Basic Mechanics and Medical Applications PDF

1. Basic concepts in Mechanics & medical applications Assoc. Prof. Mai Hong Hanh| 2023 Email: [email protected]  OUTLINE - Position, speed, velocity, & acceleration - Newton’s laws & applications - Work, energy & momentum - Rotation motion...

1. Basic concepts in Mechanics & medical applications Assoc. Prof. Mai Hong Hanh| 2023 Email: [email protected]  OUTLINE - Position, speed, velocity, & acceleration - Newton’s laws & applications - Work, energy & momentum - Rotation motion 1 very important 2 important 3 for specialists POSITION 1 To locate an object means to find its position relative to some reference points  DISTANCE AND DISPLACEMENT 1 Distance is the magnitude between two interval points. Distance is not a vector quantity Vector: -Size, or magnitude -Direction Displacement is the change in the position of an object. Displacement is a vector quantity Δx : displacement xf : final position (B) ∆𝒙 = 𝒙𝒇 − 𝒙𝒊 xi : intial position (A) SPEED AND VELOCITY 1 Velocity is the vector quantity that signifies the magnitude of the rate of change of position and also the direction of an object’s movement. Speed is the scalar quantity that signifies only the magnitude of the rate of change of an object’s Unit: m/s movement.  ACCELERATION 1 When an object’s velocity and direction change, the object is said to undergo acceleration (or to accelerate). ACCELERATION 1 For motion along an axis, the average acceleration aavg over a time interval Δt is 𝒗𝒇 − 𝒗𝒊 ∆𝒗 𝒂𝒂𝒗𝒈 = = 𝒕𝒇 − 𝒕𝒊 ∆𝒕 where the object has velocity vi at time ti and then velocity vf at time tf. The instantaneous acceleration (or simply acceleration) is 𝒅𝒗 𝒂= Unit: m/s2 𝒅𝒕 FORCE 1 The force is said to act on the object to change its velocity.  FORCE and MASS 1 A force is a push or a pull. Arrows are used to represent forces. The length of the arrow is proportional to the magnitude of the force. Contact forces arise from physical contact. Action-at-a-distance forces do not require contact and include gravity and electrical forces.  FORCE and MASS 1 Mass is a measure of the amount of “stuff” contained in an object. The mass of a body is the characteristic that relates a force on the body to the resulting acceleration. GRAVITY 1  NEWTON’S FIRST LAW 1 Newton’s First Law: An object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. 𝑭𝒏𝒆𝒕 = 𝟎 Individual Forces Net Force The net force is the vector sum of all the forces acting on an object.  NEWTON’S SECOND LAW 1 Newton’s Second Law: To get the wagon to The net force on a body accelerate, you have to apply a PULL (Force). is equal to the product of the body’s mass and its acceleration. If the MASS of the wagon increases, a 𝐹𝑛𝑒𝑡 = 𝑚𝑎Ԧ greater PULL is necessary to accelerate it. Unit: Newton (N)  Problem 1 – Biomedical example 2 Microtubules are assembled from protein molecules. Microtubules help cells maintain their shape and are responsible for various kinds of movements within cells, such as pulling apart chromosomes during cell division. Measurements show that microtubules can exert forces from a few pN (1pN=10-12 N) up to hundreds of nN (1 nN=10-9 N) Microtubules (shown in green within a fertilized sea urchin egg undergoing A particular bacterial chromosome has a mass of 2.00×10-17 division) are protein molecules found kg. If a microtubule applies a force of 1.00 pN to the within cells. They help cells hold their chromosome, what is the magnitude of the chromosome’s shape and are responsible for various kinds of movements within cells. acceleration? (Ignore any other forces that might act on the chromosome.) Problem 1 – Biomedical example 2 This acceleration is about 5000 times g, the acceleration due to gravity!  