14:440:222 Dynamics Student takeaway note sheet Lecture 10: Introduction to kinetics using Newtonian mechanics & Free-body-diagram & Equation of motion in rectangular coordinates In today’s lecture, you will learn: Newton’s laws & free-body-diagram How to write Newton’s second law using rectangular coordinates? Conceptual questions and mathematical problems Rutgers University Mechanical and Aerospace Engineering Prof. Annalisa Scacchioli, Ph.D. 14:440:222 Dynamics Student takeaway note sheet Introduction to kinetics using Newtonian mechanics History Galileo Galilei (d. 1642) in Italy began the modern development of the science of mechanics In 1687 Isaac Newton (b. 1642), in UK, published “Principia Mathematica” containing laws of motion and law of universal gravitation Newton’s laws: Kinetics: study of how forces relate to the motion Newton’s first law: a particle at rest or moving in a straight line with a constant velocity is in equilibrium ∑ 𝐹⃗ = 0 (law of inertia) Newton’s second law: when an imbalanced force (∑ 𝐹⃗ ≠ 0) acts on a particle, particle experiences an acceleration in the same direction of the force impressed 𝑎⃗ = 𝐹⃗ 𝑚 ∑ 𝐹⃗ = 𝑚𝑎⃗ (equation of motion) Newton’s third law: to every action there is an opposed an equal reaction. 𝐹⃗𝐴𝐵 = −𝐹⃗𝐵𝐴 (action-reaction) Newton’s universal gravitation law: every pair of particles in the universe exerts on one other a mutual gravitational force of attraction 𝑚 𝑚 𝐹𝑔 = 𝐺 12 2 𝑟12 Solving particle kinetics problems using Newtonian mechanics: Type A problem: given forces, use kinetics to find acceleration and then use kinematics to find velocity, space, and time Type B problem: given velocity, space, and time, use kinematics to find acceleration and then use kinetics to find forces Rutgers University Mechanical and Aerospace Engineering Prof. Annalisa Scacchioli, Ph.D. 14:440:222 Dynamics Student takeaway note sheet Modeling elastic force and tension Friction force 𝐹𝑓 : We use a linear model to describe the friction force Ff = μN, where 𝜇 is the friction coefficient and 𝑁 is the normal force Spring force 𝐹𝑠 : It is opposite to the force that produced the deformation 𝐹𝑠 = 𝑘∆𝑥 = 𝑘(𝑥 − 𝑥0 ) Tension T: It is the force transmitted by a rope when forces are applied to it It acts in the direction along the medium and it is constant If strings and pulleys are massless the tension 𝑇 is the same on either side of the pulley Solving problems using FBD and rectangular coordinates 1. Draw a picture indicating all key features in the problem identifying givens and unknowns 2. For each object draw the free-body-diagram (FBD): Draw all the forces acting on it: do not include any internal forces or forces exerted by the body on some other body! Select a coordinate system and show it in the FBD when the direction of the acceleration is known in advance, chose that direction as +x-axis can choose different reference frames for each body-all must be must be inertial! Determine components of the forces with reference to these axes 3. If there are geometrical relationship between two or more bodies, relate them algebraically 4. Write down Newton’s equation of motion for each body and solve for unknowns: 𝐹⃗ = 𝑚𝑎⃗ Equations of motion in rectangular coordinates: ∑ 𝐹𝑥 = 𝑚 𝑎𝑥 ∑ 𝐹𝑦 = 𝑚 𝑎𝑦 Rutgers University Mechanical and Aerospace Engineering Prof. Annalisa Scacchioli, Ph.D. 14:440:222 Dynamics Student takeaway note sheet FBD dos and don’ts: orienting the body FBD does and don’ts: orienting the coordinate system FBD dos and don’ts: redundant forces Rutgers University Mechanical and Aerospace Engineering Prof. Annalisa Scacchioli, Ph.D.