F. Oscillations II
13.1 Simple Harmonic Motion
1-8, 11-16, 21, 24
12: Vmax = (A)
13: Also, what is this speed in miles/hr. a) T = 1/f.
b) Vmax = (A) = wA = 2pfA (MPH answer: 141 MPH)
14: F = kx at each displacement position. Also, a = F/m
21: Notice that in graph A the y-axis is in cm and in graph b, the y-axis is in m. First, convert the y-axis into meters. Next, determine the period of each graph. The quickest way to do this is to realize that every third time the line crosses the x-axis, that is one oscillation.
Since you have no spring constant (k) given to you in this problem, you need to use these two equations in concert so that you can substitute for (k).
Equation 1: E = 1/2kA2 (solve for k so that k = 2E/A2)
Equation 2: T = 2p (substitute 2E/A2 for k in this equation)
24: E = 1/2kA2 = U = mgh. Realize that h = 5.0 m + 0.3m. Realize too that the maximum displacement of the trampoline (0.3m) is the value for A.
Set U = 0 at the point at which h = 5.3 m. Then, solve for k. Once you have k, it is good in all problems.
For part a) you need the value for k. You will have to solve for both A and h in this case (remember h = 8.0 m + some unknown value which is the same value for A). SOUNDS LIKE YOU’RE GOING TO HAVE TO DIG OUT THAT OLD FASHIONED QUADRADIC FORMULA TO ME!
13.2 Equations of Motion
26, 27, 30, 32, 33, 36, 40, 46, 48
30: Look at the equation for the period of a pendulum. What happens to gravity as you accelerate upward? Careful. While it is true you are leaving the surface of the Earth and g decreases, what happens to the “effective” g within the elevator as it accelerates upwards?
36. y = Asinwt. a) this is the value for A. b) 100 = w = 2pf solve for f. c) T = 1/f
40. Set the Period of a spring and the period of a pendulum equal to each other and solve for L.
46: Look at your answer for #40
48: Find the period, frequency and Amax from the figure on p. 447. Remember, every third time the line crosses the x axis, you have one oscillation. Remember that w = 2pf so that when you find f, you can find w and plug it into the basic equation for finding displacement based on time: y = A sin(wt + d).
4: increases because KE is greatest at this point. KE is lowest at +/-Amax.
8: 2.0 Hz
12: 0.22 m/s
14: a) F = 22.5 N; a = 45.0 m/s2 b) F = 7.50 N; a = 15.0 m/s2 c) F = 0 ; a =0
16: 0.25 kg
24: a) 0.38 m b) 8.5 x 10-3 m
30: decreases since as you accelerate upward you effectively increase the gravitational pull you are under (g + whatever you are accelerating upward at). Since g is in the denominator of the equation, as “effective” g increases, the T will decrease.
32: a) 0.31 s b) 3.2 Hz
36: a) 0.10 m b) 16 Hz c) 0.063 sec
40: Do your own proof. It’s easy!
46: 2.5 N/m
48: a) y = 0.10 m sin (10.5t + p) b) k = 38 N/m