1. Hang a mass on a vertical spring: imagine an origin at the place where the mass is at rest with i directed down as shown in Figure 4. Now set the mass into oscillation in the vertical direction by lowering the mass a distance A and letting it go. The subsequent motion is called simple harmonic oscillation and will be investigated in some detail in Chapter 7. 0 We shall see that the position vector of a particle executing such one-dimensional oscillatory motion is given by the expression 1=LAcos(or)Ji, where A is expressed in meters, and o is expressed Fig 4 in radians per second; both are constants (a)Find the velocity vector and the acceleration vector as function of time (b) What are the greatest magnitudes of the velocity and acceleration vectors? (c)What is the earliest (nonnegative)time that the position vector attains maximum magnitude? When the position vector has its greatest magnitude, what is the magnitude of the velocity vector? What is the magnitude of the acceleration vector at the same time? (d) At what time(t>0s) does the position vector first attain a magnitude of 0 m? At this time, what are the magnitudes of velocity and acceleration vectors? (a) The velocity vector is v(=dro-dIAcos(oni=-oAsin(oni The acceleration vector is a(t) dv(o d dt d LOAsin(oD)i=[-@'Acos(or)] Vmax=[-oAsin(nmax = OA amax=[o Acos(onI (c)Using F(()=[Acos(@D], so when cos(ot=1, we have Imax=[Acos(@D)Imax=A Thus ot=±krk=0,1,2 k=0,1,2, Thinking the subsequent motion from beginning, so we neglect fOs. The earliest time is o(k=1) that the position vector attains maximum magnitude At the same time, the magnitude of the velocity vector is 下(o)=-asi(o)l=asn( =0m/s The magnitude of the acceleration vector is1. Hang a mass on a vertical spring; imagine an origin at the place where the mass is at rest with i ˆ directed down as shown in Figure 4. Now set the mass into oscillation in the vertical direction by lowering the mass a distance A and letting it go. The subsequent motion is called simple harmonic oscillation and will be investigated in some detail in Chapter 7. We shall see that the position vector of a particle executing such one-dimensional oscillatory motion is given by the expression r t A t i ˆ ( ) =[ cos(ω )] r , where A is expressed in meters, and ω is expressed in radians per second; both are constants. (a) Find the velocity vector and the acceleration vector as function of time. (b) What are the greatest magnitudes of the velocity and acceleration vectors? (c) What is the earliest (nonnegative) time that the position vector attains maximum magnitude? When the position vector has its greatest magnitude, what is the magnitude of the velocity vector? What is the magnitude of the acceleration vector at the same time? (d) At what time (t ≥ 0 s) does the position vector first attain a magnitude of 0 m? At this time, what are the magnitudes of velocity and acceleration vectors? Solution: (a) The velocity vector is A t i A t i t t r t v t ˆ [ sin( )] ˆ [ cos( )] d d d d ( ) ( ) = = ω = −ω ω r v . The acceleration vector is A t i A t i t t v t a t ˆ [ cos( )] ˆ [ sin( )] d d d d ( ) ( ) 2 = = −ω ω = −ω ω v v (b) vmax = [−ωAsin(ωt)]max = ωA v a A t A2 max 2 max = [−ω cos(ω )] = ω v (c) Using r t A t i ˆ ( ) =[ cos(ω )] r , so when cos(ωt) = 1, we have rmax =[Acos(ωt)]max = A . Thus ωt = ±kπ k = 0,1,2,... and = ± k = 0,1,2,... k t ω π Thinking the subsequent motion from beginning, so we neglect t=0s. The earliest time is t = (k = 1) ω π that the position vector attains maximum magnitude. At the same time, the magnitude of the velocity vector is ( ) = − sin( ) = − sin( ) = 0m/s ω π v t ωA ωt ωA ω v . The magnitude of the acceleration vector is Fig.4 x 0 0 A x i ˆ