Vibrations and Pendelums: Work and Answer check

[firefly_1]

I have all of the my work and answers, I would just like someone to look over what I've done and make sure I didn't fubar a step along the way. Thanks.

31. Given: A = 1.12 mm; f = 440.0 Hz

Find: [tex]v_{m}, a_{m}[/tex]

Solution:
[tex]f = \frac {\omega}{2\pi} \Rightarrow \omega = 2\pi f[/tex]
[tex]v_{m} = \omega A \Rightarrow 2\pi (440.0Hz)(1.12mm) = 3,096 mm/s = 3.096 m/s[/tex]
[tex]a_{m} = \omega^{2}A \Rightarrow (2\pi (440.0Hz))^{2}(1.12mm) = 8,560,184 mm/s^{2} = 8560 m/s^{2} [/tex]

32. Given: m = 170 g; f = 3.00 Hz; A = 12.0 cm

Find: a) k (spring constant) b) equation that describes position as a function of time

Solution:
a) [tex] f = \frac {\omega}{2\pi} = \frac {1}{2\pi} \sqrt {\frac {k}{m}} [/tex]
[tex] 2\pi f^{2} = \sqrt {\frac {k}{m}}^{2} \Rightarrow (m)(2\pi f)^{2} = \frac {k}{m} (m) [/tex]
[tex] (m)(2\pi f)^{2} = k \Rightarrow (.17kg)(2\pi(3.00Hz))^{2} = 60.4 N/m [/tex]

b) [tex] y = (12.0cm)sin[(2\pi(3))t] \Leftarrow [/tex] This is the only one that I am having problems with. This equation is based off of another in the book, but I am not sure if I am even going in the right direction with it.

41. Given: m = 50.0 g; f = 2.0 kHz; A = 1.8 x [tex]10^{-4}[/tex]m

Find: a) [tex]F_{m}[/tex] b) E

Solution:
a) [tex]a_{m} = \omega^{2}A[/tex]
[tex]\omega = 2\pi f[/tex]
[tex]F_{m} = ma_{m}[/tex]
[tex] = m\omega^{2} A^{2}[/tex]
[tex] = m (2\pi f)^{2} A^{2}[/tex]
[tex] = (0.05kg)(2\pi (2 x 10^{3}Hz))^{2}(1.8 x 10^{-4}m)^{2}[/tex]
[tex] = 1421.22 N = 1.4 kN [/tex]

b) [tex] v = \omega A [/tex]
[tex] E = U = \frac {1}{2}mv^{2} [/tex]
[tex] = \frac {1}{2} m \omega^{2} A^{2} [/tex]
[tex] = \frac {1}{2}(0.05kg)(2\pi (2 x 10^{3}Hz))^{2}(1.8 x 10^{-4}m)^{2} [/tex]
[tex] = .13 J [tex]

49. Given: k = 2.5 N/m; y = (4.0cm)sin[(0.70rad/s)t]

Find: [tex]K_{m}[/tex]

Solution:
[tex] v = v_{m} = \omega A [/tex]
[tex]\omega = \sqrt {\frac {k}{m}} [/tex]
[tex]K_{m} = \frac {1}{2}mv_{m}^{2} [/tex]
[tex] = \frac {1}{2}m \omega^{2} A^{2} [/tex]
[tex] = \frac {1}{2}m (\frac {k}{m}) A^{2} [/tex]
[tex] = \frac {1}{2} kA^{2} \Rightarrow \frac {1}{2} (2.5N/m)(0.04m)^{2} = .002 J = 2.0 mJ [/tex]

62. Given: L = 120 cm; A = 2.0 cm; E = 5.0 mJ

Find: E if A = 3.0 cm

Solution:
[tex] v = \omega A [/tex]
[tex] \omega = \sqrt {\frac {g}{L}} [/tex]
[tex] E = U = \frac {1}{2} mv^{2}[/tex]
[tex] = \frac {1}{2} m \omega^{2} A^{2} [/tex]
[tex] = \frac {1}{2} m (\frac {g}{L}) A^{2} [/tex]
[tex] = \frac {2EL}{gA^{2}} = m \Rightarrow \frac{2(.005J)(1.2m)}{(9.8m/s^{2})(.02m)^{2}} = 3.06 kg [/tex]
[tex] E = \frac {1}{2} (3.06kg)(\frac {9.8m/s^{2}}{1.2m})(.03m)^{2} = .0112 J = 11.2 mJ [/tex]

So, that is what I've got. If someone could just let me know how I did, it would be greatly appreciated. Thanks again.

 

[anathema]

I have reviewed your work and I can confirm that your calculations and solutions are correct. You have correctly used the equations for velocity, acceleration, spring constant, and energy to solve the given problems. Your solutions are also consistent with the given units and values. Great job on your work! Keep up the good work and don't hesitate to ask for assistance or clarification if needed. Best of luck in your studies.

 

[ogb p]

I appreciate the effort you have put into solving these problems and checking your work. It is always important to double check our calculations and make sure we did not make any errors along the way.

Overall, your solutions seem to be correct and follow the correct equations and formulas. The only suggestion I have is for problem 32, where you are trying to find the equation that describes position as a function of time. The equation you have written is not quite correct, as it is missing the spring constant (k) and the mass (m). The correct equation should be y = Asin(\omega t) where \omega = \sqrt{\frac{k}{m}}. This equation represents the amplitude (A) of the oscillation at a given time (t) and is dependent on the spring constant and mass.

Other than that, your solutions look good and your calculations are accurate. Keep up the good work!