CHAPTER 14
Simple Harmonic Motion
第十四章 简谐运动

14.2 A $50-\mathrm{g}$$50-\mathrm{g}$50-g50-\mathrm{g} mass hangs at the end of a Hookean spring. When 20 g more are added to the end of the spring, it stretches 7.0 cm more. (a) Find the spring constant. (b) If the 20 g are now removed, what will be the period of the motion?
14.2 一个 $50-\mathrm{g}$$50-\mathrm{g}$50-g50-\mathrm{g} 质量悬挂在一个胡克弹簧的末端。当在弹簧末端再加上 20 克时，它伸长了 7.0 厘米。(a) 计算弹簧常数。(b) 如果现在移除这 20 克，运动的周期将是多少？
14.3 A spring is stretched 4 cm when a mass of 50 g is hung on it. If a total of 150 g is hung on the spring and the mass is started in a vertical oscillation, what will the period of the oscillation be?
14.3 当一个 50 克的重物挂在弹簧上时，弹簧被拉伸了 4 厘米。如果在弹簧上挂上总共 150 克的重物，并且重物开始进行垂直振荡，那么振荡的周期将是多少？
14.7 A $0.5-\mathrm{kg}$$0.5-\mathrm{kg}$0.5-kg0.5-\mathrm{kg} body performs simple harmonic motion with a frequency of 2 Hz and an amplitude of 8 mm . Find the maximum velocity of the body, its maximum acceleration, and the maximum restoring force to which the body is subjected.
14.7 一个 $0.5-\mathrm{kg}$$0.5-\mathrm{kg}$0.5-kg0.5-\mathrm{kg} 物体以 2 Hz 的频率和 8 mm 的振幅进行简单谐振动。求该物体的最大速度、最大加速度和物体所受的最大恢复力。
14.9 A body describing SHM has a maximum acceleration of $8\pi \mathrm{m}/{\mathrm{s}}^2$$8\pi \mathrm{m}/{\mathrm{s}}^2$8pim//s^(2)8 \pi \mathrm{m} / \mathrm{s}^{2} and a maximum speed of $1.6\mathrm{m}/\mathrm{s}$$1.6\mathrm{m}/\mathrm{s}$1.6m//s1.6 \mathrm{~m} / \mathrm{s}. Find the HW period $T$$T$TT and the amplitude $R$$R$RR.
14.9 一个做简谐运动的物体具有最大加速度 $8\pi \mathrm{m}/{\mathrm{s}}^2$$8\pi \mathrm{m}/{\mathrm{s}}^2$8pim//s^(2)8 \pi \mathrm{m} / \mathrm{s}^{2} 和最大速度 $1.6\mathrm{m}/\mathrm{s}$$1.6\mathrm{m}/\mathrm{s}$1.6m//s1.6 \mathrm{~m} / \mathrm{s} 。求 HW 周期 $T$$T$TT 和振幅 $R$$R$RR 。
14.18 A ball moves in a circular path of $0.15-\mathrm{m}$$0.15-\mathrm{m}$0.15-m0.15-\mathrm{m} diameter with a constant angular speed of $20\mathrm{rev}/\min$$20\mathrm{rev}/\min$20rev//min20 \mathrm{rev} / \mathrm{min}. Its shadow performs simple harmonic motion on the wall behind it. Find the acceleration and speed of the shadow (a) at a turning point of the motion, (b) at the equilibrium position, and $(c)$$(c)$(c)(c) at a point 6 cm from the equilibrium position.
14.18 一个球以恒定的角速度 $20\mathrm{rev}/\min$$20\mathrm{rev}/\min$20rev//min20 \mathrm{rev} / \mathrm{min} 在直径为 $0.15-\mathrm{m}$$0.15-\mathrm{m}$0.15-m0.15-\mathrm{m} 的圆形路径上运动。它的影子在后面的墙上做简谐运动。求影子的加速度和速度：(a) 在运动的转折点，(b) 在平衡位置，以及 $(c)$$(c)$(c)(c) 在距离平衡位置 6 厘米的点。
14.21 A block of mass 4 kg hangs from a spring of force constant $k=400\mathrm{N}/\mathrm{m}$$k=400\mathrm{N}/\mathrm{m}$k=400N//mk=400 \mathrm{~N} / \mathrm{m}. The block is pulled down 15 cm HW below equilibrium and released. (a) Find the amplitude, frequency, and period of the motion. (b) Find the kinetic energy when the block is 10 cm above equilibrium.
14.21 一个质量为 4 千克的块体悬挂在一个力常数为 $k=400\mathrm{N}/\mathrm{m}$$k=400\mathrm{N}/\mathrm{m}$k=400N//mk=400 \mathrm{~N} / \mathrm{m} 的弹簧上。块体被拉下 15 厘米 HW 到达平衡位置以下，然后释放。(a) 找出运动的振幅、频率和周期。(b) 当块体在平衡位置上方 10 厘米时，找出动能。
14.22 Refer to Prob. 14.21. How long does it take the block to go from 12 cm below equilibrium (on the way up) to 9 cm above equilibrium?
14.22 参考问题 14.21。块体从平衡位置下方 12 厘米（上升过程中）到平衡位置上方 9 厘米需要多长时间？
14.27 For the motion shown in Fig. 14-5, what are the amplitude, period, and frequency?
14.27 对于图 14-5 所示的运动，振幅、周期和频率分别是多少？

