DO PHYSICS ONLINE FORCES & NEWTON S LAWS OF MOTION

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1 DO PHYSICS ONLINE ORCES & NEWTON S LAWS O MOTION The study of what causes changes in the motion of an object is known as dynamics. The central concept of dynamics is the physical quantity force. The term force can t be defined or explained in a simple sentence. You will learn what a force is and the ideas associated with forces however, you just will not be able to state what a force is in a clear brief statement. Sometimes force is said to be a pull or push, but a force is more than just a push or pull. orces acts in pairs it is not possible to have a single force acting. The interaction between objects is often described in terms of forces. or example, (1) two objects will attract each other because of the gravitational force between them, and (2) two electrons will repel each other because electrons are negatively charged. The principles of dynamics are described by the three Newton s laws of motion. These three laws are very simple to state, but extremely difficult to interpret and apply. This is because of your experience in walking, throwing, pushing, etc has given you mental models of how things work based upon common sense. But, the way the world works is not necessarily based upon logic, intuition or common sense. After careful observation and measurement, physicists have developed scientific principles which often are at odds to our mental models based upon our experience of the physical world. Newton s laws are not the product of mathematical derivations, but rather they have been synthesized from a multitude of experimental data about how objects move. These laws are truly fundamental because they can t be deduced from other principles.

2 PROPERTIES O ORCES A force is a push or pull. A force is an interaction between two objects. orces can be attractive or repulsive. orces act in pairs. An interaction between two objects gives rise to two forces of equal magnitude but which act in opposite directions (Newton s 3 rd law). A force is a vector quantity with magnitude and direction. Multiple forces can be replaced by a single force using vector addition to give the resultant or net force acting on an object. A force can be resolved into two components at right angles to each other. If the resultant force on an object is zero, the object will remain stationary or move with a constant velocity (Newton s 1 st law: 0 a 0). If the resultant force on an object is non-zero, the object will experience an acceleration in the direction of the force (Newton s 2 nd law: 1 a m ). The S.I. Unit for force is the newton [N] 1 N = 1 kg.m.s -2

3 NEWTON S LAWS O MOTION In studying motion the first step was defining a frame of reference. The fame of reference chosen is one in which its acceleration is zero. This is called an inertial frame of reference. You are an observer in a car and the car represents your frame of reference. If you have your eyes closed, then you can t feel how fast you re travelling. This is an inertial frame of reference. You feel the effects of acceleration, if the car suddenly speeds up you feel pushed backwards, if the car brakes abruptly you will be thrown for forward and if the car turns sharply then you appear to be thrown sideways. When the car accelerates, it is a non-inertial frame of reference. The Earth although it is orbiting the Sun and rotating about its axis, these effects are small and so we can take the Earth s surface to be an inertial frame of reference. or our study of the motion we always take an inertial frame of reference. Newton s 1 st law of motion In an inertial frame of reference, if the resultant (net) force acting upon an object is zero, then the object will be at rest or moving with a constant velocity. This law implies that the natural state of motion of an object is that it has zero acceleration (a = 0), therefore, the object is at rest (v = 0) or moving in a straight line with a constant speed (v = constant). The tendency for an object to remain at rest or keep moving is called inertia. When the resultant (net) force acting on a particle is zero, the object is in equilibrium x y x y equilibrium (1) a 0 a 0 a 0

