Mechanics and Motion: Kinematics and Dynamics of Particles, Lecture notes of Physics

Download Mechanics and Motion: Kinematics and Dynamics of Particles and more Lecture notes Physics in PDF only on Docsity! CHAPTER TWO KINEMATICS AND DYNAMICS OF PARTICLES 1 2  Mechanics ;-is the study of the physics of motions and how it relates to the physical factors that affect them, like force, mass, momentum and energy.  Dynamics ;- which deals with the motion of objects with its cause – force;  kinematics ;- describes the possible motions of a body or system of bodies without considering the cause.  Alternatively, mechanics may be divided according to the kind of system studied.  The simplest mechanical system ;-is the particle, defined as a body so small that its shape and internal structure are of no consequence in the given problem.  More complicated ;- is the motion of a system of two or more particles that exert forces on one another. Average and Instantaneous Accelerations If the velocity of a particle changes with time, then the particle is said to be accelerating. Average acceleration: is the change in velocity (∆𝑣 ) of an object divided by the time interval during which that change occurs. av𝑎 = ∆𝑣 ∆𝑡 𝑡𝑓−𝑡𝑖 ……………………………….2.1.5 Instantaneous acceleration: -The limit of average acceleration as △t approaches zero. 5 2.1.2. Motion with Constant Acceleration For motion with constant acceleration,  The velocity changes at the same rate throughout the motion.  Average acceleration over any time interval is equal to the instantaneous instant of time. acceleration at any 𝑎 = ∆𝑣 =∆𝑡 𝑡 𝑣𝑓 −𝑣𝑖 i, assuming t = 0 By definition, 𝑣 𝑎𝑣 = ∆𝑟 ∆𝑡 for t=0 then, 𝑎𝑣∆𝑟 = 𝑣 t Rearranging this equation gives, 𝑣 = 𝑣 𝑖 + 𝑎 𝑡  For motion with constant acceleration, average velocity can be written as: 𝑎𝑣𝑣 = 𝑣𝑓+ 𝑣𝑖 2 1 2 𝑟 _f − 𝑟 𝑖 = 𝑣 𝑖𝑡 + 𝑎 𝑡2 6 𝑎𝑣  Again, ∆𝑟 = 𝑣 t 𝑎𝑣but, 𝑣 = 𝑣𝑓+ 𝑣𝑖 2 and 𝑡 = 𝑣𝑓−𝑣𝑖 𝑎 after substituting,  For 2D motion, 𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗, 𝑣 𝑓 = 𝑣𝑥𝑓𝑖 + 𝑣𝑦𝑓𝑗, 𝑣 𝑖 = 𝑣𝑥𝑖 + 𝑣𝑦𝑖𝑗 ∆𝑟 = 𝑣𝑓+𝑣𝑖 𝑣𝑓−𝑣𝑖 2 𝑎 𝑣𝑓 2 = 𝑣𝑖2 + 2𝑎∆𝑟 7 10 2.1.3. Free Fall Motion  The motion of an object near the surface of the Earth under the only control of the force of gravity is called free fall.  In the absence of air resistance, all objects fall with constant acceleration, g, toward the surface of the Earth.  The acceleration due to gravity varies with latitude, longitude and altitude on Earth„s surface. 2.1.4. Projectile Motion • Projectile is any object thrown obliquely into the space. • The object which is given an initial velocity and afterwards follows a path determined by the gravitational force acting on it is called projectile and the motion is called projectile motion. • A stone projected at an angle, • a bomb released from an aero plane, • a shot fired from a gun, • a shot put or javelin thrown by the athlete are examples for the projectile.  Consider a body projected from a point 'O' with velocity 'u'.  The point 'o' is called point of projection and 'u' is called velocity of projection.  Velocity of Projection (u): the velocity with which the body projected.  Angle of Projection (𝜽): The angle between the direction of projection and the horizontal plane passing through the point of projection is called angle of projection.  Trajectory (OAB): The path described by the projectile from the point of projection to the point where the projectile reaches the horizontal plane passing through the point of projection is called trajectory.  The trajectory of the projectile is a parabola. 11  For projectile motion ay = -g ax= 0 (Because there is no force acting horizontally) 12 Home Activities 1. A ball is thrown with an initial velocity of 𝑢 = (10i+15j ̂) m/s. When it reaches the top of its trajectory, neglecting air resistance, what is its a) velocity? b) Acceleration? 2. An astronaut on a strange planet can jump a maximum horizontal distance of 15m if his initial speed is 3m/s. What is the free fall acceleration on the planet? 15 16 2.2. Particle Dynamics and Planetary Motion 2.2.1. The Concept of Force as A Measure of Interaction  In physics, any of the four basic forces; gravitational, electromagnetic, strong nuclear and weak forces govern how particles interact.  All other forces of nature can be traced to these fundamental interactions. The fundamental interactions are characterized on the basis of the following four criteria: o The types of particles that experience the force, o The relative strength of the force, o The range over which the force is effective, and o The nature of the particles that mediate the force. 17 2.2.2. Types of Forces  Forces are usually categorized as contact and non-contact. i) Contact Force  It is a type of force that requires bodily contact with another object. And it is further divided into the following. 1. Muscular force ; exists only when it is in contact with an object. 2. Frictional Forces ; is the resisting force that exists when an object is moved or move on a surface. 