Physics 170 Week 7, Lecture 2

Transcription

1 Physics 170 Week 7, Lecture 2 goronws/170 Physics Week 7, Lecture 2 1

2 Textbook Chapter 12:Section Physics Week 7, Lecture 2 2

3 Learning Goals: Learn about the kinematics of a particle moving in three imensional space by escribing the position as a three-imensional vector. Become familiar with calculus operations applie to vectors to relate position, velocity an acceleration so that you unerstan the following table: v(t) = t r(t), a(t) = t v(t) r(t 2 ) r(t 1 ) = t2 t 1 t v(t), v(t 2 ) v(t 1 ) = t2 t 1 t a(t) Unerstan the example of a particle moving in a circle with constant spee. Solve an unerstan the example of projectile motion: the special case where the acceleration is a constant. Physics Week 7, Lecture 2 3

4 Review: For motion in one imension, position s(t), velocity v(t) an acceleration a(t) are escribe by functions of time. They are relate by calculus operations v(t) = t s(t), a(t) = t v(t) s(t 2 ) s(t 1 ) = t2 t 1 t v(t), v(t 2 ) v(t 1 ) = t2 t 1 t a(t) The erivative of a function at time t is the slope of its graph at t. The efinite integral of a function from t 1 to t 2 is the area uner its graph between t 1 an t 2. In this lecture we will generalize to motion in three imensions. Physics Week 7, Lecture 2 4

5 Derivatives an integrals of some common functions: t tα = αt α 1, t t α = tα+1 α constant t t t ln t = 1 t t eβt = βe βt, sin(ωt) = ω cos(ωt), cos(ωt) = ω sin(ωt),, t 1 t = ln t + constant t e βt = eβt β + constant t cos(ωt) = sin(ωt) ω t sin(ωt) = cos(ωt) ω + constant + constant Physics Week 7, Lecture 2 5

6 Some other properties of erivatives an integrals If a is a constant an f(t) is a function, then t (af(t)) = a t f(t), t af(t) = a t f(t) Derivation an Integration are linear operations The Prouct Rule The Chain Rule t (f(t) + g(t)) = t f(t) + t g(t) t (f(t) + g(t)) = tf(t) + tg(t) t (f(t)g(t)) = ( ) ( ) t f(t) g(t) + f() t g(t) t f[g(t)] = g f[g] t g(t) Physics Week 7, Lecture 2 6

7 Position In three-imensional space, the position of a particle can be specifie by giving its position vector, r = xî + yĵ + zˆk If the particle moves, its position changes with time. This is escribe by a time-epenent vector r(t) = x(t)î + y(t)ĵ + z(t)ˆk In the Cartesian representation, the three coorinates (x(t), y(t), z(t)) are each time-epenent an escribe how the components of the position vector change with time. Note that the Cartesian basis vectors, î, ĵ, ˆk o not epen on time. Physics Week 7, Lecture 2 7

8 r = x(t)î + y(t)ĵ + z(t)ˆk is the equation of a line in three imensional space. This is the line along which the particle moves. It is calle the trajectory. For example, a particle moving in a circle of raius R centere at 0 in the xy-plane has trajectory r(t) = R sin(ωt) î + R cos(ωt) ĵ What is the raius of the circle? What is the perio of the motion? R T = 2π/ω What is the value of ω if the frequency is 100Hz? ω = 2π(100Hz) Physics Week 7, Lecture 2 8

9 Distance Remember that the isplacement vector r 12 from one position r 1 to another position r 2 is given by The istance from r 1 to r 2 is r 12 = r 2 r 1 12 (t) = r 12 (t) = x 12 (t) 2 + y 12 (t) 2 + z 12 (t) 2 For the particle moving in a circle, the istance of the particle from the origin is given by r0 = r(t) 0 = R sin(ωt) î + R cos(ωt) ĵ = (R sin(ωt)) 2 + (R cos(ωt)) 2 = R (since cos 2 θ + sin 2 θ = 1) Physics Week 7, Lecture 2 9

10 Velocity The velocity is foun by taking a erivative of the position vector by time: v(t) = t r(t) The erivative by time of a Cartesian vector is that vector whose components are erivatives by time of the components v(t) = t x(t) î + t y(t) ĵ + z(t) ˆk t Physics Week 7, Lecture 2 10

