A spring with a 4 kg mass is resting on a horizontal frictionless table and is attached to a wall at the left end. It will stretch 0.5 m beyond its natura length when a force of 14 N is applied. If the spring begins at the equilibrium position and is pushed to the right with an initial velocity of 2 m/s, fin the position of the mass after t seconds. (Assume that movement to the right is the positive x direction.) x(e) 0.756 sin(2.6461) × Graph the position function of the mass. Χ x
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- The force of a spring obeys Hooke's law: F = -kx for some positive constant > 0, where x = x(t) measures the displacement from its rest position of a particle of mass m > 0 attached to the spring. i. Use Newton's Second Law to show that the equation of motion of the particle is +w²x=0. Determine w in terms of k and m. Find the solution to this equation if the particle is released at time t 0 from position x = 4 with = velocity 2w. ii. Show that = 1 w2 E = + 2 is a conserved quantity of the motion. What is the value of E for the above solution. What is the meaning of the two terms on the right hand side?A mass of 5 kg is attached to a spring with spring constant 25N/m. The mass is acted on by an external force of 10 sin(t) Newtons and moves in a medium that imparts a viscous force (i.e. friction) proportional to the velocity with constant 10Ns/m. If the mass is set in motion from its equilibrium position with an initial velocity of 60 cm/s, formulate the initial value problem describing the motion of the mass. Then solve this initial value problem. HINT: watch out for the units! Centimeters are not meters!Two masses are connected to a string and hung on a pulley to make an atwood machine. The masses are different values such as M > m. The setup is shown. Assume the string is massless and the pulley is ideal. When the masses are gently released, there is a resulting acceleration. Which of the following equations will calculate the mass of m in the system if the big mass and acceleration are known? A) m = 2M(a/g). B) = m 2M(g/a). C) m = M(g- a/g+a). D) m = M(g+a/g -a).
- In the figure below, a metal bar sitting on two parallel conducting rails, connected to each other by a resistor, is pulled to the right at a constant speed. The resistance R = 6.00 Q, the distance between the rails is { = 1.20 m, and a uniform 2.30 T magnetic field is directed into the page. At what speed (in m/s) should the bar be moved to produce a current of 0.500 A in the resistor? R Fapp m/sShown at right are two boxes (mass m and 2m) attached to identical springs, and attached to each other by a string running over a pulley. The springs are initially neither stretched nor compressed and the masses are initially at rest. The blocks then speed up and eventually slow down again until each block has moved a distance s, at which point both blocks are momentarily at rest. Determine a symbolic expression in terms of given variables for the spring constant of each identical spring. No explanation necessary. We'll use the work-energy theorem with a system consisting of both blocks, the string, both springs, and the Earth. The net external work on this system is zero (the various normal forces all act on points of contact that are not moving). The initial and final kinetic energy for both blocks is zero because they are not moving 2m m ΔΕ = 0 ks² · − 0 + ¼½ks² − 0 + mgs − 0 + (−2mgs) − 0 = 0 k = mg SIn the diagram shown below, the four currents are assumed to be held in place at their current locations. The currents have the following values: IA = 3.000 A, IB = 3.000 A, Ic 1.000 A and ID 2.000 A. Let d = 12.500 cm and assume standard x- and y-axes at point P1. (Be careful about signs!) AO 3d The magnitude of the total magnetic field at point P1, in μT and to three decimal places, is d P1
- = The figure shows a horizontal beam of mass M = 59.1 kg and length L = 6.4 m supported at its left end by a frictionless pin and at the other end by an ideal cable attached to wall h 4 m above the beam. A mass m = 22.1 kg is suspended from the beam a distance d = 2 m from the wall. Find the tension in the cable. T h d→ m M L = _ N Report your numerical answer below, assuming three significant figures. Remember to include a as necessary.A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0, 5, 1), and back to the origin under the influence of the force field F(x, y, z) = z2i+ 3xyj + 3y2k. Use Stokes' Theorem to find the work done.Physics
- An electron is accelerated through a potential difference of 1.5 MV and hence gains 1.5 MeV of energy. Find its final speed and momentum. (me 9.11^10-31 kg) explain step, by step and remember it is a relativistic problemFour long, parallel conductors carry equal currents of = 1.00 A. The figure below is an end view of the conductors. The current direction is into the page at points A and B and out of the page at C and D. A × C B(x (a) Calculate the magnitude of the magnetic field at point P, located at the center of the square of edge length = 0.200 m. μεThe two-dimensional wave equation describing the vibrations of an infi- nite string is Θη Ot2 2021 მ2 where n = n(x,t), — -x 0 is a constant. Its general solution can be written as n(x,t) = F(x-ct) + G(x + ct), with F and G arbitrary smooth functions. If at time t = 0 the shape of the string is n(x, t = 0) at 9 = 5+x4 and the string is released with velocity (x, t = 0) = 0, determine n(x,t) for t > 0.