This Week Forces on an object Newton s laws Relating force to acceleration Mass and Weight What s the difference? Friction good and bad A necessary evil. Reaction /action What makes us walk or a car move Riding in an elevator What we feel going up and down Sailing Up Wind 1/25/2016 Physics 214 Spring 2016 1
What causes motion? Inspecting Galileo s equations: i x(t) = v t (uniform linear motion, no acceleration) i a 2 x(t) = v t + t (accelerating linear motion, acceleration is a ) 2 We note that neither the mass of the moving object m nor the origin of the acceleration a are mentioned. For this reason Galileo s equations are called Kinematic Equations of Motion. Newton (1642-1727) was born after Galileo s (1564-1642) death. Newton generalized Galileo s equations by proposing two hypothesis. 1.) Newton s First Law of Motion (Law of inertia): An object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. The net force is the vector sum of all the forces. 2.) Newton s Second Law of Motion (Law of acceleration): If a net external force Fi acts i on an object of mass m, an acceleration that results is equal to a 1/25/2016 Physics 214 Spring 2016 2 a i m F i
Newton s equations are called the dynamical equations of motion These everyday observed accelerations are produced just by two forces: Gravity acts between masses Electric field acts between charges m1 m2 F = Gm 1 m 2 /r 2 q 1 q2 q q 1 2 F = kq 1 q 2 /r 2 The nuclear forces (strong and weak) will be discussed later. 1/25/2016 Physics 214 Spring 2016 3
Second Law Newton's Second and First Law The acceleration of an object is directly proportional to the magnitude of the imposed force and inversely proportional to the mass. The acceleration is in the same direction as the force F ma F and a are vectors unit of Force is a Newton (or pound) 1lb = 4.448N First Law An object remains at rest or in uniform motion in a straight line unless it is acted on by an external force. F 0 a 0 so v = constant 1/25/2016 Physics 214 Spring 2016 4
M The mass of the earth E creates a force at the earth surface on an object with mass m. r E Earth F = GM E m/r E 2 But according to Newton s second law: F, a, g F ma mg so g = GM E /r E 2 point radially to the center of the Earth r E Attracted mass Note g does not depend on m! Famous experiment by Cavendish Measured F = Gm 1 m 2 /r 2 using two known masses in the laboratory and so measured G Then using g = GM E /r E 2 he determined the mass of the earth 1/25/2016 Physics 214 Spring 2016 5
Mass and weight Newton's second law enables us to measure relative mass. Consider a scale. If the weights on the pans are equal the scale is in equilibrium. W m g, W m g 1 1 2 2 Balance of the scale means W W m m 1 2 1 2 Thus we put the unknown weight into one pan end increase the known weight in the pan on the other side until equilibrium is reached. When equilibrium is reached m g m g unknown m unknown m known known 1/25/2016 Physics 214 Spring 2016 6
Inertia Inertia = tendency of an object to resist changes in its velocity. Since F = ma and a = Δv/t then Ft = mδv So if a force acts for a time t the change in velocity will be smaller for larger masses so it is the mass that determines inertia. In particular if t is very small and m is large then F can also be large but Δv can still be very small. 1/25/2016 Physics 214 Spring 2016 7
1F-04 Brass Rod (Inertia) How to remove the paper without toppling the rod? One needs to remove the paper quickly so that the frictional force only lasts for a short time and the inertia of the rod prevents it from toppling over. 1/25/2016 Physics 214 Spring 2016 8
