Mechanics & Motion of the Body PDF

Mechanics & Motion of the body Momentum Mechanics & Motion of the body Fundamental forces in nature: Force (F): In science, force is the push or pull on an object with mass that causes it to change velocity (to accelerate). Force represents as a vector, which means it has both magnitude and directio...

Mechanics & Motion of the body Momentum Mechanics & Motion of the body Fundamental forces in nature: Force (F): In science, force is the push or pull on an object with mass that causes it to change velocity (to accelerate). Force represents as a vector, which means it has both magnitude and direction. There are four fundamental forces of nature, and they govern everything that happens in the universe. Modern physics attempts to explain every observed physical phenomenon by these fundamental interactions. Mechanics & Motion of the body Fundamental forces in nature: Four fundamental forces : Gravity. The weak force. Electromagnetism. The strong force. Mechanics & Motion of the body Fundamental forces in nature: Gravity: Is the attraction between two objects that have mass or energy, whether this is seen in dropping a rock from a bridge, a planet orbiting a star or the moon causing ocean tides. Gravity is probably the most intuitive and familiar of the fundamental forces. Isaac Newton was the first to propose the idea of gravity, supposedly inspired by an apple falling from a tree. He described gravity as a literal attraction between two objects. Mechanics & Motion of the body Fundamental forces in nature: Gravity: Albert Einstein suggested, through his theory of general relativity, that gravity is not an attraction or a force. Instead, it's a consequence of objects bending space time. A large object works on space-time a bit like how a large ball placed in the middle of a sheet affects that material, deforming it and causing other, smaller objects on the sheet to fall toward the middle. Mechanics & Motion of the body Fundamental forces in nature: Fig 2: The gravitational force. Where: F is the gravitational force between the bodies of masses m1&m2. G is gravitational constant = 6.67430 ×10¯¹¹ m³.kg¯¹.s¯² Mechanics & Motion of the body Fundamental forces in nature: Weak force: Also called the weak nuclear interaction is responsible for particle decay. This is the literal change of one type of subatomic particle into another. So , for example, a neutrino that strays close to a neutron can turn the neutron into a proton while the neutrino becomes an electron. Mechanics & Motion of the body Fundamental forces in nature: Fig 3: weak force. Mechanics & Motion of the body Fundamental forces in nature: Electromagnetic force: Consists of two parts, the electric force and the magnetic force. We described these forces as separate from one another, but researchers later realized that the two are components of the same force. Acts between charged particles, like negatively charged electrons and positively charged protons. Opposite charges attract one another, while like charges repel. The greater the charge, the greater the force. This force can be felt from an infinite distance. Mechanics & Motion of the body Fundamental forces in nature: Electromagnetic force: The electric component acts between charged particles whether they're moving or stationary, creating a field by which the charges can influence each other. But once set into motion, those charged particles begin to display the second component, the magnetic force. The particles create a magnetic field around them as they move. So when electrons zoom through a wire to charge your computer or phone or turn On your TV, for example, the wire becomes magnetic. Mechanics & Motion of the body Fundamental forces in nature: Coulomb's law: Quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force. 