Physics For Engineers E-notes

LESSON 1: VECTORS GRAPHICAL ADDITION AND SUBTRACTION OF TWO OR MORE VECTORS USING CLOSE POLYGON METHOD PHYSICAL QUANTITIES • • Involved in explaining the existence of physical phenomenon. It is all around us, example is the distance we travel every time work is done. SCALAR QUANTITY VECTOR QUANTITY • Quantity that can be specified by a single number (magnitude) with unit. • It requires not just a number and a unit (magnitude), but also direction. • Examples are Length, Area, Distance, Time, Energy, Work, Power, and Speed. • Examples are Force, Velocity, Displacement, Acceleration, and Momentum. RESULTANT VECTORS • • • While scalar quantities obey ordinary laws of addition and subtraction, vectors on the other hand are manipulated following the rules of vector algebra. In vectors, addition and subtraction is treated in special form, wherein the sum of two or more vectors is represented by a single vector called “Resultant.” The resultant vector may be found using “Graphical Method” and “Analytical Method”. Right Triangle Oblique Triangle Two or More Vectors • • ANALYTICAL ADDITION OF VECTORS (RIGHT TRIANGLE) • • • • Component Method • • • Arrowhead Tip – represents the direction of vectors. Arrows Length - represents the vector’s magnitude. Tail - represents the point of application. Scale – represents the actual scale of vectors. GRAPHICAL ADDITION OF VECTORS • • • • This method begins in beginning at any convenient point and drawing each vector arrow in turn. The order may be taken in any order since vector addition is commutative and associative. To add them, draw the first vector and draw the second vector with its tail connected to the tip of the first vector. After that, draw a line from the tip of last vector to the origin and then measure the resultant vector. Use scale factor to represent magnitude of vectors and converting the length of the resultant vector to its magnitude. GRAPHICAL ADDITION OF TWO VECTORS • • As the angle between two concurrent vectors increases, the magnitude of resultant vector increases too. You can use parallelogram method wherein you will draw an imaginary line to complete the four sides and then draw a line from origin to the other corner, that line is your resultant vector. Sum of two vectors is determined using Law of Sines and Law of Cosines. Use Law of Sines when there is only one side missing and you have one side and two angles given. Use Law of Cosines when you have (SAS) to find the missing side or (SSS) to find the missing angle. ANALYTICAL ADDITION OF TWO OR MORE VECTORS (COMPONENT METHOD) • • • • GRAPHICAL REPRESENTATION OF VECTORS • • • • Sum of the two vectors is determined using Trigonometric Functions and Pythagorean Theorem. For Trigonometric Functions, use SOH-CAH-TOA For Pythagorean Theorem, use a2 = b2 + c2 ANALYTICAL ADDITION OF VECTORS (OBLIQUE TRIANGLE) • Pythagorean Theorem • Trigonometric Functions • Law of Sines • Law of Cosines To add vectors, draw the forces simultaneously while keeping the tip and the tail connected, after connecting all the forces, make a line from the origin to the tip of the last force, that line is the resultant force. To subtract vectors, the procedure is the same for addition, but the difference is that the force being subtracted by change from it the direction to the opposite, meaning when the force is on to the east, the drawing must be on the west. • It is also known as “Rectangular Resolution Method”. Step 1: Find the X and Y component of the given vectors. Step 2: Get the summation of forces on X and Y axes. Step 3: Find the resultant force using Pythagorean Theorem. Step 4: Use arctan (Summation of Forces on Y divided by Summation of Forces on X) to get the angle. EQUILIBRANT FORCE • • • It happens when a body has two or more forces acting on it and it does not move because it’s balance. It is a force that brings a body into mechanical equilibrium. It is equal in magnitude and opposite in direction to the resultant of all other forces acting on a body. SYSTEM OF VECTORS COPLANAR COPLANAR FORCE Refers to the number of forces which remains in single plane. NON-COPLANAR FORCE Refers to the forces that acts at a different plane. CONCURRENT CONCURRENT FORCE SYSTEM Set of two or more forces whose line of action intersects at a point at the same time. NON-CONCURRENT FORCE SYSTEM Forces do not meet at one point of intersection while acting on the same object. PARALLEL NEWTON’S FIRST LAW OF MOTION • • PARALLEL FORCE SYSTEM It happens when the line of action of forces forming the system are parallel. Can be in the same or opposite direction. • NON-PARALLEL FORCE SYSTEM It happens when the line of action of forces forming the system does not form parallel system. • • COLLINEAR COLLINEAR FORCE SYSTEM INERTIA These are forces which acting along a straight line. • NON-COLLINEAR FORCE SYSTEM These are forces which does not act along a straight line. • • • 𝑅 = ∑𝐹 = √(∑𝐹𝑋 2 + (∑𝐹𝑦 ) + (∑𝐹𝑍 )2 LESSON 2: STATIC EQUILIBRIUM • • • Equilibrium is a fundamental concept when dealing with forces. If the size and direction of forces acting on a body are balanced, then there is no net force, and the object is in the state of equilibrium. Static Equilibrium is a state of equilibrium in which net external force and torque acting on a system is zero, while Dynamic Equilibrium is a state of equilibrium in which net external force and torque acting on a moving system with constant velocity is zero. FORCE • • It is a push or