H203 Dynamics - 1. Notes (1718) [For Upload]

DYNAMICS Content  Newton’s Law of motion  Linear momentum and its conservation Learning Outcomes Candidates should be able to: (a) state each of Newton’s laws of motion. (b) show an understanding that mass is the property of a body which resist s change in motion (inertia). (c) describe and use the concept of weight as the effect of a gravitational field on a mass. (d) define and use linear momentum as the product of mass and velocity. (e) define and use impulse as the product of force and time of impact. (f) Relate resultant force to the rate of change of momentum. (g) recall and solve problems using the relationship F = ma, appreciating that resultant force and acceleration are always in the same direction. (h) state the principle of conservation of momentum. (i) apply the principle of conservation of momentum to solve simple problems including inelastic and (perfectly) elastic interactions between two bodies in one dimension. (Knowledge of the concept of coefficient of restitution is not required.) (j) Show an understanding that, for a (perfectly) elastic collision between two bodies, the relative speed of approach is e qual to the relative speed of separation. (k) show an understanding that, whilst the momentum of a closed system is always conserved in interactions between bodies, some change in kinetic energy usually takes place. References: Physics for Scientists and Engineers. Serway. College Physics. Sears and Zemansky. Useful applet: Url : http://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejss_model_Momentum1D01 /Momentum1D01_Simulation.xhtml Look under Collision Carts, Atwood Machines, Newton Cradle Or https://phet.colorado.edu/en/simulation/legacy/collision-lab

Concept Map Newton’s Laws of Motion Newton’s 3rd Law Newton’s 1st Law 𝑭 𝒏𝒆𝒕 = ∆(𝒎𝒗 ) 𝒕 𝑭 𝒏𝒆𝒕 = 𝒎𝒂 𝑭 𝒏𝒆𝒕 = 𝟎 ⟺ ∆(𝒎𝒗) = 𝟎 Newton’s 2nd Law 𝒅𝑷 𝒅𝒕 ∝ 𝑭 𝒓𝒆𝒔𝒖𝒍𝒕𝒂𝒏𝒕 Inelastic Collisions Elastic Collisions Collisions 𝑽𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒐𝒇 𝑨𝒑𝒑𝒓𝒐𝒂𝒄𝒉 = 𝑽𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒐𝒇 𝑺𝒆𝒑𝒂𝒓𝒂𝒕𝒊𝒐𝒏 𝑷 𝒊 = 𝑷 𝒇 𝑬𝑲𝒊 = 𝑬𝑲𝒇 & 𝑷 𝒊 = 𝑷 𝒇 𝑬𝑲𝒊 > 𝑬𝑲𝒇 & Principle of Conservation of Momentum ∆𝑷 = 𝑭𝒏𝒆𝒕 dt

3.0 Introduction Dynamics is the branch of mechanics where the forces that act on a body are not in equilibrium. The vector sum of the forces gives a resultant force that causes the body to accelerate. This resultant force causes change in motion. In Kinematics, we learned how a body would move under constant acceleration, if we combine the knowledge from Dynamics and Kinematics, we will be able to predict the motion of a body when we know the forces acting on the body. In the seventeenth century, Sir Isaac Newton formulate d the three Newton’s laws of motion and it is the basis behind Newtonian Mechanics. Today, Newtonian mechanics is useful for many engineering efforts in our everyday scale, like how an artillery shell travels in air, and it explains many phenomena observed. 3.1 Newton’s 3rd Law of Motion Newton’s Third Law of Motion states that when body A exerts a force on body B, body B will exert an oppositely directed force of equal magnitude on body A. It should be noted that: 1. Newton’s Third Law of Motion involves two different bodies; the ‘action and reaction’ pair of forces as stated within the law acts on separate bodies (if the ‘action’ acts on body A, then the ‘reaction’ must act on body B). 2. The pair of forces is equal in magnitude and opposite in direction; they must also be of the same type /nature. In other words, if the force that A exerts on B is a gravitational force, then the equal and opposite force exerted by B on A is also a gravitational force. In summary: The action and reaction pair must  act on different bodies  be of the same type/nature  have equal magnitude and act in opposite direction to one another Examples of Newton’s Third Law ‘Action and Reaction’ Pair of Forces

Magnetic Force FA by B: Magnetic force on magnet A due to magnet B. FB by A : Magnetic force on magnet B due to magnet A. Electrostatic force FA by B : electrostatic force on electric charge A due to electric charge B. FB by A : electrostatic force on electric charge B due to electric charge A. Gravitational force FE by M, Gravitational force on Earth due to Moon. FM by E, Gravitational force on Moon due to Earth. Contact force Block resting on a slope. Consider contact forces only: The resultant of the frictional force and the normal contact force acting on a body is known as contact force. A B FA by B FB byA N N S S + + + + A B FE by M FM by E N, normal contact force on block due to floor f, frictional force on block due to floor N, normal contact force on floor due to block. f, frictional force on floor due to block FA by B FB byA Earth g Moon

3.1.1 Identifying Action-Reaction Forces For a book which is lying flat on a table, is the weight of the book and the force which the table acts on the book an action-reaction pair, and why? So, what is the action-reaction pair for weight of the book? What is the action-reaction pair for force on book due to the table, F B by T? Force on book due to table, FB by T Weight of Book, WB Ans: No 1) WB and FB by T are both acting on the same single body, i.e. the book itself! (Not on 2 different bodies) 2) WB = gravitational force, and FB by T = contact force. (Not same type of force) Weight of book or Force on book due to earth FB by E Force on earth due to earth, F E by B Action-Reaction Pair By Newton’s 3 rd law, the action - reaction pair must be a gravitational force also, Force on earth by book, FE by B In this context, by Newton’s 3 rd law, the action and reaction pair must also be a contact force, that is the Force on table by book, FT by B. Force on table due to book, F T by B

Key Question Although weight of the book is not force by book on table, can their magnitudes be the same? Why? 3.2 Newton’s First Law of Motion Newton’s First Law of Motion states that an object continues to be in a state of rest or in motion with a constant velocity, unless acted upon by a net external force. This law gives rise to the idea inertia as the resistance to change in the condition of rest or motion of a body. The inertia of a body can be considered as the reluctance of the body to start moving, as well as its reluctance to stop once it has begun moving. The mass of a body, quantified with a SI unit of kilogram (kg) is a measure of its inertia; t he larger the mass, the larger the inertia of the body. The weight of a body is defined as the force acting on it due to grav itational attraction by another body. Mathematically, the weight of a body at a point in space is given by W = m g, where m is the mass of the body and g the gravitational field strength at that po int in space . Hence, the mass of a body will remain constant throughout the universe but its weight is dependent on the value of g at the point at which it is placed. Yes: From FBD of book, when the book is in equilibrium, WB = FB by T . No: When you throw the book onto the table, the force by the book on table (on impact) is more than the weight of the book, i.e. FBT > WB ! Thinking question The magnitude of the force by the table on book may not always be the same as the weight of the book. Imagine a book placed on the floor of a lift that is accelerating.

3.2.1 Linear Momentum Linear momentum of a body is defined as the product of its mass and its velocity. 𝑃 = 𝑚𝑣 Linear momentum is a vector quantity and it takes the same direction as the velocity of the body. Momentum is the property of a body by virtue of its mas