4 igure (1) helps us understand Newton s 1 st law. You are standing in a tram travelling at a constant velocity when the tram suddenly brakes. You feel that someone has pushed you in the back and you topple forward. But, you do not experience a force. By Newton s 1 st law you upper part of the body will continue to move forward while your legs attached to the floor slow down with the tram. Again, you are standing in a stationary tram. The tram starts moving forward. Your feet move forward with the tram but your upper part of the body tends to remain at rest according to Newton s 1 st law. tram moving with constant velocity v a 0 tram suddenly stops a upper body keeps moving forward with velocity v feet stop with tram tram stopped v = 0 a 0 tram starts moving forward a upper body tends to stay at rest v= 0 ig. 1. Riding an accelerating tram. feet move forward with tram Many injuries suffered in car accidents can be explain by Newton s 1 st law. When a car is moving forward and then suddenly slows down as in a collision, the people in the car will tend to keep moving forward with a constant velocity until they are brought to rest by smashing into the steering wheel or the windscreen. The faster the car is travelling before a collision, the greater the speed before impact which brings the occupants to rest and the more substantial are the injuries incurred. In many such collisions occupants often suffer massive internal injuries. You internal organs must obey Newton s 1 st law they continue moving forward damaging the diaphragm and the organs themselves which results in excessive internal bleeding. Many young drivers have died from internal injuries in car accidents you should always remember Newton s 1 st law the faster you are going before an accident the more likely you are to die slowing down does save lives. Seat belts, collapsible steering wheels and air bags are used to limit the injuries in accidents caused by inertia.

5 Similarly, a car moving off suddenly from rest can cause a person s upper parts of their body namely the head to remain stationary whilst the lower parts of the body seated and in contact with the car floor move forward. Serve neck injuries are very common in rear end collisions when the neck muscles are severely stretched. The stretched muscles act a like a spring and the head is whipped back to catch-up with the rest of the body. This is often referred to whip-lash injuries of the neck and back. Head restraints on car seats are used to carry the head forward to prevent this type of injury to the neck. ig.2. Seat belts, air bags and head restraints can reduce injuries in a collision. Sir Isaac Newton ( ) was by many standards the most important person in the development of modern science. Newton, who was born the same year that Galileo Galilei ( ) died, built on Galileo's ideas to demonstrate that the laws of motion in the heavens and the laws of motion on the Earth were one and the same. The work of Galileo and Newton would banish the notions of Aristotle that had stood for 2000 years. Galileo s ideas challenged the views of the Church on the Aristotelian conception of the Earth centred Universe. He got him into trouble with the Inquisition and was forced to recant publicly his Copernican views. In the later years of his life he was under house arrest. His story is one of many that highlighted the conflict between the scientific approach and "science" based on unquestioned authority. Even today, there still are many forces in modern society (religious, cultural, political, etc) that are against the scientific approach of open enquiry. Perhaps Galileo's greatest contribution to physics was his formulation of the concept of inertia: an object in a state of motion possesses an inertia that causes it to remain in that state of motion unless a resultant force acts on it. Most objects in a state of motion do not remain in that state of motion. or example, a block of wood pushed across table quickly comes to rest when we stop pushing. Thus, Aristotle held the view that objects in motion did not remain in motion unless a force acted constantly on them. Galileo, by virtue of a series of experiments (many with objects sliding down inclined planes),

6 realized that the analysis of Aristotle was incorrect because it failed to account properly for a hidden force: the frictional force between the surface and the object. As we push the block of wood across the table, there are two opposing forces that act: the force associated with the push, and a force that is associated with the friction and that acts in the opposite direction. Galileo deduced that as the frictional forces are decreased (for example, by placing oil on the table) the object would move further and further before stopping. rom this he abstracted a basic form of the law of inertia: if the frictional forces could be reduced to exactly zero a moving object would continue moving at a constant speed in a straight line unless a new force acts on it at a later time. igure (3) illustrates the difference between the Aristotle view motion and the views of Galileo and Newton. Aristotle view: an object keeps moving only if it continually pushed v Galileo s deduction: a moving object will slow down and stop because of friction v Newton s 1 st law: The natural state of motion: object at rest or moving with constant velocity when 0 v 0 f friction v constant frictionless surface between box & floor no air drag, etc ig.3. Aristotle view of motion compared to the view of Galileo and Newton. Web search 1. View several movie clips on Newton s 1 st law of motion. 2. ind at least eight examples of physical situations that can be explained by Newton s 1 st law, e.g. spin dryer, sauce bottle, John Ogborn (Difficulties in dynamics, Physics at Secondary School: Physics and Applications Vol II, Proceedings for the Conference Institute of Physics, Slovak Academy of Sciences.) has summarised peoples difficulties with Newton's Laws of Motion: the first law is unbelievable the second law is incomprehensible the third is merely religious incantation

