LEC Newton's Laws of Motion Part I .pdf

Newton’s Laws of Motion  Dynamics Motion Forces  Recall: Particle and Rigid Body Particle is an idealized body that occupies only a single point in space, has mass and has no internal structure. Rigid body is a collection of...

Newton’s Laws of Motion  Dynamics Motion Forces  Recall: Particle and Rigid Body Particle is an idealized body that occupies only a single point in space, has mass and has no internal structure. Rigid body is a collection of n particles linked by a light a rigid framework. w Newton’s Laws of Motion I. Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. II. The change of motion is proportional to the motive force impressed and is made in the direction of the line in which that force is impressed. III. To every action there is always imposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and directed to contrary parts.  Newton’s First Law An object at rest stays at rest unless acted on by an external force. An object in motion continues to travel with constant velocity unless acted on by an external force. !F = 0 The tendency of a body to keep moving once it is set in motion, or the tendency of a body at rest to remain at rest, results from a property called inertia. Constant velocity means zero net force Maserati Granturismo Volkswagen Beetle Constant speed of 250 km/h Constant speed of 75 km/h Which car has the greater net force? When is Newton’s First Law Valid?  Inertial Reference Frame Reference frame: rigid body (e.g., Earth) whose particles can be labeled to create reference points. It is simplified by introducing a coordinate system (e.g., Cartesian). Inertial reference frame: If no forces act on a body, any reference frame with respect to which acceleration of the body remains zero. Newton’s laws of motion are valid only in inertial reference frame (non-accelerating and non-rotating).  Newton’s Second Law A force is a push or a pull. A force is an interaction between two objects or between an object and its environment. A force (F) is a vector quantity, with magnitude and direction. Superposition of Forces Any number of forces applied at a point on a body have the same effect as a single force equal to the vector sum of the forces. This principle is called superposition of forces. R = F$ + F% Any force can be replaced by its component vectors, acting at the same point. F = F& + F' Thus, R = F$ + F% + F( + ⋯ = ∑ F (vector sum of the forces or net force) R & = ∑ F& and R ' = ∑ F'  Example Workmen are trying to free an SUV stuck in the mud. To extricate the vehicle, they use three horizontal ropes, producing the force vectors shown in the figure below. (a) Find the x- and y- components of each three pulls. (b) Use the components to find the magnitude and direction of the resultant of the three pulls.  Example (a) Find the x- and y- components of each three pulls.  Example (b) Use the components to find the magnitude and direction of the resultant of the three pulls.  Newton’s Second Law If a net external force acts on a body, the body accelerates. The direction of acceleration is the same as the direction of the net force. The magnitude of force is the product of the mass of an object and the magnitude of its acceleration F∝a *constant 𝐹⃗ = constant 𝑎⃗ ∑ F = ma  Mass and Acceleration The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed. Recall, ∑ F = ma. Let’s say, F! = F" = F. 𝑚! 𝑎! = 𝑚" 𝑎" 𝑚! 𝑎" = 𝑚" 𝑎! Mass: measures the object’s inertia, in kilogram (kg).  Mass and Weight Weight of an object is the gravitational force that the Earth exerts on it. The weight W of an object of mass m is: w = mg The value of g depends on altitude. On other planets, g will have an entirely different value than on the Earth (9.81 m⁄s ! ). Mass is an intrinsic property of an object which does not depend on its location. Meanwhile, weight is not an intrinsic property of an object since it varies with the location of an object.  Newton’s Second Law of Motion The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object. " 𝐹! = 𝑚𝑎! 𝑑𝑝⃗ ! 𝐹⃑ = = 𝑚𝑎⃑ ⟹ " 𝐹" = 𝑚𝑎" 𝑑𝑡 " 𝐹# = 𝑚𝑎# The SI unit for force is the Newton (N): 1 N = 1 kg · m/s2 System of Units We will use the SI system. In the British system, force is measured in pounds, distance in feet, and mass in slugs. In the CGS system, mass is in grams, distance in centimeters, and force in dynes.  Newton’s Third Law of Motion If you exert a force on a body, the body always exerts a force (the “reaction”) back upon you. Figure below shows “an action-reaction pair.” A force and its reaction force have the same magnitude but in opposite directions. These collinear forces act on different bodies. 𝐹⃗' )* + = −𝐹⃗+ )* '  Contact Forces Involves direct contact between two bodies. Reaction forces Restoring force Retarding forces Applied forces Normal Coulomb Stokes Newtonian force Spring force Push friction frictional force frictional force Tensional force Static Sliding Pull friction friction Buoyant force Rolling friction  Force exerted on an object by any surface with which it is Normal Force in contact. Normal means that the force always act ⊥ to the contact surface, regardless of the surface angle.  