𝑞₂ 𝑞₁ Fe= Ke 𝑟² Where: Fe is the force. Ke is the coulombs constant (8.9× 10⁹ N.m².C¯² ) q₁&q₂ Signed magnitudes of the charges. r is the distance between the charges. Mechanics & Motion of the body Fundamental forces in nature: Fig 4: electrical force & magnetic force Mechanics & Motion of the body Fundamental forces in nature: Strong nuclear force: Is the strongest of the four fundamental forces of nature. And that's because it binds the fundamental particles of matter together to form larger particles. Is holds together the quarks that make up protons and neutrons, and part of the strong force also keeps the protons and neutrons of an atom's nucleus together. Mechanics & Motion of the body Fundamental forces in nature: Strong nuclear force: The strong force operates only when subatomic particles are extremely close to one another. They have to be somewhere within 10¯¹⁵ meters from each other, or roughly within the diameter of a proton. Mechanics & Motion of the body Fundamental forces in nature: Fig 5: nuclear force. Mechanics & Motion of the body Fundamental forces in nature: Table1: Show interaction force. Mechanics & Motion of theof body Mechanics & Motion the body Types of force: There are a variety of types of forces. Types were placed into two broad category headings on the basis of whether the force resulted from the contact or non-contact of the two interacting objects. Types of forces: Contact Forces: Frictional Force ,Tensional Force ,Normal Force ,Air Resistance Force, Applied Force ,Spring Force Action-at- Distance Forces : Gravitational Force, Electrical Force, Magnetic Force Mechanics & Motion of the body Work, Mass and Energy. Work: Measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement. The work is calculated by multiplying the force by the amount of movement of an object. W = F. d A force of 10 newton’s, that moves an object 3 meters, does 30 n-m of work. Mechanics & Motion of the body Work, Mass and Energy. Work: A newton-meter is the same thing as a joule, so the units for work are the same as those for energy – joule. 1 joule = (1 newton) (1meter) = 1 n. m Work done = force × distance moved in the direction of the force. W= ∆𝐸 = F×d W= work done (j), ∆E= energy transferred (j), F= force(N), d=distance in direction of force. If the force F acts in angle with the direction of movement then the component of that force in the direction of displacement will acts: W= f. d cos 𝜃 Mechanics & Motion of the body Work, Mass and Energy. Work: Example 1 A force of 50 N acts on the block at the angle 30 degree. The block moves a horizontal distance of 3.0 m. How much work is done by the applied force? Answer: W = F × d × cos (𝜃) W = (50 N) × (3 m) × cos (30 degrees) = 129.9 Joules Mechanics & Motion of the body Work, Mass and Energy. Work: Example 2 Apply the work equation to determine the amount of work done by the applied force in each of the three situations described below: Diagram A A 100 N force is applied to move a 15 kg object a horizontal distance of 5 meters at constant speed. Diagram B A 100 N force is applied at an angle of 30° to the horizontal to move a 15 kg object at a constant speed for a horizontal distance of 5 m. Diagram C An upward force is applied to lift a 15 kg object to a height of 5 meters at constant speed. Mechanics & Motion of the body Work, Mass and Energy. Work: Diagram A Answer: W = (100 N) × (5 m) × cos (0 degrees) = 500 J The force and the displacement are given in the problem statement. That the force and the displacement are both rightward. Since F and d are in the same direction , the angle is 0 degrees. Mechanics & Motion of the body Work, Mass and Energy. Work: Diagram B Answer: W = (100 N) × (5 m) ×cos (30 degrees) = 433 J The force and the displacement are given in the problem statement. It is said that the displacement is rightward. It is shown that the force is 30 degrees above the horizontal. Thus, the angle between F and d is 30 degrees. Mechanics & Motion of the body Work, Mass and Energy. Work: Diagram C Answer: W = (147 N) × (5 m) × cos (0 degrees) = 735 J The displacement is given in the problem statement. The applied force must be 147 N since the 15-kg mass (F grav =147 N) is lifted at constant speed. Since F and d are in the same direction, the angle is 0 degrees. Mechanics & Motion