7 Learning or teaching Physics is not a simple task. A Koyre (Journal of Historical Ideas, 4,400,p.405, 1943) has made an appropriate comment: "What the founders of modern science, among them Galileo, had to do, was not to criticise and combat faulty theories, but to replace them by better ones. They had to do something quite different. They had to destroy one world and replace it by another. They had to reshape and reform its concepts, to evolve a new approach to being, a new concept of knowledge, a new concept of science - and even to replace a pretty natural approach, that of common sense, by another which is not natural at all." Newton s 2 nd law of motion In an inertial frame of reference, the resultant (net) force acting upon an object produces an acceleration of the object in the direction of the force. The acceleration a is proportional to the resultant force the object. (2) Equation (2) tells us: and inversely proportional to the mass m of ma a Newton s 2 nd law m The greater the resultant (net) force acting on an object the greater its acceleration. The greater the mass of the object the less its acceleration. A definition of the concept of force. If the object of mass m has an acceleration a then the force acting on it is defined as the product of its mass and acceleration. The product m a is not a force but gives a number equal to the magnitude of the force.

8 Newton s 3 rd law of motion In an inertial frame of reference, if an object A exerts on object B a force BA, then object B exerts on object A a force AB which is equal in magnitude but opposite in direction (3) BA AB Newton s 3 rd law Newton s 3 rd law is illustrated by the example of the repulsion between like charges and the attraction between opposite charges as shown in figure (4). Whenever a force acts on an object there must be simultaneously be an equal in magnitude and opposite direction force acting on some other object. orces always act in pairs. Note: this pair of forces act on different objects. Action and reactions are equal and opposite. AB A B BA AB BA repulsion between two negatively charged objects A AC CA AC CA + attraction between a negative charge and a positive charge C ig. (4) charges. Newton s 3 rd law repulsion and attraction between Discuss how Newton s 3 rd law explains the following examples: The rotation of a lawn sprinkler. A boat being propeller through water. The recoil of a firing gun. The operation of a rocket and jet engine. Walking. Driving a car. orces between Sun, Moon and Earth give rise to the tides of the oceans. A plane flying through the air. The following list are common statements of Newton's Third Law found in Physics textbooks, all of which are misleading. Every action has an equal opposite reaction. To every force there is an equal and opposite force. Action and reaction are equal and opposite. Why are these three statements misleading?

9 The donkey and cart problem If the above are statements of Newton's Third Law are true, how the donkey pull the cart? can Hit by a truck problem You have just been hit by a speeding bus. What force does the bus exert on you? What force do you exert on the bus? Head-On problem Consider the consequences of driving a car into a head-on collision with an identical car travelling toward you at the same speed, as opposed to colliding at the same speed into a massive concrete wall. Which of these two situations would result in the greatest impact force? (Lewis Epstein & Paul Hewitt, Thinking Physics) Newton's laws of motion: weight If you review many introductory Physics textbooks you will find two conflicting definitions of weight. What are these two definitions? Use these two definitions to answer the following questions. How can you explain the "weightlessness" of astronauts orbiting the earth? What is the weight of a high jumper when standing on the ground and in flight? You are in a lift and the cable snaps. What is your weight? A mass is suspended from a spring balance? What is the force on the spring balance? Will the balance read different values at different latitudes? Discuss these questions. How can we walk across the room? How can an automobile move forward or stop?

10 TYPES O ORCES Electrostatic force (long range: action at a distance) the attractive or repulsion between two stationary charged objects. AB A B BA AB BA repulsion between two negatively charged objects AC A AC CA CA + C attraction between a negative charge and a positive charge Gravitational force (long range: action at a distance) attraction between two objects because of their mass. Weight (long range: action at a distance) the gravitational force acting on an object due to its attraction to the Earth. Near the Earth s surface the weight of an object is (4) G mg weight of an object near Earth s surface where m is the mass of the object and +y g is the local acceleration due to gravity at the Earth s surface. box G = mg Beware: there are other definitions of weight.

