87.3 a vertically oriented spring 1. Is the motion of this system simple harmonic? Fares x=0 m New equilibrium position: l= mg total =-k(xe+x)i+ mgi 87. 3 a vertically oriented spring Ftotal=-k(x '+r)i+mgi=-kxi Kxi= ma,I=m-t dt dx k +x=0 dt m The origin ofx is the new equilibrium position. 6
6 §7.3 A vertically oriented spring 1. Is the motion of this system simple harmonic? New equilibrium position: kx mg e ′ = F k x x i mgi e ˆ ˆ ( ) total = − ′ + + r §7.3 A vertically oriented spring i t x kxi max i m ˆ d d ˆ ˆ 2 2 − = = F k x x i mgi kxi x e ˆ ˆ ˆ ( ) ,total = − ′ + + = − r m k x m k t x = + = ω 0 d d 2 2 The origin of x is the new equilibrium position
87.3 A vertically oriented spring Example 1: as shown in Fig. 1, n when the block of mass m falls freely and make a completely h unelastic collision with the plate of mass m, the system will oscillate up and down. Find the k T,Aandφ of the motion. 1g. Solution: The system is composed of 2 m and k. The angular frequency and the period are respectively 2m T=2丌 2m k 7.3 A vertically oriented spring From the initial condition ≈、mg∠0 k h m2gh=2m→V >0 t=0 2 We can obtain the amplitude k A=1x6+2 42 mg 1+ kh mg
7 §7.3 A vertically oriented spring Solution: The system is composed of 2 m and k. , 2m k ω = k m T 2 = 2π The angular frequency and the period are respectively m m h k Example 1: as shown in Fig. 1, when the block of mass m falls freely and make a completely unelastic collision with the plate of mass m, the system will oscillate up and down. Find the T, A and φ of the motion. Fig. 1 §7.3 A vertically oriented spring 0 2 0 = > gh v 0 0 2 2 0 m gh mv k mg x = =− < m m h k x o t =0 0 v r x0 From the initial condition mg kh k mg k mgh k v m g A x = + = + = + 1 2 2 2 2 2 2 0 0 ω We can obtain the amplitude
87.3 A vertically oriented spring From the initial condition x 小==<0 >兀 vn=- Dosing>0→sinφ<0 B=arct(--0)+r=arct 十兀 g 87.4 simple harmonic motion and the uniform circular motion I. Gelileo's observation of the moons of Jupiter 5 an.15202530Feb.510152025Mar.1 What can you imagine from the results? 8
8 π π ω φ = − + = + mg kh x v arctg( ) arctg 0 0 sin 0 cos 0 0 0 = − > = < ω φ φ v A A x sinφ < 0 φ > π From the initial condition §7.3 A vertically oriented spring §7.4 simple harmonic motion and the uniform circular motion 1. Gelileo’s observation of the moons of Jupiter What can you imagine from the results?
87.4 simple harmonic motion and the uniform circular motion >0 vKO KO 0 Simple harmonic motion can be described as the projection of a uniform circular motion along a diameter of the circle. A is called rotating vector. 7. 4 simple harmonic motion and the uniform circular motion v(1) x(t)=Acos(@t+o) v(t)=- Ao sin(ot+φ)
9 §7.4 simple harmonic motion and the uniform circular motion Simple harmonic motion can be described as the projection of a uniform circular motion along a diameter of the circle. A is called rotating vector. r A r ϕ π 2 3 = ϕ =π ϕ = 0 ϕ = π 2 0 0 < > v x 0 0 > > v x 0 0 > < v x 0 0 < < v x ω A r §7.4 simple harmonic motion and the uniform circular motion A ωA x(t) = Acos(ωt +φ ) v(t) = −Aω sin(ωt +φ )