NEWTON’S THIRD LAW 1 Newton’s Third Law: Whenever one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body. We can write this law as the scalar relation: 𝐹𝐴𝐵 = 𝐹𝐵𝐴 or as the vector relation 𝐹𝐴𝐵 = −𝐹𝐵𝐴  NEWTON’S THIRD LAW 1 Newton’s Third Law and Propulsion FORCES ON SOLIDS AND THEIR ELASTIC RESPONSE 1 The amount a copper wire stretches when a force is applied to its ends is: (1) proportional to the applied force, (2) proportional to the original length, (3) inversely proportional to the cross- sectional area. Interestingly, if we remove the added weight, the wire returns to its original length. Such a stretch with a return to the original form is called an elastic deformation.  FORCES ON SOLIDS AND THEIR ELASTIC RESPONSE 1 The rule for elastic deformation: 𝐹 ∆𝐿 =𝑌 𝐴 𝐿 where F is the applied force, A is the cross-sectional area, L is the original length, and ΔL is the stretch. The constant of proportionality is called Young’s modulus. The left-hand side of the equation, F/A, is called the applied stress (in this case tensile stress) measured in N/m2, or pascal (1 Pa 1 N/m2), whereas the ratio ΔL /L is the resulting (dimensionless) strain produced.  FORCES ON SOLIDS AND THEIR ELASTIC RESPONSE 1 - In biology there are a number of structural solids whose properties are fundamental to the life processes of the organism such as bone, soft tissues…. Imagine putting the femur bone (the long bone of the thigh) under tension by exerting forces on either end along the long axis of the bone and away from the bone’s center FORCES ON SOLIDS AND THEIR ELASTIC RESPONSE 1 The curve is the typical response of bone and the lower curve shows the plastic behavior of some other materials. WORK (W) 1 “Describes what happens when a force is exerted on an object as it moves” WORK (W) 1 Work done by a force in the direction of displacement: W: work done on an object 𝑾 = 𝑭𝒅 F: magnitude of the constant force d: magnitude of the displacement Unit: Joul (J) You do 1 J of work when you exert a 1N push on an object as it moves through a distance of 1 m  WORK (W) 1 (a) The muscles of your body are composed of many individual muscle fibers. (b) The internal structure of a muscle fiber when relaxed (left) and contracted (right). WORK (W) 1 How can we calculate the work done by a constant force that is not in the direction of the object’s motion? 𝑾 = (𝑭𝒄𝒐𝒔𝜽)𝒅 𝜃 : angle between the direction of F and d  KINETIC ENERGY (K) 1 Kinetic energy K is the energy associated with the state of motion of an object. 𝟏 𝑲 = 𝒎𝒗𝟐 Unit: Joule (J) equals 1 kg-m2/s2 𝟐 The faster the object moves, the greater is its kinetic energy. When the object is stationary, its kinetic energy is zero. Kinetic energy is a scalar quantity. Work-energy theorem 1 1 𝑊𝑛𝑒𝑡 = 𝑚𝑣𝑓 − 𝑚𝑣𝑖2 2 When an object undergoes a displacement, 2 2 the work done on it by the net force equals the object’s kinetic energy at the end of the displacement minus its kinetic energy at the beginning of the displacement. 𝑊𝑛𝑒𝑡 = 𝐾𝑓 − 𝐾𝑖  POTENTIAL ENERGY (U) 1 A stationary object can also have the ability to do work. We use the term “potential energy” to refer to an ability to do work that’s related to an object’s position. Unit: Joule (J) GRAVITATIONAL POTENTIAL ENERGY (U) 1 𝑊 = 𝐹ℎ = 𝑚𝑔ℎ 𝑊𝑔𝑟𝑎𝑣 = − 𝑚𝑔 ℎ = −𝑚𝑔ℎ 𝑊𝑤𝑒𝑖𝑔ℎ𝑡 𝑙𝑖𝑓𝑡𝑒𝑟 = +𝑚𝑔ℎ 𝑼𝒈𝒓𝒂𝒗 = 𝒎𝒈𝒉 We call 𝑼 = 𝑾 = 𝒎𝒈𝒉 the gravitational potential energy since it arises from the weight lifter doing work against the gravitational force. ELASTIC POTENTIAL ENERGY (U) 1 𝟏 𝟐 𝑼 = 𝒌𝒙 𝟐 k: Spring constant 𝑥 1 2 x: Displacement 𝑊 = න 𝑘𝑥𝑑𝑥 = 𝑘𝑥 Potential energy is a scalar quantity 2 0  LAW OF