A mass, m, on a spring with spring constant kı obeys the equation of motion d2x...
1) Amass, m, on a spring with spring constant k obeys the equation of motion Where-1 kg. Andk is assigned a value 1 (in Sl units) What are the units of the spring constant? Assuming that at time O, the mass mis at rO traveling with a velocity of 1 m/s Work out the maximum displacement of the mass in subsequent oscillations Can you find an alternative way of getting this answer? 2) Amass,, on a spring with spring constant...
1) When a mass of 3 kilograms is attached to a spring whose constant is 48 N/m, it comes to rest in the equilibrium position. Starting at t = 0, a force equal to f(t) = 180e−4t cos(4t) is applied to the system. Find the equation of motion in the absence of damping. x(t) = 2) Solve the given initial-value problem. d2x dt2 + 9x = 5 sin(3t), x(0) = 6, x'(0) = 0 x(t) =
Consider a mass m suspended from a massless spring that obeys Hooke's Law (i.e. the force required to stretch or compress it is proportional to the distance stretched/compressed). The kinetic energy T of the system is mv2/2, where v is the velocity of the mass, and the potential energy V of the system is kr-/2, where k is the spring constant and x is the displacement of the mass from its gravitational equilibrium position. Using Lagrange's equations for mechanics (with...
(1 point) A mass m = 4 kg is attached to both a spring with spring constant k = 325 N/m and a dash-pot with damping constant c=4N s/m. The mass is started in motion with initial position Xo = 1 m and initial velocity vo = 9 m/s. Determine the position function z(t) in meters. x(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t)...
The position x of a mass m attached to a spring obeys the differential equation i + yi + w?x = 0 where y 2w. a) (2 marks) Write down expressions for the forces on the mass due to (i) the spring, and (ii) damping. (3 marks) Using a trial solution x = Ae"', show that a = --y/2 ± (y2/4 - «2)1/2 b) c) (4 marks) Show, by finding wd, that the solution is a damped oscillation of the...
Differntial Equations Forced Spring Motion 1. A 1 kg mass is attached to a spring of spring constant k = 4kg/82, The spring-mass system is attached to a machine that supplies an external driving force of f(t) = 4 cos(wt). The systern is started from equilibrium i.e. 2(0) = 0 and z'(0) = 0. There is no damping. (a) Find the position x(t) of the mass as a function of time (b) write your answer in the form r(t)-1 sin(6t)...
I need to rescale (4) from the first page to the equation on the second page. 2.[60pts.] A bead of mass m is constrained to slide along a straight rigid horizontal wire. A spring with natural length Lo and spring constant k is attached to the bead and to a support point a distance h from the wire. See Figure 1. Let z(t) denote the position of the bead on the wire at time t. (Note that x is measured...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26, under the influence of an external force F(t) = 82 cos(4t). a) (8 points) Find the position u(t) of the mass at any time t, if u(0) = 6 and u 0 (0) = 0. b) (4 points) Find the transient solution uc(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time?...
ONLY attempt to solve if you know what you are doing. A mass of 1 kg is attached to a spring whose constant is 5 N/m. Initially, the mass is released 1 m below the equilibrium position with a downward velocity of 5 m/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocit y. a) Find the equation of motion if the mass is driven...
One end of a spring with a force constant of k 10.0 N/m is attached to the end of a long horizontal frictionless track and the other end is attached to a mass m = 2.20 kg which glides along the track. After you establish the equilibrium position of the mass-spring system, you move the mass in the negative direction (to the left), compressing the spring 1.73 m. You then release the mass from rest and start your stopwatch, that...