1F-05 Coin, Hoop & Milk Bottle (Inertia) How can you get the coin into the bottle without touching it? This is actually a trick which depends on hitting the ring so that the top deflects down and the coin is free to drop 1/25/2016 Physics 214 Spring 2016 9
1F-06 Inertial Ball Which string breaks first? Case 1: Place the aluminum rod in the lower loop and pull SLOWLY downward. Case 2: Use the wooden mallet to strike a sharp blow to the aluminum rod. IF IT IS DONE SLOWLY THE UPPER STRING BREAKS FIRST BECAUSE THE TENSION IN THAT STRING WILL BE THE WEIGHT OF THE BALL PLUS THE TENSION IN THE LOWER STRING. IF THE LOWER STRING IS STRETCHED SUFFICIENTLY RAPIDLY, IT WILL REACH ITS BREAKING POINT BEFORE THE BALL HAS A CHANCE TO MOVE APPRECIABLY. 1/25/2016 Physics 214 Spring 2016 10
1F-02 Stack of Washers This is a demonstration of Inertia where a washer can be removed from a stack if the blow is fast. Why does it work less well as the stack gets shorter? Strike the stack quickly. So the friction will be very short-lived and the stack will not gain speed before the force is gone. THIS TENDENCY TO RESIST CHANGES IN THEIR STATE OF MOTION IS DESCRIBED AS INERTIA. MASS IS A MEASURE OF THE AMOUNT OF INERTIA. SO FOR A FIXED FRICTIONAL FORCE ACTING FOR A SHORT TIME. THE BIGGER THE MASS, THE LESS IT WILL MOVE. 1/25/2016 Physics 214 Spring 2016 11
1F-07 Table Cloth Jerk Can the table cloth be removed without breaking any dishes? THERE IS A FORCE ACTING ON THE DISHES, BUT IT LASTS FOR A VERY SHORT TIME. COMBINED WITH THE RELATIVELY LARGE MASS OF THE DISHES, THIS FORCE IS OVER SO QUICKLY AND IS SO SMALL THAT THE DISHES HARDLY MOVE. 1/25/2016 Physics 214 Spring 2016 12
1F-03 Egg drop Is it possible to get the eggs in the beakers without touching them? IF THE PAN IS HIT SHARPLY A FORCE WILL ACT ON THE EGGS FOR A VERY SHORT TIME AND THEY WILL NOT MOVE HORIZONTALLY. THE PAN HAS TO BE HIT HARD ENOUGH SO THAT HAS MOVED OUT OF THE WAY BEFORE THE EGGS DROP ANY APPRECIABLE DISTANCE 1/25/2016 Physics 214 Spring 2016 13
Friction In our everyday world any object which is moving feels a force opposing the motion --- this is friction. An object which is sliding The air resistance on your car These types of friction result in energy being lost and minimizing friction is very important. But: Friction is also useful and essential since with no friction a car would not move but just spin it s wheels a car would not be able to turn a corner we would not be able to walk objects would slide off surfaces unless perfectly horizontal 1/25/2016 Physics 214 Spring 2016 14
When we are analyzing a particular object we have to take into account all Force diagrams the forces acting on the body both in magnitude and direction. The acceleration of the object is equal to where is the net force acting. F F/m in the direction of As in the example above we know that if we want to move an object that there is a force called friction which opposes what we want to do. In the case shown the 10N force is + and the 2N force is- so the net force is 8N 8 m mkg F net = +8N and 8N = 5kg a resulting = a (N ) 2 2 5kg 5 s s F 1/25/2016 Physics 214 Spring 2016 15
Terminal velocity As an object moves through the atmosphere the air exerts a frictional retarding force F r which increases with velocity. So an object that is dropped from a great height first accelerates at 9.8m/s 2 but this acceleration decreases until the retarding force becomes equal to mg F r F and at that point the acceleration becomes zero and the object has it s terminal velocity. Could a penny dropped from the top of the Empire State building Kill you? How about a pen? r mg F r mg v mg - F = ma r When a car is moving at constant speed on the highway the forward force produced by the engine is balanced by the frictional force of the air. F tire F engine F air engine 1/25/2016 Physics 214 Spring 2016 16 F - F = ma air