A pulling force exerted on an object by ideal strings Tension Force (massless, frictionless, unbreakable, and inextensible) and is always measured ∥ to the string on which it applies. In a string or a chain, tension is only In a rod or a stick, extensional. tension can be extensional or compressional or both.  Force exerted on an object by a surface acts ∥ to the Frictional Force surface, in the direction that opposes sliding, or the tendency to slide. Friction between two surfaces arises from When a body rests or slides on a surface, interactions between molecules on the the friction force is parallel to the surface. surfaces  Static Friction Static friction (⃗f) ) acts when there is no relative motion between bodies. The static friction force can vary between zero and its maximum value (⃗f),+,& ), depending on how hard you push an object. By definition, ⃗f),+,& = µ) n In general, ⃗f) ≤ µ) n Laws of Limiting friction (maximum static friction, f),+,& ) 1. The magnitude of limiting frictional force is proportional to the normal force at the contact surface. 2. The magnitude of limiting frictional force is independent of area of contact between the surfaces.  Kinetic (Sliding) Friction Kinetic friction (⃗f- ) acts when a body slides over a surface. The coefficient of kinetic friction µ- is the ratio of the magnitudes of the ⃗f- and the n. ⃗f- = µ- n Experimentally, 𝜇. < 𝜇/ Rolling friction (⃗f0 ) is the opposing force that comes into existence when one object rolls over the surface of another object. The coefficient of rolling friction µ0 is the ratio of the magnitudes of the ⃗f0 and the n. ⃗f0 = µ0 n µ$ = 0.01 to 0.02 (rubber tires on concrete) and 0.001 to 0.002 (steel wheels on steel rails) Approximate Values of Frictional Coefficients Materials 𝝁𝒔 𝝁𝒌 Materials 𝝁𝒔 𝝁𝒌 Steel on steel 0.74 0.57 Glass on glass 0.94 0.40 Aluminum on steel 0.61 0.47 Copper on glass 0.68 0.53 Copper on steel 0.53 0.36 Teflon on Teflon 0.04 0.04 Brass on steel 0.51 0.44 Teflon on steel 0.04 0.04 Zinc on cast iron 0.85 0.21 Rubber on dry 1.0 0.8 concrete Copper on cast iron 1.05 0.29 Rubber on wet 0.30 0.25 concrete  Drag Force or Retarding Force Drag force (𝐟⃗𝐃 ) or retarding force on a body depends on the shape of the object, the properties of the fluid and the speed of the object relative to the fluid. ⃗f2 = bv 3 ⃗f2 = Dv % (gas (air); higher speed) ⃗f2 = kv (liquid; low speed) A falling body reaches its terminal speed, v4 when the resisting force equals the weight of the body. bv43 = mg mg $/3 v4 = ( ) b  Free-Body Diagram (FBD) Free-Body Diagram (FBD) A single body or a subsystem of bodies isolated from its surroundings showing all the external forces acting on it is its free body diagram. Steps for Free-Body Diagram Step 1: Identify the object or system and isolate it from other objects clearly, specify its boundary. Step 2: First draw non-contact external force in the diagram. Generally, it is weight. Step 3: Draw contact forces which acts at the boundary of the object or system. Contact forces are normal, friction, tension and applied force.  Examples of FBD (1) The dog in front pulls on a rope attached to the sled with a horizontal force causing the sled to gain speed. (2) A book is at rest on a tabletop.  Example of FBD (3) A gymnast holding onto a bar, is suspended motionless in mid-air. The bar is supported by two ropes that attach to the ceiling. (4) A skydiver is descending with a constant velocity. Consider air resistance.  Examples of FBD (5) A block having 𝑚$ sits at rest on a horizontal surface. A second block 𝑚% sits on top of the first block. (6) A crate having mass 𝑚$ sits on a frictionless incline plane that makes an angle of 30° with the horizontal. A rope attached to 𝑚$ passes over a pulley at the top of the incline and has a second mass 𝑚% attached to the other end. FBD: Activity F,66 FBD: Activity FBD: Activity  References 1. Luna, Reynold V. (2018). Lecture Slides on Newtonian Mechanics. Polytechnic University of the Philippines. Date Retrieved: April 2021 2. Chow, T. (2013) Classical Mechanics, 2e, Taylor & Francis Group, LLC, CRC Press 3. Kleppner, D. and Kolenkow, R. (2010), An Introduction to Mechanics, Cambridge University Press. 4. Strauch, D. (2009), Classical Mechanics, Springer-Verlag Berlin Heidelberg. 5. Gregory, D. (2006) Classical Mechanics: An Undergraduate Text, Cambridge University Press 6. Fowles, G. and Cassiday G. (2005) Analytical Mechanics, 7e, Brooks/Cole Thomson Learning 7. Thornton, S. and Marion J. (2004) Classical Dynamics of Particles and Systems 5e, Brooks/Cole Thomson Learning 8. Young, H., Freedman, R. and Ford, A. (2016) University Physics with Modern Physics, 14e, Pearson “Don’t let what you cannot do interfere with what you can do.” © John Wooden