of the body Work, Mass and Energy. Mass and weight: Mass (m) of an object is a measure of the object's inertial property, or the amount of matter it contains F = ma where a is acceleration. Weight (w) of an object is a measure of the force exerted on the object by gravity, or the force needed to support it. W = mg g = 9.8m/s2 gravitational acceleration The difference between mass & weight is that mass is the amount of matter in a material, while weight is a measure of the force of gravity acts upon that mass. Mechanics & Motion of the body Work, Mass and Energy. Energy: In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity. It is the capacity to do work or the ability to cause change. Mechanics & Motion of the body Work, Mass and Energy. Conservation of energy principle: In physics, the law of conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time. Energy can be neither created nor destroyed, but can change form. The first law of thermodynamics is the application of the conservation of energy principle to heat and thermodynamic processes: The first law of thermodynamics for a closed system or fixed mass was expressed: ΔE = Q – W Mechanics & Motion of the body Work, Mass and Energy. Conservation of energy principle: The first law of thermodynamics defines the internal energy (E) as equal to the difference of the heat transfer (Q) into a system and the work (W) done by the system. E2 - E1 = Q – W………….. (1) Q = net heat transfer across system boundaries (=Ʃ Qin –Ʃ Q out) W = net work done in all forms (=ƩW out –Ʃ Win) ΔE = net change in total energy The total energy E of a system is considered to consist of 3 parts: ΔE tot = ΔU+ ΔE kin+ ΔE pot ………… (2) Substitute Eq 2 in Eq 1 We get: Q – W= ΔU+ ΔE kin+ ΔE pot ………… (3) Mechanics & Motion of the body Work, Mass and Energy. Conservation of energy principle: Where: ΔU= change in internal energy = m (u2-u1) ΔE kin= change in kinetic energy = ½ m (v²₂ – v²₁). ΔEpot = change in potential energy=mg (Z2 –Z1). In SI base units: joule = J = kg m² s¯² Mechanics & Motion of the body Work, Mass and Energy. Power: The rate of doing work or of transferring heat. The amount of energy transferred or converted per unit time. It is a scalar quantity. P = 𝑤/𝑡=∆𝐸/𝑡 Where: P=Power (watt).W= Work done (J). ∆E=Energy transferred (J).t=Time (s). In SI base units: power is the joule per second (J/s), known as the watt (W). Motion. Mechanics & Motion of the body Motion: Straight line (linear motion): The equation of a straight line is usually written this way: y = mx + b where: 1. m is the slope or gradient of the line, m= ∆𝒚 ∆𝒙 = 𝒚₂−𝒚₁ 𝒙₂−𝒙₁ 2. b is the y-intercept of the line. 3. x is the independent variable of the function y = f(x). 4.y is dependent variable. Mechanics & Motion of the body Motion: Fig 6: the slope of straight line. Mechanics & Motion of the body Straight line (linear motion): a- Displacing time and average velocity. Displacement: Is the vector difference between the ending and starting positions of an object. Velocity: Is the rate at which displacement changes with time. It is a vector. The average velocity over some interval is the total displacement during that interval, divided by the time. Vav = 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒊𝒎𝒆 = ∆𝒅 ∆𝒕 Mechanics & Motion of the body Straight line (linear motion): Fig7: the average velocity Mechanics & Motion of the body Straight line (linear motion): b- Instantaneous velocity. The velocity of an object in motion at a specific point in time. Or specific point along the path. We use the velocity at very short time intervals: V = lim∆𝑡→0= ∆𝒙 ∆𝒕 = 𝒅𝒙 𝒅𝒕 Note: Instantaneous velocity cannot be negative Mechanics & Motion of the body Straight line (linear motion): c- Average velocity & average speed. Average velocity = v avg =𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 / 𝑡𝑖𝑚𝑒 = ∆x/∆𝑡 Average speed = 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 / 𝑡𝑖𝑚𝑒 =𝐷 /∆t Mechanics & Motion of the body Straight line (linear motion): c- Average velocity & average speed: Example 3: A car travels