11 Contact forces occur when two objects are in direct contact with each other. Three examples of contact forces are: normal force, friction and tension. +y Normal force the force acting on an object in contact with a box surface which acts in a direction at right angles to the surface. N +y N riction force the force acting on the object which acts in a direction parallel to the surface. A simple model for friction f is that it is proportional to the normal force N and the constant of proportionality is called the coefficient of friction. The maximum frictional force is (5) fmax simple model for friction N v box f friction +y N normal max frictional force fmax constant of proportionality is the coefficient of friction between the two surfaces N front wheel drive car v N normal front driving wheels f friction f driving Equation (5) is only valid for a simplified model. There are two coefficients of friction known as the static coefficient of friction s (no movement between surfaces in contact) and the kinetic coefficient of friction (relative movement between the surfaces) where s k. Usually the difference between the values for s and k is only small. k

12 In many texts, it is stated that friction opposes the motion of all objects and eventually slows them down. But this type of statement is very misleading. How can we walk across the room, how can cars move? It is because of friction. In walking, you push against the floor and the frictional force of the floor on you is responsible for the forward movement. The drive wheels of a car turn because of the engine. The driven wheels exert a force on the road, and the road exerts a forward force on the car because of friction between the types and road. In icy conditions the driven wheels spin so that that there is very little friction between the tyres and the road and the car can t be driven safely. When the foot is taken off the accelerator peddle, a car will slow down because of friction. Tension the force acting along a stretched rope which is connected to an object. rope pulling box along floor +y tension T box tension a pulling force exerted by a rope, chain, chord, etc box rope under tension pulley changes the direction of the force (tension) +y box being pulled by the rope tension in the rope tension T tension T

13 Springs a common instrument for measuring force magnitudes is the spring balance. When an object is connected to a spring it will experience a force when the spring is compressed or extended. A simple model for a spring is that the force s exerted on the object is proportional to the compression or extension x of the string from its natural length. This force is known as the elastic restoring force. The constant of proportionality is the spring constant k. This relationship is known as Hooke s law (6) s k x Hooke s law elastic springs k x = 0 s = 0 x x s = -kx s < 0 x > 0 s = -kx s > 0 x < 0

14 Example 1 We will consider the forces acting on a stationary book on a floor and the reaction forces. The book is at rest, therefore, from Newton s 1 st law the sum of all the forces acting on the book must be zero. The book pushes down on the floor due to its weight B and the floor deflects slightly pushing back on the book B. The book is attracted towards the centre of the Earth with a force BE and the Earth is pulled by the book with a force EB. book at rest on floor 0 book +y N floor pushes up on book N = B Earth pulls on book weight G = BE N G G orces acting ON book normal force N weight G N = G N and G are not a action / reaction pair since they act upon the same object book pushed down on floor G = B Action / reaction pair B B Book pulls on Earth EB Action / reaction pair EB BE ig. 5. orces involved in a book resting on a floor.

15 Example 2 You need to study all the steps in this example of the box on an inclined surface as shown in figure (6). Note the procedure. Choose the x-axis parallel to the include (up) and the y-axis at right angle to the incline (up). The object is represented as a dot. Show all the forces which act only on the box. Resolve all these forces into components parallel and at right angles to the incline. Apply Newton s 2 nd law to the forces acting along each coordinate axis to find the acceleration up the incline a x. The box only moves along the surface therefore, a y = 0. The diagram showing the forces are often referred to as free-body diagram. To gain the most from this very important example, work through it many times after you have completed this Module on orces and Newton s law.