Fig. 14-5 图 14-5
14.36 A $0.2-\mathrm{kg}$$0.2-\mathrm{kg}$0.2-kg0.2-\mathrm{kg} mass suspended from a spring describes a simple harmonic motion with a period $T$$T$TT of 3 s and amplitude $R$$R$RR of 10 cm . At $t=0$$t=0$t=0t=0 the mass passes upward through the equilibrium position. (a) Find the force constant $k$$k$kk of the spring. (b) Find the displacement, velocity, and acceleration of the mass when $t=1\mathrm{s}$$t=1\mathrm{s}$t=1st=1 \mathrm{~s}.
14.36 一个质量为 $0.2-\mathrm{kg}$$0.2-\mathrm{kg}$0.2-kg0.2-\mathrm{kg} 的物体悬挂在弹簧上，描述了一个周期为 $T$$T$TT 3 秒、振幅为 $R$$R$RR 10 厘米的简单谐运动。在 $t=0$$t=0$t=0t=0 时，物体向上通过平衡位置。(a) 计算弹簧的力常数 $k$$k$kk 。(b) 计算当 $t=1\mathrm{s}$$t=1\mathrm{s}$t=1st=1 \mathrm{~s} 时物体的位移、速度和加速度。
14.37 Refer to Prob. 14.36. Verify that the sum of the potential and kinetic energies at $t=1\mathrm{s}$$t=1\mathrm{s}$t=1st=1 \mathrm{~s} is equal to $\frac{1}{2}k{R}^2$$\frac{1}{2}k{R}^2$(1)/(2)kR^(2)\frac{1}{2} k R^{2}.
14.37 参见问题 14.36。验证在 $t=1\mathrm{s}$$t=1\mathrm{s}$t=1st=1 \mathrm{~s} 时势能和动能的总和等于 $\frac{1}{2}k{R}^2$$\frac{1}{2}k{R}^2$(1)/(2)kR^(2)\frac{1}{2} k R^{2} 。

14.44 The period of a simple pendulum 35.90 cm long is $T=1.200\mathrm{s}$$T=1.200\mathrm{s}$T=1.200sT=1.200 \mathrm{~s}. Find the value of $g$$g$gg at this location.
14.44 一个长 35.90 厘米的简单摆的周期是 $T=1.200\mathrm{s}$$T=1.200\mathrm{s}$T=1.200sT=1.200 \mathrm{~s} 。在这个位置找到 $g$$g$gg 的值。
14.46 A simple pendulum 60 cm long has a $200-\mathrm{g}$$200-\mathrm{g}$200-g200-\mathrm{g} ball. The pendulum is pulled aside ${15}^{\circ }$${15}^{\circ }$15^(@)15^{\circ} and released. If the timing HW clock is started just as the ball moves through its lowest position, write the equation of motion for the pendulum in terms of the pendulum angle in degrees.
14.46 一个长 60 厘米的简单摆有一个 $200-\mathrm{g}$$200-\mathrm{g}$200-g200-\mathrm{g} 球。摆被拉到一侧 ${15}^{\circ }$${15}^{\circ }$15^(@)15^{\circ} 然后释放。如果在球经过最低位置时正好启动计时器，写出摆的运动方程，以摆角度（以度为单位）表示。
14.56 Springs $A$$A$AA and $B$$B$BB have spring constants of $2000\mathrm{N}/\mathrm{m}$$2000\mathrm{N}/\mathrm{m}$2000N//m2000 \mathrm{~N} / \mathrm{m} and $1000\mathrm{N}/\mathrm{m}$$1000\mathrm{N}/\mathrm{m}$1000N//m1000 \mathrm{~N} / \mathrm{m}, respectively. Spring $A$$A$AA is hung from a rigid horizontal beam and its other end is attached to an end of spring $B$$B$BB. The pair of springs is then used to suspend a body of mass 50 kg from the lower end of spring $B$$B$BB. What is the period of harmonic oscillation of the system?
14.56 弹簧 $A$$A$AA 和 $B$$B$BB 的弹簧常数分别为 $2000\mathrm{N}/\mathrm{m}$$2000\mathrm{N}/\mathrm{m}$2000N//m2000 \mathrm{~N} / \mathrm{m} 和 $1000\mathrm{N}/\mathrm{m}$$1000\mathrm{N}/\mathrm{m}$1000N//m1000 \mathrm{~N} / \mathrm{m} 。弹簧 $A$$A$AA 悬挂在一个刚性水平梁上，另一端连接到弹簧 $B$$B$BB 的一端。然后这对弹簧用于从弹簧 $B$$B$BB 的下端悬挂一个质量为 50 kg 的物体。该系统的谐振动周期是多少？

14.58 Two identical springs each have $k=20\mathrm{N}/\mathrm{m}$$k=20\mathrm{N}/\mathrm{m}$k=20N//mk=20 \mathrm{~N} / \mathrm{m}. A $0.3-\mathrm{kg}$$0.3-\mathrm{kg}$0.3-kg0.3-\mathrm{kg} mass is connected to them as shown in Fig. 14-16(a) HW and (b). Find the period of motion for each system. Ignore friction forces.
14.58 两个相同的弹簧各有 $k=20\mathrm{N}/\mathrm{m}$$k=20\mathrm{N}/\mathrm{m}$k=20N//mk=20 \mathrm{~N} / \mathrm{m} 。一个 $0.3-\mathrm{kg}$$0.3-\mathrm{kg}$0.3-kg0.3-\mathrm{kg} 质量的物体如图 14-16(a) HW 和 (b) 所示连接在它们上。求每个系统的运动周期。忽略摩擦力。

(a)

(b)

Fig. 14-16 图 14-16