Wooden cube, density 620kg/m', is 0.08m tall, 0.12m wide, and 0.12m thick. It floats in a liquid with density 900kg/m'. A string attached to the bottom of the container prevents the cube from floating up to the surface. Find the tension in the string. [8]

From a differential equations course: A mass that weight 6 lb stretches a spring 1in. The system is acted on by an external force 11* sin⁡(19.5959*t) lb.   If the mass is pushed up 4 in and then released, determine the position of the mass at any time t. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units. NOTES: Answer should be a function u(t).  The 'guess' when using the method of undetermined coefficients is: t(A*cos(19.5959*t) + B*sin(19.5959t)). I just have not been able to finish the MUC part to find the particular soluton to the Non-Homogeneous Equation. The general solution to the homogeneous equation is: u_c = C1*cos(19.5959t) + C2*sin(19.5959t)

Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable…

A laminar boundary layer profile may be assumed to be approximately of the form u/U₁ = f(n) = f(y/6) i) Use an integral analysis with the following two-segment velocity profile, f(n)=(n/6)(10-3n-13), for 0≤17 ≤0.293 and ƒ (7) = sin (лn/2) for 0.293≤ n ≤1, to find expressions for the displacement thickness &, the momentum thickness e, the shape factor H, the skin-friction coefficient a, and the drag coefficient CD.

Coupled Harmonic Oscillators X2 :0 eeeee Leeee NM ) Write down the 2nd law for each of the masses. Use coordinates x, aud x, for m and M, and i,, *2 and a, a2. Hirnt: the force from spring I on monly depends on x, , but the ferce from spring 2 on m depand s on (x2-x,) two differantial 2) To simplify, let k, -k,k and m-M. Rewrite egg equatious from i) right above each other. your 3) Detine two new variables, X (Greek"chi") - x,+X2 and Ax = X,-x 2 two di ferential equations to produce Then add and subtract your very simple (harmanic) ones. 4) Write down the solutiou to bothe equations using Wask+2k2 ond evefficients A, B, C, and D two new , but w. : un 5) Now assune k-10k2 (this means that the two masses are "weakly coupled"). Also assume X,(0) = -10cm, x,(0)=0, xz (0)= o, x,(0)=0. Solve for A,B, C, D, and solve for x (t).

A mass of 55 grams stretches a spring by 8 cm. (Note that this means the forces balance, and thus mg = kx where m = 55 grams is mass, g = 981 cm/s is acceleration due to gravity, k is the spring constant, and x = 8 cm is the displacement.) The mass is set in motion from this equilibrium position with an initial downward velocity of 23 cm/s, and there is no damping. Find the position u (in cm) of the mass at any time t (in s). (Assume that position is measured upward from the equilibrium position.) u(t) Find the frequency (in radians per second), period (in seconds), and amplitude (in cm) of the motion. Frequency is Period is Amplitude is

Full solution for the whole qn please :) Some ans: b) sq rt 5 gL

Why is Logger Pro set up to report the time between every other blocking of the Photogate? Why not the time between every block? Using either Graphical Analysis or graph paper, plot a graph of pendulum period vs. amplitude in degrees. Scale each axis from the origin (0,0). Does the period depend on amplitude? Explain. Using either Graphical Analysis or graph paper, plot a graph of pendulum period T vs. lengthl. Scale each axis from the origin (0,0).  Does the period appear to depend on length? Using either Graphical Analysis or graph paper, plot the pendulum period vs. mass. Scale each axis from the origin (0,0). Does the period appear to depend on mass? Do you have enough data to answer conclusively? To examine more carefully how the period T depends on the pendulum length l, create the following two additional graphs of the same data: T 2 vs. l and T vs. l2 . Of the three period-length graphs, which is closest to a direct proportion; that is, which plot is most nearly a straight line…

In the solution here to problem 3.37 Griffiths, the first step is to use Eqn 3.73* Can you show me why <d(xp)/dt> = 0  or why it isn't in the RHS of the equation?  (see my image)   * Equation 3.73:   d<Q>/dt = (i/h_bar) <[H,Q]> + <dQ/dt>

Problem 3: The motion of critically damped and overdamped oscillator systems is hardly "oscillatory". (a) To illustrate this, prove that a critically damped oscillator passes through the originx = 0 at most once, and determine the relationship between the initial conditions To and vo that is required for the oscillator to pass through the origin. (b) Do the same thing for the overdamped oscillator.