of IRF) If you can eliminate all forces, then an IRF is a reference frame in which a mass moves with a constant velocity. (alternative definition of IRF)

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Is Waterloo a good IRF? Is Waterloo accelerating? Is Waterloo accelerating? YES! YES! Waterloo is on the Earth. Waterloo is on the Earth. The Earth is rotating. The Earth is rotating. What is the centripetal acceleration of Waterloo? What is the centripetal acceleration of Waterloo? T = 1 day = 8.64 x 10 4 sec, T = 1 day = 8.64 x 10 4 sec, R ~ R E = 6.4 x 10 6 meters. R ~ R E = 6.4 x 10 6 meters. Plug this in: a U =.034 m/s 2 ( ~ 1/300 g) Plug this in: a U =.034 m/s 2 ( ~ 1/300 g) Close enough to 0 that we will ignore it. Close enough to 0 that we will ignore it. Therefore Waterloo is a pretty good IRF. Therefore Waterloo is a pretty good IRF.

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Newton’s Second Law For any object, F NET = F = ma. For any object, F NET = F = ma. The acceleration a of an object is proportional to the net force F NET acting on it. The acceleration a of an object is proportional to the net force F NET acting on it. The constant of proportionality is called “mass”, denoted m. The constant of proportionality is called “mass”, denoted m. This is the definition of mass and force. This is the definition of mass and force. The mass of an object is a constant property of that object, and is independent of external influences. The mass of an object is a constant property of that object, and is independent of external influences. The force is the external influence The force is the external influence The acceleration is a combination of these two things The acceleration is a combination of these two things Force has units of [M]x[L / T 2 ] = kg m/s 2 = N (Newton) Force has units of [M]x[L / T 2 ] = kg m/s 2 = N (Newton)

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Newton’s Second Law... What is a force? What is a force? A Force is a push or a pull. A Force is a push or a pull. A Force has magnitude & direction (vector). A Force has magnitude & direction (vector). Adding forces is just adding force vectors. Adding forces is just adding force vectors. FF1FF1 FF2FF2 a FF1FF1 FF2FF2 a F F NET Fa F NET = ma

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Newton’s Second Law... Components of F = ma : Components of F = ma : F X = ma X F X = ma X F Y = ma Y F Y = ma Y F Z = ma Z F Z = ma Z Suppose we know m and F X, we can solve for a X and apply the things we learned about kinematics over the last few lectures: (if the force is constant) Suppose we know m and F X, we can solve for a X and apply the things we learned about kinematics over the last few lectures: (if the force is constant)

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Example: Pushing a Box on Ice. A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50 N in the x direction. If the box starts at rest, what is its speed v after being pushed a distance d = 10 m? A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50 N in the x direction. If the box starts at rest, what is its speed v after being pushed a distance d = 10 m? F v = 0 m a x

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Example: Pushing a Box on Ice. A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50 N in the x direction. If the box starts at rest, what is its speed v after being pushed a distance d = 10m ? A skater is pushing a heavy box (mass m = 100 kg) across a sheet of ice (horizontal & frictionless). He applies a force of 50 N in the x direction. If the box starts at rest, what is its speed v after being pushed a distance d = 10m ? d m m F v a x

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Example: Pushing a Box on Ice... Start with F = ma. Start with F = ma. a = F / m. a = F / m. Recall that v 2 - v 0 2 = 2a(x - x 0 )(Last Yeat) Recall that v 2 - v 0 2 = 2a(x - x 0 )(Last Yeat) So v 2 = 2Fd / m So v 2 = 2Fd / m d F v m a i

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Example: Pushing a Box on Ice... Plug in F = 50 N, d = 10 m, m = 100 kg: Plug in F = 50 N, d = 10 m, m = 100 kg: Find v = 3.2 m/s. Find v = 3.2 m/s. d F v m a i

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Question Force and acceleration A force F acting on a mass m 1 results in an acceleration a 1. The same force acting on a different mass m 2 results in an acceleration a 2 = 2a 1. A force F acting on a mass m 1 results in an acceleration a 1. The same force acting on a different mass m 2 results in an acceleration a 2 = 2a 1. l If m 1 and m 2 are glued together and the same force F acts on this combination, what is the resulting acceleration? (a) (b) (c) (a) 2/3 a 1 (b) 3/2 a 1 (c) 3/4 a 1 Fa1a1 m1m1 Fa 2 = 2a 1 m2m2 Fa = ? m1m1 m2m2