pull, which produces motion and tends to stop motion. Forces can be contact or non-contact. o Contact Force ▪ Applied Force ▪ Drag Force ▪ Spring Force ▪ Frictional Force ▪ Normal Force. o Non-Contact Force ▪ Gravitational Force ▪ Electric Force ▪ Magnetic Force • One of the more common events in our surroundings. It is the change in objects position. • • It is to what an observer compares the position of an object. For an observer to say that an object has moved, he must view object with respect to another object. In physics and engineering, and FBD is a graphical illustration used to visualize the applied forces, movements, and resulting reactions on a body in a given condition. NORMAL FORCE (FN) • • It is the force exerted by one surface on another in contact with it. It is always perpendicular to the contact area. WEIGHT (W) • • • • It is the force of gravity acting on an object due to its mass. Formula: W = mg The normal force on the flat surface is equal to the weight. The normal force on an inclined plane is given by: N = W cos (theta) = mg cos (theta) TENSION FORCE • • • FRAME OF REFERENCE • It states that whenever one body exerts a force on the second body, the second body exerts an opposite directed force of equal magnitude on the first body. In simple terms, for every action, there is an equal but opposite reaction. FREE BODY DIAGRAM MOTION • • It is the tendency of an object to remain in the same state of rest or motion. The inertia of the body is closely related to its mass. The larger the mass, the larger the inertia. NEWTON’S THIRD LAW OF MOTION For the Concurrent Force System and Non-Coplanar Force System, Resultant Force can be determined by: )2 It is also called as Law of Inertia. The law of inertia states that, if the forces acting upon an object are balanced, then the acceleration of an object will be 0 m/s2. Objects at equilibrium will not accelerate, it will only accelerate if there is unbalanced force acting upon it. The presence of unbalanced force may change its speed, direction, or both. In other words, if the body stays at rest, it will stay at rest. If it is moving, it will continue to move with the same velocity. Defined as the force that is transmitted through a rope, string, or wire when pulled by forces acting from opposite sides. Tension force is directed over the length of the wire and pulls energy equally on the bodies at the ends. The cord and pulley are often said to be massless, stretchable, and frictionless. COMPRESSION FORCE • It is an internal reaction of a rigid body to an external pushing force being exerted on the body opposing the tension force. FRICTIONAL FORCE • • It refers to an actual force that arises to oppose relative motion between contacting forces. It is in the opposite direction of the intended motion. STATIC FRICTION FORCE • Force exerted on the surface by another when there is no motion between two surfaces. This makes a heavy box start to move on a rough surface. This friction acts in response to a force trying a stationary object to move. If there is no force parallel to the ground and no tendency to move, the static friction force is zero. Fsmax = us FN Coefficient of Static Friction is the ratio of the magnitude of two forces static friction force and normal force and is unit less. If the magnitude of the component F that is parallel to the surface exceeds Fsmax, then the body begins to slide along the surface. • • • • • • MOMENT ARM • • • CENTER OF GRAVITY (CG) • • • • • Also known as dynamic or sliding friction. It is a force that acts between moving surfaces. Fkmax = uk FN If the body begins to slide along the surface, the magnitude of frictional force rapidly decreases. FIRST CONDITION OF EQUILIBRIUM • • It states that if a body is in equilibrium, the sum of forces acting on it must be ZERO. The first condition involves only forces and is therefore independent of where the origin of the reference frame is placed. ∑𝐹 = 0 ; ∑𝐹𝑥 = 0 ; ∑𝐹𝑦 = 0 ; ∑𝐹𝑧 = 0 TORQUE (ROTATIONAL EQUILIBRIUM) • To complete satisfy an object in mechanical equilibrium, the following conditions must meet: o Based on the observer, if the object is at rest, it will remain at rest. If the object is moving, it will remain moving. ∑𝐹 = 0 ; ∑𝐹𝑥 = 0 ; ∑𝐹𝑦 = 0 o If an object is in rotational equilibrium, the net torque acting on it about any axis must be zero. ∑𝑇 = 0 • • Torque is the tendency of a force to cause an object to rotate or turn. It is the product of the magnitude of force and moment arm or lever arm. T=Fxl o o o T = Torque F = Force l = Moment Arm It is the force of gravity on solid can be considered to act at a single point. It is the point at which the weight can be considered to act when calculating the torque due to the weight. It is also the point where weight is concentrated. If the body is symmetrical, its CG is in its geometric center. If the body is unsymmetrical, its CG is closer to the more massive part of the body. FINDING CENTER OF GRAVITY A. KINETIC FRICTION FORCE • • • • It is also called as LEVER ARM. It is the perpendicular distance from the axis of rotation to the line of action of the force. The torque is POSITIVE if the force tends to rotate the object CCW about the origin and NEGATIVE if it tends to rotate the object CW. B. PLUMB LINE METHOD • Suspend the object from any two close locations and drop plumb lines from the suspension points. The intersection of the two lines is the center of the mass. ANALYTICAL METHOD • It uses torque and equilibrium concepts.