SPH4U: Lecture 1 Dynamics How and why do objects move

Course Info & Advice Course has several components: Course has several components: Lecture: (me talking, demos and you asking questions) Lecture: (me talking, demos and you asking questions) Discussion sections (tutorials, problem solving, quizzes) Discussion sections (tutorials, problem solving, quizzes) Homework Web based Homework Web based Labs: (group exploration of physical phenomena) Labs: (group exploration of physical phenomena) What happens if you miss a lab or class test What happens if you miss a lab or class test Read notes from online Read notes from online What if you are excused?? (What you need to do.) What if you are excused?? (What you need to do.) That topic of your exam will be evaluated as your test That topic of your exam will be evaluated as your test The first few weeks of the course should be review, hence the pace is fast. It is important for you to keep up! The first few weeks of the course should be review, hence the pace is fast. It is important for you to keep up! Then, watch out…. Then, watch out….

How we measure things! How we measure things! All things in classical mechanics can be expressed in terms of the fundamental units: All things in classical mechanics can be expressed in terms of the fundamental units: Length: L Length: L Mass : M Mass : M Time :T Time :T For example: For example: Speed has units of L / T (e.g. miles per hour). Speed has units of L / T (e.g. miles per hour). Force has units of ML / T 2 etc... (as you will learn). Force has units of ML / T 2 etc... (as you will learn). Fundamental Units

Units... SI (Système International) Units: SI (Système International) Units: mks: L = meters (m), M = kilograms (kg), T = seconds (s) mks: L = meters (m), M = kilograms (kg), T = seconds (s) cgs: L = centimeters (cm), M = grams (gm), T = seconds (s) cgs: L = centimeters (cm), M = grams (gm), T = seconds (s) British Units: British Units: Inches, feet, miles, pounds, slugs... Inches, feet, miles, pounds, slugs... We will use mostly SI units, but you may run across some problems using British units. You should know where to look to convert back & forth. We will use mostly SI units, but you may run across some problems using British units. You should know where to look to convert back & forth. Ever heard of Google

Converting between different systems of units Useful Conversion factors: Useful Conversion factors: 1 inch= 2.54 cm 1 inch= 2.54 cm 1 m = 3.28 ft 1 m = 3.28 ft 1 mile= 5280 ft 1 mile= 5280 ft 1 mile = 1.61 km 1 mile = 1.61 km Example: convert miles per hour to meters per second: Example: convert miles per hour to meters per second:

Dimensional Analysis This is a very important tool to check your work This is a very important tool to check your work It’s also very easy! It’s also very easy! Example: Example: Doing a problem you get the answer distance d = vt 2 (velocity x time 2 ) Units on left side = L Units on right side = L / T x T 2 = L x T Left units and right units don’t match, so answer must be wrong!! Left units and right units don’t match, so answer must be wrong!!

Dimensional Analysis The period P of a swinging pendulum depends only on the length of the pendulum d and the acceleration of gravity g. The period P of a swinging pendulum depends only on the length of the pendulum d and the acceleration of gravity g. Which of the following formulas for P could be correct ? Which of the following formulas for P could be correct ? (b) (c) Given: d has units of length (L) and g has units of (L / T 2 ). (a) P = 2  (dg) 2

Solution Solution Realize that the left hand side P has units of time (T ) Realize that the left hand side P has units of time (T ) Try the first equation Try the first equation (a)(b)(c) (a) Not Right!

Solution Solution Realize that the left hand side P has units of time (T ) Realize that the left hand side P has units of time (T ) Try the second equation Try the second equation (a)(b)(c) (b) Not Right!

Solution Solution Realize that the left hand side P has units of time (T ) Realize that the left hand side P has units of time (T ) Try the first equation Try the first equation (a)(b)(c) (c) Dude, this is it

Time to Drop an Apple Want to ask myself a question: If I drop an Apple from a certain height, h, what then happens to the time, t, it takes for the apple to fall? The time t, must be proportional to the height to the power of some value. The time t may be proportional to the mass of the Apple to the power of some value. The time t, may be proportional to the acceleration due to gravity, g to some power Dimensional Analysis to the Rescue There is no M on Left side: There is no L on Left side: There is T to power 1 on Left side: Therefore