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Solution Force and acceleration Since a 2 = 2a 1 for the same applied force, m 2 = (1/2)m 1 ! Since a 2 = 2a 1 for the same applied force, m 2 = (1/2)m 1 ! m 1 + m 2 = 3m 1 /2 m 1 + m 2 = 3m 1 /2 (a) (b) (c) (a) 2/3 a 1 (b) 3/2 a 1 (c) 3/4 a 1 Fa = F / (m 1 + m 2 ) m1m1 m2m2 l So a = (2/3)F / m 1 but F/m 1 = a 1 a = 2/3 a 1

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Forces We will consider two kinds of forces: We will consider two kinds of forces: Contact force: Contact force: This is the most familiar kind. This is the most familiar kind. I push on the desk. I push on the desk. The ground pushes on the chair... The ground pushes on the chair... A spring pulls or pushes on a mass A spring pulls or pushes on a mass A rocket engine provides some number of Newtons of thrust (1 lb of thrust = mg = 2.205*9.81 = Newtons) A rocket engine provides some number of Newtons of thrust (1 lb of thrust = mg = 2.205*9.81 = Newtons) Action at a distance: Action at a distance: Gravity Gravity Electricity Electricity Magnetism Magnetism

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Contact forces: Objects in contact exert forces. Objects in contact exert forces. Convention: F a,b means acting on a due to b”. Convention: F a,b means acting on a due to b”. So F head,thumb means “the force on the head due to the thumb”. So F head,thumb means “the force on the head due to the thumb”. F F head,thumb The Force

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Action at a Distance Gravity: Gravity: Burp!

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Gravitation (Courtesy of Newton) Newton found that a moon / g = Newton found that a moon / g = and noticed that R E 2 / R 2 = and noticed that R E 2 / R 2 = This inspired him to propose the Universal Law of Gravitation: This inspired him to propose the Universal Law of Gravitation: RRERE a moon g where G = 6.67 x m 3 kg -1 s -2 And the force is attractive along a line between the 2 objects Hey, I’m in UCM! W e will discuss these concepts later

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Understanding If the distance between two point particles is doubled, then the gravitational force between them: A)Decreases by a factor of 4 B)Decreases by a factor of 2 C)Increases by a factor of 2 D)Increases by a factor of 4 E)Cannot be determined without knowing the masses

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Newton’s Third Law: Forces occur in pairs: F A,B = - F B,A. Forces occur in pairs: F A,B = - F B,A. For every “action” there is an equal and opposite “reaction”. For every “action” there is an equal and opposite “reaction”. We have already seen this in the case of gravity: We have already seen this in the case of gravity: R 12 m1m1 m2m2 F F 12 F F 21 W e will discuss these concepts in more detail later.

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Newton's Third Law... F A,B = - F B,A. is true for contact forces as well: F A,B = - F B,A. is true for contact forces as well: F F m,w F F w,m F F m,f F F f,m Force on me from wall is equal and opposite to the force on the wall from the me. Force on me from the floor is equal and opposite to the force on the floor from the me.

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block Example of Bad Thinking Since F m,b = -F b,m, why isn’t F net = 0 and a = 0 ? Since F m,b = -F b,m, why isn’t F net = 0 and a = 0 ? a ?? F F m,b F F b,m ice

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block Example of Good Thinking Consider only the box as the system! Consider only the box as the system! F on box = ma box = F b,m F on box = ma box = F b,m Free Body Diagram (next power point). Free Body Diagram (next power point). a box F F m,b F F b,m ice No ice Friction force

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block Add a wall that stops the motion of the block Now there are two forces acting (in the horizontal direction) on block and they cancel Now there are two forces acting (in the horizontal direction) on block and they cancel F on box = ma box = F b,m + F b,w = 0 F on box = ma box = F b,m + F b,w = 0 Free Body Diagram (next power point). Free Body Diagram (next power point). a box F F m,b F F b,m ice F F b,w F F w,b

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Newton’s 3rd Law Understanding Two blocks are stacked on the ground. How many action-reaction pairs of forces are present in this system? Two blocks are stacked on the ground. How many action-reaction pairs of forces are present in this system? (a) 2 (b) 3 (c)4 (d)5 a b

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Solution: F E,a F a,E a b F E,b a b F b,E F b,a F a,b a b F b,g F g,b a b F a,b F b,a a b contact gravity contact gravity very tiny 5

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Understanding A moon of mass m orbits a planet of mass 100m. Let the strength of the gravitational force exerted by the planet on the moon be denoted by F 1, and let the strength of the gravitational force exerted by the moon on the planet be F 2. Which of the following is true? A)F 1 =100F 2 B)F 1 =10F 2 C)F 1 =F 2 D)F 2 =10F 1 E)F 2 =100F 1 Newton’s Third Law

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Flash: Newton’s 1 St Law

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Flash: Newton’s 2 nd Law

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Flash: Newton’s 3 rd Law

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Flash: Applications of Newton

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