distance of 70 km in 2 hours, What is the average speed? Solution: average speed = 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 / 𝑡𝑖𝑚𝑒=70𝑘𝑚 / 2ℎ𝑟𝑠 = 35 km/hr Mechanics & Motion of the body Straight line (linear motion): c- Average velocity & average speed: Example 4: A boy walks 10 km east in 2 hours & then 2.5 km west in 1 hour. Calculate the total average velocity of the body? Solution: V = 10−2.5 / 2+1 = 7.5 / 3 = 2.5 km/hr. Note: velocity with direction, but speed without direction. Mechanics & Motion of the body Straight line (linear motion): d- Average acceleration: The change of velocity by time interval during changes. a =∆𝑣/ ∆𝑡 = 𝑣₂−𝑣₁ /𝑡₂−𝑡₁ Fig 8: Time acceleration graph: Mechanics & Motion of the body Straight line (linear motion): e- General equation: There are some equations can be used to calculate quantities related to the motion of particles in straight line these are: 1. V₂ = v₁ + a t 2. X = V₀ + (1/ 2) a t² 3. 𝑣²₂ = 𝑣 ²₁ + 2 ax If it is projectable then: 1. X = V₀ t cos α 2. Y = V₀ t sin α - (1/ 2) g t² 3. Vx = V₀ cos α 4. Vy = V₀ sin α – g t Mechanics & Motion of the body Nonlinear motion: a- Pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. This is Called simple harmonic motion. Mechanics & Motion of the body Nonlinear motion: A- Pendulum The period (T): Is the time for one complete cycle, a left swing and a right swing. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing. The frequency (F): Is the number of times the pendulum swings back and forth per second. 𝟏 Frequency and period are inversely related that is, 𝑻 =. if the angle of 𝒇 displacement is small, the period is given by: 1 𝑇 = = 2𝜋 𝑔/𝑙 𝑓 Where 𝑔 is the gravitation acceleration and 𝑙 is the length of the pendulum arm Mechanics & Motion of the body Straight line (linear motion): Fig 9: simple pendulum Mechanics & Motion of the body Nonlinear motion: B - Centrifuge: Is a laboratory device that is used for the separation of fluids, gas or liquid, based on density. Separation is achieved by spinning a vessel containing material at high speed; the centrifugal force pushes heavier materials to the outside of the vessel. The centrifuge works using the sedimentation principle, where the centrifugal acceleration causes denser substances and particles to move outward in the radial direction. Mechanics & Motion of the body Nonlinear motion: B - Centrifuge: At the same time, objects that are less dense are displaced and move to the center. In a laboratory centrifuge that uses sample tubes, the radial acceleration causes denser particles to settle to the bottom of the tube, while low-density substances rise to the top. Fig 10: Show blood centrifugal machine. Mechanics & Motion of the body Stokes law: Stokes law: The force that retards a sphere moving through a viscous fluid is directly proportional to the velocity of the sphere, the radius of the sphere, and the viscosity of the fluid. The force of viscosity on a small sphere moving through a viscous fluid is given by: Fd = 6 π a η v Where: Fd : is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle. η : is the viscosity. a : is the radius of the spherical object v : is the flow velocity relative to the object. In SI units: Fd is given in newton’s. η in Pa· s, a in meters, and v in m/s. Mechanics & Motion of the body Stokes law: Stokes law: When the particle moving at constant speed, the retarding force (Fd) is in equilibrium with the difference between the download gravitational force and upward buoyant force. The 𝟒 force gravity 𝑭𝒈 = 𝝅𝒂³𝝆𝒈 𝟑 𝟒 buoyant force 𝑭𝒃 = 𝝅𝒂³𝝆₀𝒈 𝟑 𝝆: fine particles of density The 𝝆₀: liquid of density The retarding force 𝑭𝒅 = 𝟔 𝝅 𝒂 𝜼 𝒗 Stokes' drag Mechanics & Motion of the body Stokes law: Stokes law: Three forces actin on a rigid particle moving in a fluid: 1. External force, gravitational force or centrifugal 2. Buoyant force, which acts parallel with external force but in the opposite direction. 