11 The velocity vector is tangent to the trajectory. Physics Week 7, Lecture 2 11

12 This can be seen by consiering v(t) = t r(t) = lim ɛ 0 r(t+ɛ) r(t) ɛ Physics Week 7, Lecture 2 12

13 For the example of a particle moving in a circle, where r(t) = R sin(ωt) î + R cos(ωt) ĵ v(t) = Rω cos(ωt) î Rω sin(ωt) ĵ (We have use t cos(ωt) = ω sin(ωt) an sin(ωt) = ω cos(ωt).) t Notice that r(t) v(t) = 0 - the tangent to a circle is orthogonal to the raius. Physics Week 7, Lecture 2 13

14 Spee The spee is the magnitue of the velocity. If v(t) = v x (t)î + v y (t)ĵ + v z (t)ˆk the spee is v(t) = vx(t) 2 + vy(t) 2 + vz(t) 2 For particle motion in a circle, v(t) = Rω cos(ωt) î Rω sin(ωt) ĵ an v(t) = (Rω cos(ωt)) 2 + (Rω sin(ωt)) 2 = Rω The spee is constant. This is (the special case of a) particle moving in a circle at constant spee. Displacement is the integral of velocity: r(t 2 ) r(t 1 ) = t2 t 1 t t r(t) = t2 t 1 t v(t) Physics Week 7, Lecture 2 14

15 Distance is the integral of spee: For an infinitesimal isplacement, = r(t + ɛ) r(t) = ɛ r(t + ɛ) r(t) ɛ = ɛ v(t) For a finite isplacement, the istance that the particle actually moves is = t2 t 1 t v(t) Physics Week 7, Lecture 2 15

16 Example of motion in a circle As an example, consier the motion of a particle on a circle r(t) = R sin(ωt) î + R cos(ωt) ĵ in the time interval 0 t 2π/ω. The particle moves aroun the circle once, so its isplacement is zero. However, the istance travele is one unit of circumference, = 2πR. Let us recover this result using calculus: We showe above that the spee of the particle is a constant v(t) = Rω. The istance travele uring the time interval is the efinite integral over the interval of the spee, 2π/ω 2π/ω ( ) 2π = t v(t) = t Rω = Rω = 2πR ω 0 0 Physics Week 7, Lecture 2 16

17 Acceleration The acceleration is a vector which is gotten by taking a time erivative of the velocity vector, a(t) = t v(t) = t v x(t)î + t v y(t)ĵ + t v z(t)ˆk Remembering that the velocity is a time erivative of the position, a(t) = r(t) = x(t)î + y(t)ĵ + t2 t2 t2 t 2 z(t)ˆk This is the secon erivative by time. The magnitue of the acceleration is given by a(t) = a 2 x(t) + a 2 y(t) + a 2 z(t) Physics Week 7, Lecture 2 17

18 For the example of motion in a circle, a(t) = a x (t)î + a y (t)ĵ + a z (t)ˆk = t v(t) = t (Rω cos(ωt)) î + ( Rω sin(ωt)) ĵ ( t ) = ω 2 R sin(ωt) î + R cos(ωt) ĵ = ω 2 r(t) The acceleration of the particle is irecte opposite to the raial irection - to the insie of the circle. It has magnitue a(t) = ω 2 R = v 2 /R. This is the centripetal acceleration. Physics Week 7, Lecture 2 18

19 Derivatives an Integrals We have seen how to go from position to velocity to acceleration by taking successive time erivatives v(t) = t r(t), a(t) = t v(t) We can go in the other irection by oing time integrals, r(t) = t v(t) + c 1, v(t) = t a(t) + c 2 where c 1 an c 2 are constant vectors. Alternatively, we can fin the change of velocity an the isplacement over a time interval (t 1, t 2 ) by oing a efinite integral r(t 2 ) r(t 1 ) = t2 t 1 t v(t), v(t 2 ) v(t 1 ) = t2 t 1 t a(t) Physics Week 7, Lecture 2 19

20 For the next lecture, please rea Textbook Chapter 12:Section Physics Week 7, Lecture 2 20