16 We are interested only in the forces that are acting on the box being pulled up the inclined surface due to the object attached to the pulley system. The forces acting on the box are its weight G = mg, the tension T due to the rope and the contact force C between the box and surface. The tension is due to the object at the end of pulley rope, T = Mg. The box is replaced by a particle represented as a dot. Choose a coordinate system with the x-axis acting up the inclined surface and the y-axis acting upward and at right angles to the surface. The contact surface can be resolved into two components: the normal force N at right angles to the incline and the friction f parallel to the surface. The frictional force is related to normal force f = N where is the coefficient of friction. The weight can also be resolved into its x and y components, Gx and Gy Gx = G sin = mg sin Gy = G cos = mg cos The acceleration a x up the incline can be found from Newton s 2 nd law m M +y T C G = mg +y T N f G = mg +y T N f y = N Gy = N mg cos = ma y a y = 0 N = mg cos f = N = mg cos Gy Gx x = T - f Gx = ma x ma x = Mg - mg cos - mg sin a x = (M/m - cos - sin)g ig. 6. orces acting on a box accelerating up the inclined surface due to the weight of the object at the end of the rope of the pulley system. Newton s 2 nd law can be used to find the unknown acceleration of the box up the incline.

17 Example 3 A flower pot which has a mass of 5.55 kg is suspended by two ropes one attached horizontally to a wall and the other rope sloping upward at an angle of 40 o to the roof. Calculate the tension in both ropes. Solution How to approach the problem Identify Setup Execute Evaluate Visualize the situation write down all the given and unknown information. Draw a free-body diagram forces. Draw a free-body diagram x and y components for the forces. orces can be added using the head-to-tail method but it is best to solve the problem using x and y components. Object at rest: apply Newton s 1 st law to the x and y components. Solve for the unknown quantities. physical picture 40 o T1?N 40 o +y free-body diagrams T2?N m 5.55kg g 9.81m.s -2 Object is stationary: Newton s 1 st law T T G G T 2 T 1 40 o T cos 40 1 o x G free-body diagrams G T sin 40 1 mg o T y Applying Newton s 1 st law to the x and y components for the forces: T T T cos40 0 T T cos40 x o o mg (5.55)(9.81) y T1sin40 mg 0 T N o o sin40 sin40 T o o 2 1cos40 (84.7)cos N o

18 Example 4 You are driving a car along a straight road when suddenly a car pulls out of a driveway of 50.0 m in front of you. You immediately apply the brakes to stop the car. What is the maximum velocity of your car so that the collision could be avoided? Consider two cases: (1) the road is dry and (2) the road is wet. mass of car m = 1500 kg coefficient of friction rubber on dry concrete Solution rubber on wet concrete How to approach the problem Identify Setup Execute Evaluate Visualize the situation write down all the given and unknown information. Draw a diagram of the physical situation showing the inertial frame of reference. Type of problem forces and uniform acceleration. Draw a free-body diagram showing all the forces acting on the car. Use Newton s 2 nd law to find normal force and frictional force and acceleration. Car slows down due to constant frictional force acceleration of car is uniform use equations of uniform acceleration to find initial velocity of car. Solve for the unknown quantities. u =? m.s -1 v = 0 m.s -1 x = 0 x = 50.0 m motion map velocity vectors a =? m. s-2 constant acceleration N m = 1500 kg f = - N = - m g G = mg (dry) = (wet) = N = G = mg g = 9.81 m.s -2 rom Newton s 2 nd law the acceleration of the car is f mg 0 mg y N N mg N f x f max ma a g m

19 Constant acceleration 2 2 v u 2a s 2 u v 2a s Maximum initial velocity to avoid collision dry conditions =0.800 u = m.s -1 u = +(28.0)(3.6) km.h -1 = +101 km.h -1 wet conditions =0.300 u = m.s -1 u = +(17.2)(3.6) km.h -1 = km.h -1 When you are driving, whether an event occurs that could lead to a fatal accident is very dependent upon the road conditions. In this example, to stop and avoid the collision, there is a difference of 40 km.h -1 in the initial velocities. Many occupants inside a car have been killed or severely injured because the driver did not slow down in poor road holding conditions. Also, in this example we ignored any reaction time of the driver. If the reaction time of the driver was just 1 s, then a car travelling at 100 km.h -1 (28 m.s -1 ) would travel 28 m before breaking. Predict Observe Explain Image that you are standing on a set of bathroom scales in an elevator (lift). Predict how the scale reading changes when the elevator accelerates upwards, downwards and travels with a constant velocity. Write down your predictions and justify them. Observe: work through Example 5. Explain any discrepancies between the answers given in Example 5 and your prediction. Example 5 A 60 kg person stands on a bathroom scale while riding an elevator. What is the reading on the scale in the following cases: (1) The elevator is at rest. (2) The elevator is going up at 2.0 m.s -1. (3) The elevator is going up at 4.0 m.s -1. (4) The elevator is going down at 2.0 m.s -1. (5) The elevator starts from rest and goes up reaching a speed of 2.0 m.s -1 in 1.8 s. (6) The elevator is moving up and accelerates from 2.0 m.s -1 to 4.0 m.s -1 in 1.8 s. (7) The elevator starts from rest and goes down reaching a speed of 2.0 m.s -1 in 1.8 s. (8) The elevator is moving down and accelerates from 2.0 m.s -1 to 4.0 m.s -1 in 1.8 s. (9) The elevator is moving up and slows from 4.0 m.s -1 to 2.0 m.s -1 in 1.8 s. (10) The elevator is moving down and slows from 4.0 m.s -1 to 2.0 m.s -1 in 1.8 s.