3. Drag force, which appears whenever there is relative motion between the particle and fluid (frictional resistance). Drag: The force in direction of flow exerted by the fluid on the solid. Mechanics & Motion of the body Stokes law: Fig 11: Drug force, buoyant force & External force. Mechanics & Motion of the body Stokes law: Stokes law: In another case we can increase (g) by means of centrifuge, which provides an effective acceleration (g eff): g eff = 4π² f² r Where: f is the rotation in revolutions per second & r is the position on the radius of the centrifuge. Mechanics & Motion of the body Newton's laws of motion. Are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a quantitative measure of the force, and the third asserts that a single isolated force doesn't exist. These three laws have been expressed in several ways. Mechanics & Motion of the body Newton's laws of motion: Newton's first law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. ෍ Fig 12: Newton’s 1st law. 𝑑𝑣 𝐹 = 0↔ = 𝑜 𝑑𝑡 Mechanics & Motion of the body Newton's laws of motion: Newton's second law The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. 𝑭 = 𝒎𝒂 Fig 13: Newton’s 2nd law. Mechanics & Motion of the body Newton's laws of motion: Newton's second law Differential form: force change of momentum with change of time, 𝑭 = 𝒅 𝒎𝒗 𝒅𝒕 Force = change in mass × velocity with time, F = (m₁ v₁ – m₀ v₀) ̸ (t₁ – t₀) With mass constant: force = mass × acceleration, F = ma Force, acceleration. Momentum and velocity are all vector quantities. Each has both a magnitude & a direction. Mechanics & Motion of the body Newton's laws of motion: Newton's third laws: Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it. Force equals mass times acceleration. For every action there is an equal and opposite reaction. 𝑭𝑨 𝒐𝒏 𝑩 = − 𝑭𝑩 𝒐𝒏 𝑨 Fig 14: Newton’s 3rd law. Mechanics & Motion of the body Fractional force: Refers to the force generated by two surfaces that contacts and 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. The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: 𝑭𝒇 = µ𝒌 𝑭𝒏 Mechanics & Motion of the body Fractional force: Where: 1. Ff: is the force of friction exerted by each surface on the other. It is parallel to the surface, in a direction opposite to the net applied force. 2. µk: is the coefficient of friction, which is an empirical property of the contacting materials. 3. Fn: is the normal force exerted by each surface on the other, directed perpendicular (normal) to the surface Fig 15: frictional force. Mechanics & Motion of the body Fluid resistance and terminal speed: Friction: Is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other Fluid (a gas or liquid). Fig 16: Fluid friction. Mechanics & Motion of the body Fluid resistance and terminal speed: Friction: By newton 3rd law, the fluid pushes back the body with equal and opposite proportional to viscosity, but at high speed it will proportional to the velocity squared of the body as: f = KV at low speed f = DV² at high speed Where : 1. K is proportionality constant, depend on shape and size of the body &proportional to the fluid. 2. D depends on shape and size of the body &on density of the fluid. Mechanics & Motion of the body Fluid resistance and terminal speed: Friction: The unite of K is N.s/m= (kg/s), but the unite of D is N.s²/m²= (Kg/m). For bodies falling through fluids the terminal speed are: Vt = 𝑚𝑔 / 𝑘 Vt = (fluid resistance f = KV) 𝒎𝒈 (fluid resistance f = DV² ) 𝒅 Mechanics & Motion of the body Fluid resistance and terminal speed: Viscosity: Resistance of a fluid (liquid or gas) to a change in shape, or movement of neighboring portions relative to one another. Viscosity denotes opposition to flow. Fluid Viscosity sometimes referred to as dynamic viscosity or absolute viscosity is the fluid's resistance to flow, which is caused by a shearing stress within a flowing fluid and between a flowing fluid and its container. For example, syrup has a higher viscosity than water. Mechanics & Motion of the body Fluid resistance and terminal speed: Viscosity: The SI unit of dynamic viscosity is the pascal-second, the customary unit of viscosity is called the poise. 