Chapter 4 The Laws of Motion “If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent.” -Sir Isaac Newton Classical Mechanics Describes the relationship between the motion of objects in our everyday world and the forces acting on them Conditions when Classical Mechanics does not apply very tiny objects (< atomic sizes) objects moving near the speed of light Forces Force is a push or pull on an object Force is Vector quantity Force can be a contact force or a field force Contact forces result from physical contact between two objects Field forces act between disconnected objects Also called “action at a distance” Examples of this force are A.Contact force I. Pulling a spring II. Kicking a ball III. Pushing a wheelbar B.Field force I. Elctric force i.e force of charge II. Gravitation force i.e force which cause an object to move down III. Force of magnetic bar Contact and Field Forces Fundamental Forces Types of Forces Strong nuclear force-between sub atomic particles Electromagnetic force-between electric charges Weak nuclear force -arises in certain radioactive decay processes Gravity between objects Classical physics, however, deals only with gravitational and electromagnetic forces, which have infinite range. Units of Force SI unit of force is a Newton (N) kg m 1N 1 2 s Newton’s First Law An object at rest tends to stay at rest and an object in motion tends to stay in motion at a constant velocity unless acted upon by an unbalanced force. It takes force to change the motion of an object. The net force is defined as the vector sum of all the external forces exerted on the object First Law Example A soccer ball is sitting at rest. It takes an unbalanced force of a kick to change its motion. External and Internal Forces External force Any force that results from the interaction between the object and its environment Internal forces Forces that originate within the object itself They cannot change the object’s velocity Inertia Is the tendency of an object to continue in its original state of motion The First Law states that all objects have inertia. Mass A measure of the resistance of an object to changes in its motion due to a force The more mass an object has, the more inertia it has (and the harder it is to change its motion). Scalar quantity SI unit is kilogram (kg) Newton’s Second Law Acceleration of an object is direct proportion to its force and inversely to its mass A= 𝒎 𝒇 The net force of an object is equal to the product of its mass and acceleration F = ma F and a are both vectors When mass is in kilograms and acceleration is in m/s/s, the unit of force is in newtons (N). One newton is equal to the force required to accelerate one kilogram of mass at one meter/second/second. Second Law Cont. F = ma basically means that the force of an object comes from its mass (m) and its acceleration (a). Something very massive (high mass) that’s changing speed very slowly (low acceleration), like a glacier, can still have great force. Second Law Cont. Something very small (low mass) that’s changing speed very quickly (high acceleration), like a bullet, can still have a great force. Something very small changing speed very slowly will have a very weak force. Sir Isaac Newton 1642 – 1727 Formulated basic concepts and laws of mechanics Universal Gravitation Calculus Light and optics Gravitational Force Mutual force of attraction between any two objects in the universe. Expressed by Newton’s Law of Universal Gravitation: m1 m2 Fg G 2 r where G = 6.67x10‒11 N. m2/kg2 is the universal gravitation constant Gravitational Force Cont. Newton’s law of universal gravitation states that every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. Weight The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the object w = mg is a special case of Newton’s Second Law g is the acceleration due to gravity g can also be found from the Law of Universal Gravitation Weight Cont. We know that 𝑤 = 𝑚𝑔 And 𝑀𝐸 𝑚 𝑤=𝐺 2 𝑟 Comparing the two equations, we get: 𝑀𝐸 𝑔=𝐺 2 𝑟 Where 𝑀𝐸 is the mass of the Earth. More about weight Weight is not an inherent property of an object mass is an inherent property Weight depends upon location Newton’s Third Law For every action force acting on body 2 by body 1 there is an equal and opposite reaction force acting on body 1 by body 2. F12 F21 Equivalent to saying a single isolated force cannot exist Newton’s Third Law cont. F12 may be called the action force and F21 the reaction force Actually, either force can be the action or the reaction force The action and reaction forces act on different objects Third Law Example The reaction of a rocket is an application of the third law of motion. Various fuels are burned in the engine, producing hot gases. The hot gases push against the inside tube of the rocket and escape out the bottom of the tube. As the gases move downward, the rocket moves in the opposite direction. Action-Reaction Pairs n and n ' n is the normal force, the force the table exerts on the TV n is always perpendicular to the surface n 'is the reaction – the TV on the table n n ' Action-Reaction pairs Cont. Fg and Fg' F is the force the g Earth exerts on the object ' F g is the force the object exerts on the earth Fg Fg' Forces