Newton's Third Law: (reaction - action) For every force that acts on a body there is an equal and opposite reaction force. F f F F You pull on the block the block pulls back on you The floor exerts a frictional force holding the block back The block exerts a frictional force on the floor trying to move the floor to the right. F f The block accelerates providing F > F f 1/25/2016 Physics 214 Spring 2016 17
Force analysis To analyze the motion of an object we need to draw a diagram and put in all the forces that are acting on the object. We will only deal with problems that have an acceleration along a single axis. The net force along an axis perpendicular to this axis is zero F 3 F 2 F 1 F net = +F1 F2 F3 F net DRAW A FORCE DIAGRAM We normally choose + in the direction of the acceleration. We now can use the equations F = ma v = v 0 + at d = v 0 t + ½ at 2 v 2 = v 2 0 + 2ad net 1/25/2016 Physics 214 Spring 2016 18
Normal forces Normal force provided by the table For the book N t = mg Nt N mg DRAW A FORCE DIAGRAM For the table resting on the earth N earth = mg + m table g m table g mg N e For a stack of plates or a brick wall and even a mountain each layer has to support the weight of everything higher. So for a stack of 48 plates the force on the second plate from the top is mg. For the bottom plate it is 47mg but of course each plate has a net force of mg to balance it s own weight. 1/25/2016 Physics 214 Spring 2016 19
1H-02 Fan Cart (Action-Reaction) Can a fan attached to a cart propel the cart? What if the sail is removed? Action Reaction What if the sail is canted at an angle? In which direction will it move? INTERNAL FORCES IN A SYSTEM CANCEL EACH OTHER WHEN THE SYSTEM AS A WHOLE IS CONSIDERED. SO IF THE SAIL IS PERPENDICULAR THE FAN DOES NOT MOVE. IF THE SAIL IS REMOVED THE FAN MOVES IN THE OPPOSITE DIRECTION TO WHICH IT BLOWS AIR. THE FAN WOULD NOT MOVE IN OUTER SPACE. 1/25/2016 Physics 214 Spring 2016 20
1H-03 CO 2 Rocket Imagine that you are sitting in a cart with a pile of bricks. ROCKET PROPULSION! How could you use the bricks to get yourself and the cart to move? What would happen if you throw a brick out of the cart? Then you throw out another. What if you throw smaller bricks faster and more frequently? Now, if the bricks were the size of molecules... What happens when the fire extinguisher rapidly throws out CO 2 molecules? THIS Because IS DIFFERENT the repulse THAN THE FAN CART. CO 2 IS EXPELLED AT HIGH VELOCITY AND IN TERMS OF FORCES THE REACTION FORCE CAUSES THE CART TO MOVE IN THE OPPOSITE DIRECTION. THE QUANTITY CALLED MOMENTUM IS CONSERVED AND THE MASS OF THE CO 2 X AVERAGE SPEED = TOTAL MASS OF THE CART X AVERAGE VELOCITY. THIS IS HOW A ROCKET IS ACCELERATED AND IT WORKS IN OUTER SPACE. 1/25/2016 Physics 214 Spring 2016 21
1H-04 Heron's Engine A glass bulb emits steam from small nozzles What happens when the Glass Bulb begins to emit steam? Zegner s wheel Reaction = Bulb Spins Action = Ejects Steam Same Principle causes a Lawn Sprinkler to Turn. THE REACTION FORCE TO THE EJECTION OF MASS CAUSES THE OBJECT TO SPIN. THIS IS THE SAME AS THE CO 2 ROCKET IN THAT MATERIAL IS EXPELLED AT HIGH VELOCITY 1/25/2016 Physics 214 Spring 2016 22
1J-04 Scale Paradox 1 What is the reading on the scale? A Scale Measures the Force acting on it NOW, What is the reading on the scale? WALL mg mg mg T T T T mg T = mg T = mg mg mg mg THE TENSION IN THE CORD IS THE SAME FOR BOTH CASES. THE SCALE MEASURES THE TENSION IN THE CORD. FOR EXAMPLE THE TENSION IN A ROPE IS THE SAME IF TWO PEOPLE PULL ON EACH END WITH FORCE F OR IF ONE PERSON PULLS WITH FORCE F TO A ROPE TIED TO A WALL. 1/25/2016 Physics 214 Spring 2016 mg 23
