Physics 2514 Lecture 22 P. Gutierrez Department of Physics & Astronomy University of Oklahoma Physics 2514 p. 1/15
Information Information needed for the exam Exam will be in the same format as the practice with the same number of questions Bring a # 2 pencil & eraser Calculators will be allowed No cell phones, no laptops,... Only exam, pencil, eraser, calculator allowed on desk. Bring student id with you You will need to know Student id number Discussion section # Your name Physics 2514 p. 2/15
Exam 2 Exam 2 will cover material in lectures 11 through 21. This includes material in Chapters 5 through 8 Kinematics Projectile motion & general 2-d motion with constant acceleration; Circular motion; Relative motion. Dynamics Newton s 3 laws of motion. Physics 2514 p. 3/15
Formulas to be given The following formulas will be given s(t) = 1 2 at2 + v 0 t + s 0 v(t) = at + v 0 v 2 v 2 0 = 2a(s s 0 ) s = rθ v = v + V Fnet = m a f k = µn f s µn Physics 2514 p. 4/15
Review Kinematics Started with discussion of motion Displacement final minus initial position, with direction pointing from initial to final position r = r = r f r i Velocity is the rate of change in position v avg = r t v(t) = d r dt Acceleration is rate at which the velocity changes a avg = v t a(t) = d v dt Physics 2514 p. 5/15
Review 2-d Kinematics Kinematic equations for constant acceleration (Apply independently in each dimension) Position Velocity Combining equations s(t) = 1 2 at2 + v 0 t + s 0 v(t) = at + v 0 v 2 v 2 0 = 2a(s s 0 ) Find constraint that ties equations together How long does it take to fall, how far does it travel horizontally Physics 2514 p. 6/15
Steps in Problem Solving Steps in problem solving A) Rewrite the problem eliminating all extraneous information. (What are you given, what are you looking, what are the constraints); B) Draw a diagram along with a coordinate system, label each object with the variables associated with it; C) What are the known and unknown quantities, which unknowns are you solving for; D) Write down the equations associated with the problem, and solve the problem algebraically (SIMPLIFY!!!) E) Finally, substitute numbers into the equation, and calculate the numerical solution Physics 2514 p. 7/15
Example A catapult is tested by Roman legionnaires. They tabulate the results in a papyrus scroll and 2000 years later an archaeological team reads (distance translated into modern units): Range = 0.20 km; angle of launch = π/3. What is the initial velocity of launch of the boulder? Object launched at angle π/3 radians, travels 0.2 km. Determine initial velocity using the constraint, how long does it take to hit the ground? g replacements v0 θ v y θ v 0 v x Physics 2514 p. 8/15
Example Object launched at angle π/3 radians, travels 0.2 km. Determine initial velocity using the constraint, how long it take to hit the ground? g replacements v0 θ v y θ v 0 v x y f = 1 2 gt2 f + (v 0 sin θ)t f x f = (v 0 cos θ)t f t f = 2x f tan θ i 2y f g v 0 = x f gxf tan θ i gy f 2(xf sin θ y f cos θ) = 8.4 s = 47.6 m/s x f x(t f ) = 200 m, y f y(t f ) = 0 m, θ i θ(t i ) = π/3 rads Physics 2514 p. 9/15
Circular Motion Arc-Length & Angle Tangential & Angular Velocity s = rθ Uniform Motion v = constant v = ds dt = r dθ dt = rω s(t) = v 0 t + s 0 θ = ω 0 t + θ 0 v(t) = v 0 ω(t) = ω 0 Physics 2514 p. 10/15
Nonuniform Circular Motion Consider that case when the speed around a circle changes. Physics 2514 p. 11/15
Kinematic Equations The kinematic equations for uniform circular motion were derived earlier, here we consider nonuniform motion Motion along arc is 1-D with tangential acceleration and velocity determining motion s = s 0 + v ot t + 1 2 a tt 2 v t = v 0t + a t t Now divide by r 1 r (s = s 0 + v 0t t + 1 2 a tt 2 ) 1 r (v t = v 0t + a t t) θ = θ 0 + ω 0 t + 1 2 αt2 ω = ω 0 + αt α = a t /r Physics 2514 p. 12/15
Centripetal Acceleration Calculate average acceleration CB = r 2 r 1 = v 2 t v 1 t = v t Angles ) ABO: θ + α + α = 180 DAC: φ + α + α = 180 θ = φ Similar triangles CB AB = AB AO v t v t = v t r Average radial acceleration a average r = v t = v2 r a r = lim t 0 v t = v2 r Physics 2514 p. 13/15
Example A car starts from rest on a curve with a radius of 120 m and accelerates at 1.0 m/s 2. Through what angle will the car have traveled when the magnitude of its total acceleration is 2.0 m/s 2. Knowns cements a r v t a t a t = 1.0 m/s 2 a f = a 2 t + a 2 r = 2.0 m/s 2 v t0 = 0 m/s r = 120 m θ 0 = 0 rads Unknowns θ f =? rads Physics 2514 p. 14/15
Example A car starts from rest on a curve with a radius of 120 m and accelerates at 1.0 m/s 2. Through what angle will the car have traveled when the magnitude of its total acceleration is 2.0 m/s 2. Knowns a t = 1.0 m/s 2 a f = a 2 t + a 2 r = 2.0 m/s 2 v t0 = 0 m/s r = 120 m θ 0 = 0 rads How long to reach a f ) 2 a f = a 2 t + = 2.0 m/s 2 v tf = 14.4 m/s v tf = a t t f ( v 2 tf r t f = 14.4 s Unknowns θ f =? rads How far θ f = 1 2 αt2 f = 1 2 a t r t 2 f θ f = 0.864 rads Physics 2514 p. 15/15