3. Normal Force ; 4. Applied Force ; is a force that is applied to a person or object. 5. Tension Force ; Tension is the force applied by a fully stretched cable or wire on to an object. 6. Spring Force ; is Force exerted by a compressed or stretched spring. tries to anchored 7. Air Resisting Force ; is wherein objects experience a frictional force when moving through the air. 20 Newton's Second law of Motion:  The acceleration acquired by a point particle is directly proportional to the net force acting on the particle and inversely proportional to its mass and the acceleration is always in the direction of the net force.  Mathematically, Σ F = ma Newton's Third law of Motion:  States that ; “For every action there is always an equal and opposite reaction.” FBA = - FAB or FAB + FBA = 0 Note that:  Action and reaction forces are always exist in pair.  A single isolated force cannot exist.  Action and reaction forces act on different objects. 21  Frictional force refers to the force generated by two surfaces that are in contact and either at rest or slide against each other.  These forces are mainly affected by the surface texture and amount of force impelling them together.  The angle and position of the object affect the amount of frictional force.  If an object is placed on a horizontal surface against another object, then the frictional force will be equal to the weight of the object.  If an object is pushed against the surface, then the frictional force will be increased and becomes more than the weight of the object.  Generally friction force is always proportional to the normal force between the two interacting surfaces.  Mathematically ; Ffrict ∝ Fnorm Ff = μFN……………………………………………….(2.2.2) Where ; μ- is the coefficient of friction. Forces of Friction 22  frictional forces have two types; a. Static friction: exists between two stationary objects in contact to each other. Mathematically ; 𝑓𝑠 = 𝜇𝑠𝑁 b. Kinetic friction: arises when the object is in motion on the surface. 𝑓𝑘 = 𝜇𝑘𝑁 Where 𝜇𝑘 - is called the coefficient of kinetic friction. The values of 𝜇𝑘 and 𝜇𝑠 depend on the nature of the surfaces, but 𝝁 𝒌 is generally less than 𝝁𝒔 .  The coefficients of friction are nearly independent of the area of contact between the surfaces. Example A 25.0-kg block is initially at rest on a horizontal surface. A horizontal force of 75.0 N is required to set the block in motion. After it is in motion, a horizontal force of 60.0 N is required to keep the block moving with constant speed. Find the coefficients of static and kinetic friction from this information. Example A bag of cement of weight 300 N hangs from three ropes as shown in the figure below. Two of the ropes make angles of 𝜃1 = 53.0° and 𝜇2 = 37.0° with the horizontal. If the system is in equilibrium, find the tensions T1, T2, and T3 in the ropes. 25 Solution We can draw two free body diagrams jor the problem az follows a) Ty TysinS3°TsTs al « t 370 Tpein3 7) Typcos53°Tzc0s37°= — Since the system is in equiliwium EF =o > Enno From free body diagram (a) © ER, = Tzc0s37"9 — Tycos53° = oO O.8T, = 0.6 T,(a) Sa= T,sin53° + T,sin37"—T, = 0 0.8 T, + 0.6 T, = T3(b) From free body diagram (b) EF, =T3 —W=0 >T; = W = 300AN(c) Substituting(c) im(b) O87, + O67, = 300N (da) But 0.8T,=06T, — 72 = = O.75T, Substituting for Tin (d) Gives 0.8 T, + 0.6(0.757,) = 300N T, =240N Tz = 0.757, = 0.75(240N) T, = 180N 26 2.2.4. Uniform Circular Motion  Uniform Circular Motion is motion of objects in a circular path with a constant speed. Objects moving in a circular path with a constant speed can have acceleration. 𝑎 = ∆𝑣 ∆𝑡  There are two ways in which the acceleration can occur due to:  change in magnitude of the velocity  change in direction of the velocity.  For objects moving in a circular path with a constant speed, acceleration arises because of the change in direction of the velocity.  Hence, in case of uniform circular motion:  Velocity is always tangent to the circular path and perpendicular to the radius of the circular path.  Acceleration is always perpendicular to the circular path, and points towards the center of the circle. Such acceleration is called the centripetal acceleration . 27  The weight of an object mg is the gravitational force between it and Earth. Substituting mg for F in Newton„s universal law of gravitation gives; 𝑚 𝑔 = 𝐺 𝑚 𝑀 𝑟2 It gives, 𝑔 = 𝐺 𝑀 𝑟2  Substituting known values for Earth‘s mass and radius,  And we obtain a value for the acceleration of a falling body: g = 9.8m/s2 30 2.2.6. Kepler’s Laws, Satellites Motion and Weightlessness  The points F1 and F2 represented in figure are known as the foci of the ellipse.  Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.  The basic laws of planetary motion were established by Johannes Kepler (1571-1630) based on the analysis of astronomical observations of Tycho Brahe (1546−1601).  