Forces in an elevator W = mg = true weight with no acceleration N = apparent weight N mg is the net force taking + to be up + N N mg = ma is the equation of motion If N > mg a is positive and the apparent weight is > than the true weight If N < mg a is negative and the apparent weight is less than the true weight mg g IRRESPECTIVE OF THE DIRECTION OR MAGNITUDE OF THE VELOCITY 1/25/2016 Physics 214 Spring 2016 24
Connected objects T Both objects have the same acceleration The coupling pulls back on the 4kg mass but accelerates the 2kg mass. 30 8 T = 4a T 6 = 2a Or 30 8 6 = 6a a = 16/6 m/s 2 T = 68/6N If I have a freight train with 100 cars each of mass m then for the coupling between the engine and the first car T 1 = 100ma And between the last two cars T 99 = ma or 100 times less 1/25/2016 Physics 214 Spring 2016 25
Connected objects 6mg F 7mg F 3 = 3ma F 4 = 4ma In the example shown each brick has a net upward force = mg down. But each brick has to support all the bricks above it. So the 7 th brick has a downward force on it s top surface of 6mg and an upward force of 7mg on it s bottom surface. In the example shown of 7 equal mass objects being accelerated F = 7ma. The tension in each coupling pulls back on the mass ahead and accelerates the mass behind. So F 4 =4ma but the net force on block 4 is 4ma 3ma = ma. So the net force on each block is ma but the tension in each coupling reduces by ma as one goes down the chain. 1/25/2016 Physics 214 Spring 2016 26
Acceleration of a car F tire F air F reaction F air = ma F reaction Generally F air is proportional to v 2 so the difference between 55mph and 80mph is a factor of 2.16. This means you use 2.16 more gasoline to cover the same distance Travel from Indianapolis to Lafayette a distance of 60 miles Car does 30miles/gallon at 55mph gas costs $4/gallon At 55mph use 2 gallons cost 8$ time = 65.45 minutes At 80 mph use 4.32 gallons cost $17.28 time = 45 minutes Cost of saving 20.45 minutes = $9.28 1/25/2016 Physics 214 Spring 2016 27
Summary of Chapter 4 Forces are responsible for all physical phenomena Gravitation and the electromagnetic force are responsible for all the phenomena we normally observe in our everyday life. Newton s laws F = ma where F is net force v = v 0 + at d = v 0 t + ½ at 2 d = ½(v + v 0 )t v 2 = v 2 0 + 2ad Every force produces an equal and opposite reaction Weight = mg where g = 9.8m/s 2 locally Apparent weight in an elevator depends on the acceleration a up weight is higher a down weight is lower If your weight becomes zero it s time to worry because you are in free fall!! 1/25/2016 Physics 214 Spring 2016 28
Sailing Up Wind The sail and keel forces are like lift forces on an airplane wing An example is when the boat is moving perpendicular to the wind the force of the winds on the sails remains constant. The sails are set at about 45 0 to the direction of motion and the wind. The boats equilibrium speed is determined by the resistance to the motion. If the resistance can be made small the boat speed can be very high. Sail iceboats have very little resistance and reach speeds in excess of 90mph with wind speeds of 30mph The force on the sail balances out the force on the keel and leaves a component of force against the wind. Then F = ma and the boat will increase speed until the component of the wind force equals the drag force 1/25/2016 Physics 214 Spring 2016 29