1centipoise = 1 cP = 0.001Pa.s → 1Pa.s = 1000 cP. Viscosity equation for newton’s fluid: 𝑭 𝒅𝒗 = 𝒏 𝑨 𝒅𝒓 n is viscosity coefficient. 𝑑𝑣 / v𝑑𝑟 Is shear rate. For every 1 unit increase in viscosity, the shear rate must increase by a certain amount (depending on the liquid). Mechanics & Motion of the body Fluid resistance and terminal speed: Viscosity: Example: for every 1 Pascal second (Pa s) increases in viscosity of water, the shear rate must increase by 0.00089 s. So, we multiply the shear rate by 'n', which is equal to 0.00089 Pa. Fig 17: Viscosity: a) Friction between layers. b) An obstruction in the vessel produces turbulence Mechanics & Motion of the body Momentum Simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. SI unite kilogram meter per second kg⋅ m/s Then the object’s momentum (p) is: p = mv m= mass & v= velocity Momentum does not just depend on the object’s mass and speed. Because velocity is speed in a particular direction, the momentum of an object also depends on the direction of travel. Mechanics & Motion of the body Momentum This means that the momentum of an object can change if the object speeds up or slows down; or if it changes direction. When Newton’s 2nd law combined with definition of acceleration the following equalities result. ∆𝒗 𝑭 = 𝒎𝒂 = 𝒎 → 𝑭∆𝒕 = 𝒎∆𝒗 ∆𝒕 1 Newton = (1 kilogram) (1meter / second square) 1 N = 1 Kg. m/ s² Mechanics & Motion of the body Momentum: Momentum: Is a vector quantity conceptually; momentum is a characteristic of motion. That reflect how difficult it would be to stop the moving object. In physics, the quantity (Force × time) is known as Impulse Impulse = change in momentum Fig 18: impulse equal change in momentum Mechanics & Motion of the body Examples: Ex1: Find the gravitational force between two mass of 0.01 kg and 0.5kg, if the distance between center- to- center is 0.05m? Solution: 𝒎₁×𝒎₂ (𝟎.𝟎𝟏) (𝟎.𝟓) F=G = 6.67 × 10¯¹¹ = 1.3 × 10¯¹⁰ N (𝟎.𝟎𝟓)² 𝒓² Mechanics & Motion of the body Examples: Ex 2: A worker applies a constant horizontal force with magnitude 40 N to a box with mass 20 kg resting on level floor with negligible fraction, What is the acceleration of the box? Solution: ∑ 𝐹. 𝑠 = F = 40N F= m a → a = ∑ 𝐹.𝑠 / 𝑚 = 40𝑁 /20𝑘𝑔 =2 𝒌𝒈.𝒎/𝒔² =2𝑚/𝑠² 𝒌𝒈 Mechanics & Motion of the body Examples: Ex 3: What the weight of mass of 1kg & 3 kg respectively? Solution: W = mg For 1 kg 1.0 kg × 9.8 m/s² = 9.8 N For 3 kg 3.0 kg × 9.8 m/s²= 29.4 N Mechanics & Motion of the body Examples: Ex 4: A car of 2.49 × 10⁴ N weight traveling in x-direction makes fast stop, the x - direction of the net force acting its 1.83 × 10⁴ N. What is bits acceleration? Solution: W = mg → m = w/g m =2.49 ×10⁴ / 9.8 = 2540 kg , Car mass. F = ma → a = F/m a = 1.83 ×10⁴ / 2540 = 7.2 m/s² Mechanics & Motion of the body Examples: Ex 5: A 40 kg person walking at 1m/s bumps into a wall and stops in distance of 2.5 cm in about 0.05 sec. What is the force developed on impact? Solution: F = ∆(𝑚𝑣) / ∆𝑡 F = 40 / 0.05 = 800 Kg. m/ s² = 800 N where: (1 N = 1 Kg. m/ s²) Mechanics & Motion of the body Examples: Ex 6: a) A person walking at 1 m/sec hits his head on a steal beam and stops in 0.5 cm after 0.01 sec. if the mass of the head is 4 kg. What is the force developed? Solution: F = ∆(𝑚𝑣) / ∆𝑡 F = 4/0.01 = 400 Kg. m/ s² = 400 N where: (1 N = 1 Kg. m/ s²) Mechanics & Motion of the body Examples: Ex 6: b) If the steel beam has 2 cm of padding and ∆𝑡 is increased to 0.04 sec, what is the force developed? Solution: F = ∆(𝑚𝑣) / ∆𝑡 F = 4/0.04 = 100 Kg. m/ s² = 100 N Mechanics & Motion of the body Examples: Ex 7: A car weights 12000 N. if the coefficient of rolling friction is 0.015, what horizontal force is needed to make the car move? Solution: Ff = µ k Fn Ff = 0.015 × 12000 = 180 N Where (Fn) is the normal force which is the weight of the body. Mechanics & Motion of the body Examples: Ex 8: Find the terminal speed and the air resistance for 50 kg skydiver, if the value of the constant D is 0.25 kg/m? 9.8 Solution: vt= 𝒎𝒈 𝒅 = 𝟓𝟎𝒌𝒈)(𝟗. 𝟖 𝒎Τ𝒔 ² 𝟎. 𝟐𝟓𝒌 𝒈Τ𝒎 = 44 m/s f = DV² at high speed f= (0.25𝑘𝑔/𝑚)(44 m/s)² = 484 kg. m/s². Where: (1 N = 1 Kg. m/ s²) f= 484 N