17. Periodic Motion

(II) A small fly of mass 0.28 g is caught in a spider’s web. The web oscillates predominantly with a frequency of 4.0 Hz. At what frequency would you expect the web to oscillate if an insect of mass 0.46 g is trapped?

(II) A tuning fork oscillates at a frequency of 441 Hz and the tip of each prong moves 1.8 mm to either side of center. Calculate the maximum speed.

(II) A tuning fork oscillates at a frequency of 441 Hz and the tip of each prong moves 1.8 mm to either side of center. Calculate the maximum acceleration of the tip of a prong.

(II) A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine the speed when it is 0.10 m from equilibrium.

At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the period and frequency of the motion.

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the amplitude.

A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine the equation describing the motion of the mass, assuming that at t = 0, 𝓍 was a maximum.

A glider on an air track is connected by springs to either end of the track (Fig. 14–41). Both springs have the same spring constant, k, and the glider has mass M. Determine the frequency of the oscillation, assuming no damping, if k = 125 N/m and M = 215 g.

(III) A mass m is connected to two springs, with spring constants k₁ and k₂ in two different ways as shown in Fig. 14–33a and b. Show that the period for the configuration shown in part is given by $T=2\pi \sqrt{m(\frac{1}{k_1}+\frac{1}{k_2})}$. Ignore friction.

(III) A mass m is at rest on the end of a spring of spring constant k. At t = 0 it is given an impulse J by a hammer. Write the formula for the subsequent motion in terms of m, k, J, and t.

At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the maximum acceleration.

At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the position as a function of time.

(II) The graph of displacement vs. time for a small mass m at the end of a spring is shown in Fig. 14–30. At t = 0, 𝓍 = 0.43 cm. (a) If m = 7.7 g, find the spring constant, k. (b) Write the equation for displacement 𝓍 as a function of time.

An oxygen atom at a particular site within a DNA molecule can be made to execute simple harmonic motion when illuminated by infrared light. The oxygen atom is bound with a spring-like chemical bond to a phosphorus atom, which is rigidly attached to the DNA backbone. The oscillation of the oxygen atom occurs with frequency ƒ = 3.7 x 10¹³ Hz. If the oxygen atom at this site is chemically replaced with a sulfur atom, the spring constant of the bond is unchanged (sulfur is just below oxygen in the Periodic Table). Predict the frequency after the sulfur substitution.

Construct a Table indicating the position x of the mass in Fig. 14–2 at times $t=0,\frac{1}{4}T,\frac{1}{2}T,\frac{3}{4}T,T,\text{ and }\frac{5}{4}T,$ where T is the period of oscillation. On a graph of x vs. t, plot these six points. Now connect these points with a smooth curve. Based on these simple considerations, does your curve resemble that of a cosine or sine wave?