20 Solution How to approach the problem Identify Setup Execute Evaluate Visualize the situation write down all the given and unknown information. Draw a diagram of the physical situation showing the inertial frame of reference. Type of problem forces and Newton s laws. Draw a free-body diagram showing all the forces acting on the person. Use Newton s 2 nd law to give the relationship between the forces acting on the person and the acceleration of the person. Determine the acceleration of the person in each case. Solve for the unknown quantities. + up is the positive direction m N scale reading: force of scale on person a a > 0 acceleration up a < 0 acceleration down m = 60 kg g = 9.81 m.s -2 G =mg weight of person The person exerts a force on the bathroom scales and the bathroom scales exerts a force on the person. This is an action / reaction pair. But, we are only interested in the forces acting on the person which are the weight and the normal force due to the scale on the person. The scale reading N is found from Newton s 2 nd law: y N G N m( g a) N mg ma acceleration due to gravity g = 9.8 m.s -2 (scalar quantity in this example) acceleration of person a > 0 if direction up and a < 0 if acceleration down The weight of the person is G = mg = (60)(9.81) N =588.6 N We can assume when the velocity changes the acceleration a is constant and equal to the average acceleration aa avg v t In cases (1), (2), (3) and (4) there is no change in the velocity, hence v 0 a 0 N G mg

21 Therefore, the scale reading is N = N or 60 kg. or cases (5), (6) and (10) t 1.8 s v 2 0 m.s 2m.s case (5) -1-1 v 4 2 m.s 2m.s case (6) -1-1 case (10) v The acceleration is m.s 2m.s v 2.0 a m.s 1.11m.s t 1.8 The scale reading is -2-2 m g a N 655N or 67 kg N This scale reading is often called the person s apparent weight. The person feels the floor pushing up harder than when the elevator is stationary or moving with a constant velocity. or cases (7), (8) and (9) t 1.8 s v 2 0 m.s 2m.s case (7) -1-1 case (8) v m.s 2m.s v 2 4 m.s 2m.s case (9) -1-1 The acceleration is v 2.0 a m.s 1.11m.s t 1.8 The scale reading is -2-2 m g a N 522N or 53 kg N The person feels their weight has decreased. In the extreme case when the cable breaks and the elevator and the person are in free-fall and the downward acceleration is a = -g. In this case the normal force of the scales on the person is N = m(g - g) = 0 N. The person seems to be weightless. This is the same as an astronaut orbiting the Earth in a spacecraft where they experience apparent weightlessness. The astronaut and spacecraft are in free-fall and there are zero normal forces acting on the person. The astronaut still has weight because of the gravitational force acting on them.

22 The acceleration does not depend upon the direction of the velocity. What is important is the change in the velocity. A good way to understand this concept is to draw the appropriate motion maps: (1) (2) (3) (4) (5) + a = 0 a = 0 a = 0 a = 0 a > 0 (6) (7) (8) (9) (10) a > 0 a > 0 a < 0 a < 0 a < 0 red arrows: velocity vectors