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2053Lecture3

# 2053Lecture3 - Mechanical Systems Mechanical Topics of...

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Unformatted text preview: Mechanical Systems Mechanical Topics of Discussion Topics Mechanical Systems Displacement Average and Instantaneous Velocity & Average Acceleration Acceleration Average and Instantaneous Quantities Mechanical Systems Mechanical A mechanical system is a collection of bodies mechanical with each body described as having a mass m, a displacement r, a velocity v and an acceleration velocity a at any given time. The variables m, r, v and a are called the state variables and describe the state temporal evolution of the mechanical system. temporal Mechanical System x, v, a Displacement Displacement The displacement is a vector which points from The displacement an object’s initial displacement, given by the vector ri , to its final displacement, given by the vector rf . ∆r = r f – r i The magnitude | r | = r of the displacement The magnitude vector is the distance between the two positions. In SI, the unit of displacement is the meter (m). Displacement, cont. Displacement, Let be the initial location ri = x0 î + y0 ĵ Let and final location rf = xf î + yf ĵ of a particle. Then the total displacement is given by ∆r = rf – r0 . Average Velocity Average Let the initial time, the initial displacement Let and initial velocity be written as ti and ri. Let the final time and the final displacement be written as tf and rf . The average velocity, in m/s, iis given by a difference m/s s quotient. quotient ∆ r rf − ri v= = ∆ t t f − ti Instantaneous Velocity Instantaneous Suppose the time interval ∆t is very small Suppose then the instantaneous velocity v, iin m/s, iis n m/s s the definition of a derivative. rf − ri ∆r v = lim = lim ∆ t→ 0 ∆ t ∆ t→ 0 t − t f i Geometric Meaning of the Average & Instantaneous Velocity Instantaneous r r(t+ ∆ t ) d r /d t Secant Line Tangent Line ∆ r /∆ t r(t) t t+ ∆ t t Average Acceleration Average Let the initial time and the initial velocity be Let written as ti and vi. Let the final time and Let final velocity be written as tf and vf . The average acceleration, in m/s2, is m/s ∆ v v f − vi a= = ∆ t t f − ti Instantaneous Acceleration Instantaneous Suppose the time interval ∆t is very small Suppose then the instantaneous acceleration a, iin n m/s2, iis m/s s v f − vi ∆v a = lim = lim ∆ t→ 0 ∆ t t f → ti t − t f i Geometric Meaning of the Average & Instantaneous Acceleration Instantaneous v v (t+ ∆ t ) d v /dt Secant Line Tangent Line ∆ v /∆ t v (t) t t+ ∆ t t Let the initial values of the quantities be Let denoted by xi and yi. Let the final values of the Let quantities be denoted by xf and yf . Then the average quantity z is given by Average & Instantaneous Quantities Quantities ∆ y y f − yi z= = ∆ x x f − xi . If the interval ∆x is very small then the If instantaneous quantity z is y f − yi ∆y z = lim = lim ∆ x→ 0 ∆ x x f → xi x − x . f i ...
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