Questions Chapter 4 Q8 A 3-kg block is observed to accelerate at a rate twice that of a 6-kg block. Is the net force acting on the 3-kg block therefore twice as large as that acting on the 6-kg block? Explain. The net force is the same Q9 Two equal-magnitude horizontal forces act on a box as shown in the diagram. Is the object accelerated horizontally? Explain. -F F No the net force is zero Q10 Is it possible that the object pictured in question 9 is moving, given the fact that the two forces acting on it are equal in size but opposite in direction? Explain. Yes, constant velocity 1/25/2016 Physics 214 Spring 2016 30
Q18 The acceleration due to gravity on the moon is approximately one-sixth the gravitational acceleration near the earth s surface. If a rock is transported from the earth to the moon, will either its mass or its weight change in the process? Explain. It s mass will not change but it s weight wil be 6 times less Q22 The engine of a car is part of the car and cannot push directly on the car in order to accelerate it. What external force acting on the car is responsible for the acceleration of the car on a level road surface? Explain. F tire F air F reaction It s the reaction force between the tires and the road Q23 It is difficult to stop a car on icy road surface. It is also difficult to accelerate a car on this same icy road? Explain. Because of a lack of friction the wheels will skid or spin 1/25/2016 Physics 214 Spring 2016 31
Q25 When a magician performs the tablecloth trick, the objects on the table do not move very far. Is there a horizontal force acting on these objects while the tablecloth is being pulled off the table? Why do the objects not move very far? Explain. Yes but the force acts for a very short time and the objects start to move, then when the cloth is gone friction stops them. Q30 Two masses, m 1 and m 2, connected by a string, are placed upon a fixed frictionless pulley as shown in the diagram. If m 2 is larger than m 1, will the two masses accelerate? Explain. Yes m 2 will fall and m 1 will rise m 1 m 2 1/25/2016 Physics 214 Spring 2016 32
Q31 Two blocks with the same mass are connected by a string and are pulled across a frictionless surface by a constant force, F, exerted by a string (see diagram). A. Will the two blocks move with constant velocity? Explain. B. Will the tension in the connecting string be greater than, less than, or equal to the force F? Explain. F A. They will accelerate F = ma B. The tension will be less 1/25/2016 Physics 214 Spring 2016 33
Q33 If you get into an elevator on the top floor of a large building and the elevator begins to accelerate downward, will the normal force pushing up on your feet be greater than, equal to, or less than the force of gravity pulling downward on you? Explain. + N g N mg = ma but a is negative so N is smaller than mg The only force pulling you down is gravity so if you are accelerating down the force due to gravity must be larger than the reaction force N ( N is apparent weight) a mg 1/25/2016 Physics 214 Spring 2016 34
Ch 4 E4 A 2.5kg block is pulled with a force of 80N and friction is 5N a) What is the acceleration? 5 N 2.5 kg 80 N Net force = 75 N a = 75 / 2.5 = 30 m/s 2 1/25/2016 Physics 214 Spring 2016 35
Ch 4 E6 A 6kg block is being pushed with a force P and has an acceleration of 3.0m/s 2 a) What is the net force? b) If P is 20N what is F f? P 6 kg F f a = 3m/s 2 a) F = ma = 6 x 3 = 18 N b) F = P F f F f = 2N 1/25/2016 Physics 214 Spring 2016 36
Ch 4 E14 A 4kg rock is dropped and experiences air resistance of 15N a) What is the acceleration? F = 4 x 9.8 15 F = ma = 24.2 N a = 24.2/4 = 6.05m/s 2 4 kg Mg 15 N 1/25/2016 Physics 214 Spring 2016 37
Ch 4 E16 g A vertical force of 6N presses on a book. a) What is the gravitational force? b) What is the normal force? 6 N 0.4 kg N a) Gravitational Force = mg = 3.92 N b) Upward Force = 6 + 3.92 = 9.92 N 1/25/2016 Physics 214 Spring 2016 38