In 1609, Kepler formulated the first two laws. The third law was discovered in 1619. Kepler’s First Law (Law of Orbits)  States that, “The orbit of each planet in the solar system is an ellipse, the Sun will be on one focus” 31 Kepler’s Second Law (The Law of Areas)  States that ; “the radius vector connecting the centers of the Sun and the Planet sweepsout equal areas in equal intervals of time”  A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun.  Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. 32 35 NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun - 1.4957 x 1011 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x 107 seconds.  Weightlessness is the complete or near-complete absence of the sensation of weight. Or,  Weightlessness is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it.  Weightless sensations exist when all contact forces are removed.  These sensations are common to any situation in which you are in a state of free fall. Satellite motion and Weightlessness 36 2.3. Work, Energy and Linear Momentum  Work:- can be defined as transfer of energy due to an applied force.  In physics, work is done when a force acts on an object that undergoes a displacement from one position to another.  for work to be done on an object, three essential conditions should be satisfied: 1. Force must be exerted on the object 2. The force must cause a motion or displacement. 3. The force should have a component along the line of displacement  If a particle subjected to a constant force 𝐹 undergoes a certain displacement, ∆𝑟 , the work done W by the force is given by: 𝑊 = 𝐹 . ∆𝑟 = 𝐹 |∆𝑟 |𝑐𝑜𝑠𝜃 Where 𝜇 𝑖𝑠 the angle between 𝐹 𝑎𝑛𝑑 ∆𝑟 .  The work done by the applied force is positive when the projection of 𝐹 onto is in the same direction as ∆𝑟 . 37  Energy:- is defined as the capacity of a physical system to perform work. And it exists in several forms such kinetic, potential, thermal, chemical and other forms. And its SI unit is joule (J).  Power:- 40 2.3.3. Linear Momentum  Momentum is defined as the quality of a moving object to exert a force on anything that tries to stop it.  The linear momentum of a particle or an object that can be modeled as a particle of mass m moving with a velocity 𝑣 is defined to be the product of its mass and velocity: 𝑝 = 𝑚𝑣  Using Newton„s second law of motion, we can relate the linear momentum of a particle to the resultant force acting on the particle ,  Therefore, the time rate of change of the linear momentum of a particle is equal to the net force acting on the particle, ∆𝑝 = 𝐹 ∆𝑡 𝐼 𝐼𝑚𝑝𝑢𝑙𝑠𝑒 = ∆𝑝 = 𝑝 𝑓 − 𝑝 𝑖 = 𝐹 ∆𝑡  The impulse of the net force 𝐹 acting on the particle is equal to the change in momentum of the particle. 41 Conservation of Linear Momentum  It says that:- “Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant”. Let:  𝐹 21 = 𝑡𝜇𝑒 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑚1 𝑓𝑟𝑜𝑚 𝑚2  𝐹 12 = 𝑡𝜇𝑒 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑚2 𝑓𝑟𝑜𝑚 𝑚1  Then, in symbols, Newton‟s third law says,  Where, 𝑝 1𝑖 and 𝑝 2𝑖 𝑎𝑟𝑒 the initial values and 𝑝 1𝑓and 𝑝 2𝑓 the final values of the momenta for the two particles for the time interval during which the particles interact. 42 2.3.4. Collisions ? What is the difference between elastic, inelastic and perfectly inelastic collision?  Whether or not kinetic energy is conserved is used to classify collisions as either elastic or inelastic. A. Elastic collision: - An elastic collision between two objects is one in which the total kinetic energy as well as total momentum of the system is conserved. If the collision is elastic, both the momentum and kinetic energy of the system are conserved, Perfectly elastic collisions occur between atomic and subatomic particles. 45 B. Inelastic Collision: - An inelastic collision is one in which the total kinetic energy of the system is not conserved. But the momentum of the system is conserved. Therefore, for inelastic collision of two particles: C. Perfectly Inelastic Collision: -When the colliding objects stick together after the collision, the collision is called perfectly inelastic.  The two particles collide head-on, stick together, and then move with some common velocity v after the collision.  The total momentum before the collision equals the total momentum of the composite system after the collision . m1v1i + m2v2i = (m1+m2)vf It gives, 𝑣𝑓 = 𝑚1𝑣1𝑖+𝑚2𝑣2𝑖𝑚 +𝑚 1 2 46 Example An archer shoots an arrow toward a target that is sliding toward her with a speed of 2.50 m/s on a smooth surface. The 22.5-g arrow is shot with a speed of 35.0 m/s and passes through the 300-g target, which is stopped by the impact. What is the speed of the arrow after passing through the target? 47 Therefore, 𝒄 � � 𝒊 𝒎𝒊𝒓 𝒊 𝒓 = � � 50 Example A system consists of three particles with masses m1=m2= 1.0Kg and m3=2.0Kg and located as shown in the figure below. Find the center of mass of the system. 51 Solution MX, + IMNyX, + MG Xy Xen = ml, + Mm, + M1, oe (1kg)Qm) + (1kg)(2m) + (2kg)(Om) ou ikg + ikg + 2kg Xen = 0.75m _ my, + Myy2 + Msg my +m, +m, _ (kg)(Om) + (1kg)(Om) + (2kg)(2m) ‘Vou You = kg + 1kg + 2kg a im 52