Time for Apple to Drop Dimensional Analysis tells us, that if we increase the height by a factor of 100, then to time will increase by the square root of 100, or by a factor of 10 Let’s verify this by dropping an apple from 3 m then from 1.5 m and compare the times Since we double the height, the time should be longer. The experiment shows: 1.417±0.006

Vectors. A vector is a quantity that involves both magnitude and direction. 55 km/h [N35E] A downward force of 3 Newtons A scalar is a quantity that does not involve direction. 55 km/h 18 cm long

Vector Notation Vectors are often identified with arrows in graphics and labeled as follows: We label a vector with a variable. This variable is identified as a vector either by an arrow above itself : Or By the variable being BOLD: A

Displacement   Displacement is an object’s change in position. Distance is the total length of space traversed by an object. 5m 3m 1m Distance: Displacement: 6.7m Start Finish = 500 m m m 005 Distance 0 Displacement = =

Vector Addition A B C D E A B C D E A B C D E R A + B + C + D + E = Distance R = Resultant = Displacement RR

Rectangular Components A B R  Quadrant I Quadrant II Quadrant IIIQuadrant IV y -y x -x

Vectors... The components (in a particular coordinate system) of r, the position vector, are its (x,y,z) coordinates in that coordinate system The components (in a particular coordinate system) of r, the position vector, are its (x,y,z) coordinates in that coordinate system r = (r x,r y,r z ) = (x,y,z) r = (r x,r y,r z ) = (x,y,z) Consider this in 2-D (since it’s easier to draw): Consider this in 2-D (since it’s easier to draw): r x = x = r cos  r x = x = r cos  r y = y = r sin  r y = y = r sin  y x (x,y)  r where r = |r | r  arctan( y / x )

Vectors... The magnitude (length) of r is found using the Pythagorean theorem: The magnitude (length) of r is found using the Pythagorean theorem: r y x The length of a vector clearly does not depend on its direction The length of a vector clearly does not depend on its direction.

Vector Example Vector A = (0,2,1) Vector A = (0,2,1) Vector B = (3,0,2) Vector B = (3,0,2) Vector C = (1,-4,2) Vector C = (1,-4,2) What is the resultant vector, D, from adding A+B+C? (a) (3,5,-1) (b) (4,-2,5)(c) (5,-2,4) (a) (3,5,-1) (b) (4,-2,5) (c) (5,-2,4)

Resultant of Two Forces force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense. Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force. The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs. Force is a vector quantity.

Vectors Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations. Vector classifications: -Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis. -Free vectors may be freely moved in space without changing their effect on an analysis. -Sliding vectors may be applied anywhere along their line of action without affecting an analysis. Equal vectors have the same magnitude and direction. Negative vector of a given vector has the same magnitude and the opposite direction. Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature P Q P+Q P -P

Addition of Vectors Trapezoid rule for vector addition Law of cosines, Law of sines, Vector addition is commutative, Vector subtraction P Q P+Q P Q Triangle rule for vector addition P P+Q Q P Q -Q P-Q

Addition of Vectors Addition of three or more vectors through repeated application of the triangle rule The polygon rule for the addition of three or more vectors. Vector addition is associative, Multiplication of a vector by a scalar increases its length by that factor (if scalar is negative, the direction will also change.) P Q S Q+S P+Q+S P Q S P 2P -1.5P

Resultant of Several Concurrent Forces Concurrent forces: set of forces which all pass through the same point. A set of concurrent forces applied to a particle may be replaced by a single resultant force which is the vector sum of the applied forces. Vector force components: two or more force vectors which, together, have the same effect as a single force vector.

Sample Problem Two forces act on a bolt at A. Determine their resultant. Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphically evaluate the resultant which is equivalent in direction and proportional in magnitude to the diagonal. Trigonometric solution - use the triangle rule for vector addition in conjunction with the law of cosines and law of sines to find the resultant.

Sample Problem Solution Q P R Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphically evaluate the resultant which is equivalent in direction and proportional in magnitude to the diagonal.

Sample Problem Solution Trigonometric solution From the Law of Cosines, From the Law of Sines,

Dynamics How and why do objects move

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

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”

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”.

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)

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.

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)

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

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)

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

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

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

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

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

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

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

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

Action at a Distance Gravity: Gravity: Burp!

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

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

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.

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.

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

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

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

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

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

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

Flash: Newton’s 1 St Law

Flash: Newton’s 2 nd Law

Flash: Newton’s 3 rd Law

Flash: Applications of Newton