Acting on an Object Newton’s Law uses the forces acting on an object n and Fg are acting on the object ' n ' and Fgare acting on other objects Applications of Newton’s Laws Assumptions Objects behave as particles can ignore rotational motion (for now) Masses of strings or ropes are negligible Interested only in the forces acting on the object can neglect reaction forces Free Body Diagram Must identify all the forces acting on the object of interest Choose an appropriate coordinate system If the free body diagram is incorrect, the solution will likely be incorrect Free Body Diagram, Example The force is the tension acting on the box The tension is the same at all points along the rope n and Fg are the forces exerted by the earth and the ground Free Body Diagram, final Only forces acting directly on the object are included in the free body diagram Reaction forces act on other objects and so are not included The reaction forces do not directly influence the object’s motion Solving Newton’s Second Law Problems Read the problem at least once Draw a picture of the system Identify the object of primary interest Indicate forces with arrows Label each force Use labels that bring to mind the physical quantity involved Solving Newton’s Second Law Problems Draw a free body diagram Apply Newton’s Second Law If additional objects are involved, draw separate free body diagrams for each object Choose a convenient coordinate system for each object The x- and y-components should be taken from the vector equation and written separately Solve for the unknown(s) Equilibrium An object either at rest or moving with a constant velocity is said to be in equilibrium The net force acting on the object is zero (since the acceleration is zero) F 0 Equilibrium cont. Easier to work with the equation in terms of its components: F x 0 and F y 0 This could be extended to three dimensions Equilibrium Example A traffic light weighing 1.00x102 N hangs from a vertical cable tied to two other cables that are fastened to a support, as in Figure 4.14a. The upper cables make angles of 37.0° and 53.0° with the horizontal. Find the tension in each of the three cables. Equilibrium Example – Free Body Diagrams Inclined Planes Choose the coordinate system with x along the incline and y perpendicular to the incline Replace the force of gravity with its components Inclined Planes Example A sled is tied to a tree on a frictionless, snow-covered hill, as shown in Figure 4.15a. If the sled weighs 77.0 N, find the magnitude of the tension force T : exerted by the rope on the sled and that of the normal force n: exerted by the hill on the sled. Multiple Objects When you have more than one object, the problem-solving strategy is applied to each object Draw free body diagrams for each object Apply Newton’s Laws to each object Solve the equations Multiple Objects – cont. Forces of Friction When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion This is due to the interactions between the object and its environment This resistance is called friction That is resistance force of an More About Friction Friction is proportional to the normal force The force of static friction is generally greater than the force of kinetic friction (Rolling, Sliding & Fluid friction) The coefficient of friction (µ) depends on the surfaces in contact The direction of the frictional force is opposite the direction of motion The coefficients of friction are nearly independent of the area of contact Static Friction, ƒs Static friction acts to keep the object from moving If F increases, so does ƒs If F decreases, so does ƒs ƒs µsn At the verge of slipping ƒs is maximum Kinetic Friction, ƒk The force of kinetic friction acts when the object is in motion ƒ k = µk n Variations of the coefficient with speed will be ignored Block on a Ramp, Example Axes are rotated as usual on an incline The direction of impending motion would be down the plane Friction acts up the plane Opposes the motion Apply Newton’s Laws and solve equations Connected Objects Apply Newton’s Laws separately to each object The magnitude of the acceleration of both objects will be the same The tension is the same in each diagram Solve the simultaneous equations Connected Objects Cont. More About Connected Objects Treating the system as one object allows an alternative method or a check Use only external forces Not the tension – it’s internal The mass is the mass of the system Doesn’t tell you anything about any internal forces e.g Tension force Connected Objects –Problem1 A 5.00-kg object placed on a frictionless, horizontal table is connected to a string that passes over a pulley and then is fastened to a hanging 9.00-kg object, as in the Figure below. Draw free-body diagrams of both objects. Find the acceleration of the two objects and the tension in the string. Connected Objects-Problem2 (a) A block with mass m1=4.00 kg and a ball with mass m2 =7.00 kg are connected by a light string that passes over a frictionless pulley, as shown above. The coefficient of kinetic friction between the block and the surface is 0.300. Find the acceleration of the two objects and the tension in the string. (b) Check the answer for the acceleration by using the system approach.