CONSERVATION OF ENERGY 1 Principle of conservation of mechanical energy: In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change. U = 𝑚𝑔ℎ U=0 U = 𝑚𝑔ℎ 𝐾=0 1 𝐾=0 𝐾 = 𝑚𝑣 2 𝑬𝒎𝒆𝒄 = 𝑼𝒊 + 𝑲𝒊 = 𝑼𝒇 + 𝑲𝒇 2 LAW OF CONSERVATION OF ENERGY 1 𝟏 𝟐 𝑼𝒊 = 𝒌𝒙 𝟐 𝑲𝒊 = 𝟎 𝑼𝒇 = 𝟎 𝟏 𝑲𝒇 = 𝒎𝒗𝟐 𝟐 LAW OF CONSERVATION OF ENERGY 1 Kinetic and potential energy in the arteries When you feel the pulse in your radial artery, you are actually feeling spring potential energy of the arterial walls being converted to kinetic energy of the blood. In both of these situations the sum of kinetic energy and potential energy—a sum that we call the total mechanical energy—keeps the same value and is conserved. POWER (P) 1 If you walk for a kilometer, your heart rate will increase above its resting value. But if you run for a kilometer, your heart rate will increase to an even higher value. You covered the same distance and did roughly the same amount of work in both cases, so why is Running versus walking Both running a higher heart rate needed for running? and walking involve doing work; the difference is the rate at which work is done. POWER (P) 1 The rate at which energy is transferred from one place to another, or from one form to another, is called power. The quantity P in this equation is sometimes called the power delivered to the object on which work is being done. 𝑾 𝑷= 𝒕 Unit: 1 W = 1 Joule/second = 1 J/s. Other common units of power are the kilowatt (1 kW = 1000 W) and the horsepower (1 hp = 746 W is a typical rate at which a horse does work by pulling on a plow) MOMENTUM 2 MOMENTUM – LAW OF MOMENTUM CONSERVATION 2 𝑝Ԧ = 𝑚𝑣Ԧ The momentum of an object is a vector that points in the same direction as its velocity. It depends on both the mass and velocity of the object. Do not confuse momentum with kinetic energy, which is a scalar quantity. Unit: kg.m/s ROTATIONAL MOTION 2 Translation: an object as whole moves through space Rotation: an object spins around an axis. ROTATIONAL MOTION 1 By analogy to how we defined average velocity for straight-line motion we define the average angular velocity of the blade to be O 𝛥𝜃 𝜔𝑎𝑣𝑒𝑟𝑎𝑔𝑒,𝑧 = 𝛥𝑡 Speed of a point on a rotating rigid object 𝑣 = 𝑟𝜔 Unit: rad/s  TORQUE 1 Torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect. It represents the capability of a force to produce change in the rotational motion of the body TORQUE 1 𝝉 = 𝒓𝑭𝒔𝒊𝒏𝝓 τ : magnitude of the torque r: position vector (lever arm vector) Torque is a vector quantity Փ:the angle between the force F and Unit: N.m the lever arm vector  TORQUE 1 Lever arms in the human jaw and arm (a) The masseter muscle exerts a force on the lower jaw with a relatively long lever arm, so the torque on the lower jaw is quite large. (b) The lever arm for the force of the biceps muscle on the forearm is relatively short, so the torque is relatively small. CENTRIPETAL FORCE 1  CENTRIPETAL FORCE 1 The component of force acting on an object in curvature motion which is directed toward the axis of rotation or center of curvature. 𝒎𝒗𝟐 𝑭 = 𝒎𝒂𝒄 = 𝒓 ac is the Centripetal acceleration m is the mass of the object v is the speed or velocity of the object r is the radius  CENTRIPETAL FORCE 1 𝒎𝒗𝟐 𝑭=− 𝒓 Centrifugal force is equal in magnitude and opposite in direction to the centripetal force, drawing a rotating object away from the center of rotation, caused by the inertia of the object. BLOOD COMPOSITION 1 How to separate blood? CENTRIPETAL TECHNIQUE PRINCIPLE 1  References: 1. J. Newman, Physics of the Life Science. 2008. 2. Roger A. Freedman,Todd Ruskell,Philip R. Kesten, David L. Tauck - College Physics-W. H. 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