Chapter 15, Problem 5Q

A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is 2π ph/g, where h is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced water

The graph shows the displacement from equilibrium of a mass-spring system as a function of time after the vertically hanging system was set in motion at time t = 0. Assume that the units of time are seconds, and the units of displacement are centimeters. The first t-intercept is (0.75, 0) and the first minimum has coordinates (1.75,-4). (a) What is the period T of the periodic motion? T = seconds (b) What is the frequency f in Hertz? What is the angular frequency w in radians / second? f = Hertz W = radians / second (d) Determine the amplitude A and the phase angle y (in radians), and express the displacement in the form y(t) = A cos(wt - y), with y in meters. y(t) = meters (e) with what initial displacement y(0) and initial velocity y'(0) was the system set into motion? y(0) = meters y'(0) = meters / second ***

Question 16: Find the period of the pendulum 1.03 m long in elevator moving with 1.6 m/s2 up? Calculate to one decimal. Question 17:  Water ripples are produced by making a cork oscillate up and down at a frequency of 7Hz. The peaks of the ripples are0.06 meters apart. What is the speed of the water ripples? Calculate to one decimal.

A pendulum has a period of 47.9 s.  If you were to transfer this pendulum to the surface of Mars (where g is 3.71 meters per second squared), what would the period (in seconds) be?

(a) If the coordinate of a particle varies as x = -A cos wt, what is the phase constant in equation x(t) = A cos(wt + ¢)? (Use any variable or symbol stated above along with the following as necessary: Tn.) (b) At what position is the particle at t = 0? (Use any variable or symbol stated above along with the following as necessary: 1.) X =

Considering an undamped, forced oscillator (b = 0), show that equation (1) is a solution of equation (2). (1) x = A cos(wt + ¢) d²x EF = ma → Fo sin wt – bOK – kx = m9 dt? xp dt Use an amplitude given by the following equation. (Submit a file with a maximum size of 1 MB.) Fo A V (w? - w,3)? + m

Let there be a mass M = 0.5 kg on a spring. a) If we can hang the mass on the spring in a vertical position, how can we measure the spring constant, k, of the spring? Now we use this same spring in the horizontal configuration, assuming the mass slides on a frictionless horizontal surface. b) Assuming we have measured k to be 10 N/m, how many times will the system oscillate in a minute? (assume we displace the mass from equilibrium so that the system oscillates) c) If the mass passes through the equilibrium position with a velocity vo at time t = 0, what is the amplitude of the oscillation? d) Write the equation for the horizontal displacement of the block as a function of time.

On a certain planet, a simple pendulum of length 1.4 m makes three complete oscillations in five seconds. Find the gravitational acceleration on such a planet. The results of all problems should be expressed in SI units, unless the problem tells you indicate in other units.

the general solution to a harmonic oscillator are related. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t: 1. x(t) = A cos (wt + p) and 2. x(t) = C cos (wt) + S sin (wt). Either of these equations is a general solution of a second-order differential equation (F= mā); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.) Part D Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin Equation 2 (found in Part B), which are now considered as given parameters. Express the amplitude A and phase (separated by a comma) in terms of C and S. ► View Available Hint(s) Α, φ = V ΑΣΦ ?