Ch 4 E18 A 60kg person is in an elevator With an upward acceleration of 1.2m/s 2 a) What is the net force? b) What is the gravitational force? c) What is the normal force? mg m = 60 KG a = 1.2 m/s 2 a) Net Force F = Ma = 60 x 1.2 = 72 N b) mg = 60 x 9.8 = 588 N c) N = 588 + 72 = 660 N 1/25/2016 Physics 214 Spring 2016 39
Ch 4 CP4 A 60kg crate is lowered from a height of 1.4m and the tension is 500N a) Will the crate accelerate? b) What is the acceleration? c) How long to reach the floor? d) How fast does the crate hit the floor? a) Net Force = 60 x 9.8 500 60 kg 500 N g = 588 500 = 88 N b) Will accelerate down a = 88 / 60 = 1.47 m/s 2 c) d = 1/2 at 2 t = 1.38s d) v = v 0 + at v = 2.03 m/s 1/25/2016 Physics 214 Spring 2016 40
Ch 4 CP6 A 60kg person accelerating down at 1.4m/s 2 a) What is the true weight? b) What is the net force? c) What is N? d) What is the apparent weight? e) a, b, c, d with 1.4m/s 2 up? a) True weight = mg = 60 x 9.8 = 588 N b) Net Force = Ma = 84 N c) N = 588 84 = 504 N d) 504 N e) 1.4 m/s 2 1/25/2016 Physics 214 Spring 2016 41 N Mg 1.4 m/s 2 Net Force = 84 N N = 588 + 84 = 672 N W = 672 N
Review Chapters 1-4 - d + x Units----Length, mass, time SI units m, kg, second Coordinate systems Average speed = distance/time = d/t Instantaneous speed = d/δt Vector quantities---magnitude and direction Magnitude is always positive Velocity----magnitude is speed Acceleration = change in velocity/time =Δv/Δt Force = ma Newtons 1/25/2016 Physics 214 Spring 2016 42
Conversions, prefixes and scientific notation giga 1,000,000,000 10 9 billion 1 in 2.54cm mega 1,000,000 10 6 million kilo 1,000 10 3 thousand 1cm 1ft 0.394in 30.5cm centi 1/100 0.01 10 - milli 1/1000 0.00 1 2 10-3 micro 1/1,000,000 1/10 6 10 - nano 1/1,000,000,000 1/10 9 10-6 9 hundredth thousandth millionth billionth 1m 39.4in 3.281ft 1km 0.621mi 1mi 5280ft 1.609km 1lb 0.4536kg g =9.8 1kg 2.205lbs g=9.8 1/25/2016 Physics 214 Spring 2016 43
Speed, velocity and acceleration v = Δd/Δt a = Δv/Δt The magnitude of a is not related to the magnitude of v the direction of a is not related to the direction of v 1 2 3 4 v = v 0 + at constant acceleration d = v 0 t + 1/2at 2 d,v 0 v,a can be + or - d = 1/2(v + v 0 ) t v 2 = v 2 0 + 2ad 1/25/2016 Physics 214 Spring 2016 44
One dimensional motion and gravity v = v 0 + at d = v 0 t + 1/2at 2 d = ½(v + v 0 )t v 2 = v 0 2 + 2ad + g = -9.8m/s 2 + At the top v = 0 and t = v 0 /9.8 At the bottom t = 2v 0 /9.8 1/25/2016 Physics 214 Spring 2016 45
Equations v = v 0 + at d = v 0 t + 1/2at 2 d = ½(v + v 0 )t v 2 = v 0 2 + 2ad Sometimes you have to use two equations. v 0 = 15m/s v = 50m/s What is h? v = v 0 + at v 0 50 = 15 + 9.8t t = 3.57 s ` h = v 0 t + 1/2at 2 h g v h = 15 x 3.57 + 1/2x9.8x3.57 2 = 116m h = ½(15 + 50) x 3.57 = 116m 1/25/2016 Physics 214 Spring 2016 46
Projectile Motion axis 1 v 1 = constant and d 1 = v 1 t axis 2 v v = v 0v + at and d = v 0v t + 1/2at 2 g v 1 h 9.8m/s 2 v R Use + down so g is + and h is + h = v 0v t + 1/2at 2 v 0v = 0, t 2 = 2h/a R = v 1 t v = v 0v + at 1/25/2016 Physics 214 Spring 2016 47
Complete Projectile v 0v v 1 9.8m/s 2 v 1 v 1 v 0v highest point the vertical velocity is zero v v = v 0v + at so t = v 0v /9.8 h = v 0v t + 1/2at 2 end t = 2v 0v /9.8 and R = v 1 x 2v 0v /9.8 and the vertical velocity is minus v 0v 1/25/2016 Physics 214 Spring 2016 48
Newton s Second and First Law Second Law F = ma unit is a Newton (or pound) First Law F = 0 a = 0 so v = constant Third law For every force there is an equal and opposite reaction force Weight = mg N mg F f F F F = ma v = v 0 + at d = v 0 t + ½ at 2 d = ½(v + v 0 )t v 2 = v 0 2 + 2ad F f 1/25/2016 Physics 214 Spring 2016 49
Examples T + N 30 8 T = 4a T 6 = 2a 30 8 6 = 6a mg a + a N mg = ma N > mg N < mg g 1/25/2016 Physics 214 Spring 2016 50