1. c. $v_{\text{max}}=2\pi A/T$. Doubling A and T leaves $v_{\text{max}}$ unchanged.

2. d. Think of circular motion. At $45\text{°}$, the particle is in the first quadrant (positive x) and moving to the left (negative $v_x$).

3. $(v{}_{\text{max}}){}_{\text{C}}>(v{}_{\text{max}}){}_{\text{B}}>(v{}_{\text{max}}){}_{\text{A}}=(v{}_{\text{max}}){}_{\text{D}}$.

Energy conservation $\frac{1}{2}k{A}^2=\frac{1}{2}m{(v_{\text{max}})}^2$ gives $v_{\text{max}}=\sqrt{k/m}A$. K or m has to be increased or decreased by a factor of 4 to have the same effect as increasing or decreasing A by a factor of 2.

4. c. $v_x=0$ because the slope of the position graph is zero. The negative value of x shows that the particle is left of the equilibrium position, so the restoring force is to the right.

5. c. The period of a pendulum does not depend on its mass.

6. $\tau {}_{\text{D}}>{\tau }_{\text{B}}={\tau }_{\text{C}}>{\tau }_{\text{A}}$. The time constant is the time to decay to 37% of the initial height. The time constant is independent of the initial height.

1.

1. 13 cm

2. 9.0 cm

3. 0.41 s

5.

1. 20 cm

2. 0.13 Hz

3. $+120\text{°}$

7.

1. $5\pi \text{/}6$

2. $-14\text{ cm}/\text{s}$

3. 16 cm/s

9.

11. $x(t)=(8.0\text{cm})\text{cos}[(\pi \text{rad/s})t-\pi \text{rad}]$

13.

1. ${\varphi }_0=-{\displaystyle \frac{2}{3}}\pi \text{rad, or (}-120\text{°})$

2. $\varphi =-{\displaystyle \frac{2}{3}}\pi \text{rad, 0 rad,}{\displaystyle \frac{2}{3}}\pi \text{rad,}{\displaystyle \frac{4}{3}}\pi \text{rad}$

15.

1. 2.8 s

2. 1.41 s

3. 2.0 s

4. 1.41 s

17.

1. 2.0 cm

2. 0.63 s

3. 5.0 N/m

4. $-{\displaystyle \frac{1}{4}}\pi \text{rad}$

5. 14 cm/s

6. 20 cm/s

7. $1.00\times {10}^{-3}\text{J}$

8. 1.5 cm/s

19.

1. 2.2 N/m

2. 0.24 m

21.

1. 25 N/m

2. 0.90 s

3. 0.70 m/s

23.

1. 0.17 kg

2. 0.57 m/s

25. 7 times

27. 33 cm

29. 0.67 s

31. 54 cm

33. 1850, 0.78 m

35. $250\text{ N}/\text{m}$

37. $4.2\text{ N}/\text{m}$

39. 1.7%

41.

1. ${\displaystyle \frac{3}{4}}\text{,}{\displaystyle \frac{1}{4}}$

2. ${\displaystyle \frac{A}{\sqrt{2}}}$

43.

1. 6.4 cm

2. $160{\text{cm/s}}^2$

3. $x=-6.4\text{cm}$

4. 28 cm/s

45.

1. $10.1\mu \text{m}$

2. 64 m/s

49.

1. 1.125 Hz

2. 23 cm

3. 4.1 cm below

51. $9.2\text{ m}/\text{s}$

53. 66 rpm

55. 0.72

57. $8.7\times {10}^{-2}\text{kg}{\text{m}}^2$

59. $0.65\text{ m}/\text{s}$

61.

1. 0.84 s

2. $7.1\text{°}$

63.

1. 0.11 m, 1.7 s

65. $f={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{2T}{mL}}}$

67.

1. Highest

2. 2.5 Hz

69. 0.66 s

71. 6.0 cm

73. 236

75. $f=\sqrt{\frac{f_1^2{f}_2^2}{f_1^2+f_2^2}}$

77. ${\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{5}{7}}{\displaystyle \frac{g}{R}}}$

79. 1.8 Hz

81. 0.58 s