Set up and solve the following spring problems.
When a mass of 5 grams is hung vertically from a spring, at rest it stretches the spring 2.45 cm. (Gravitational acceleration is \(g = 980 cm/sec^2\).) At \(t =0\) the mass is moved from its equilibrium position with an initial velocity of 2 cm/sec. Find a solution to this initial value problem for relatively small values of \(t\) where we will assume the effect of dampening is negligible. Write your solution as \(A\cos(\omega_0t -\delta)\) where \(w_0\) is the natural frequency, and \(\delta\) is the phase shift.
When a mass of 4 grams is hung vertically from a spring, at rest it stretches the spring 9.8 cm. (Gravitational acceleration is \(g = 980 cm/sec^2\).) At \(t = 0\) the mass is stretched 2 cm from its equilibrium position and released with no initial velocity. Find a solution to this initial value problem for relatively small values of \(t\) where we will assume the effect of dampening is negligible. Write your solution as \(A\cos(\omega_0t -\delta)\) where \(w_0\) is the natural frequency, and \(\delta\) is the phase shift.
Assume a 8 lb weight stretches a spring 6 inches. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass). Find an expression which represents the position of the mass with respect to its equilibrium position if we assume the effect of dampening is negligible and there are no external forces acting on the mass, where the mass is initially stretched an addition 3 in beyond the equilibrium position an is released with an initial velocity of 2 ft/s.
Assume a 4 lb weight stretches a spring 2 feet. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass) Find an expression which represents the position of the mass with respect to its equilibrium position if we assume the effect of dampening is negligible and there are no external forces acting on the mass, where the mass is initially stretched an addition \(1/\sqrt{3}\) ft beyond the equilibrium position an is released with an initial velocity of 4 ft/s.
Suppose each of the following equations model the motion of a mass on spring. Determine if the spring is critically damped, overdamped, or underdamped.
\(4z'' + 2z' + z = 0\)
\(2u'' + 10u' + 3u = 0\)
\(2w'' + 5w' + 2w = 0\)
\(3u'' + 12u' + 12u = 0\)
\(2v'' + 20v' + 50 v = 0\)
\(3u'' +2u' + 4u=0\)
Solve the following problems
Assume a 4 lb weight stretches a spring 3 inches. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass). Suppose we also that when the mass is moving \(2 ft/sec\) the medium in which the mass moves imparts of force of 6 lbs. Is this system overdamped, critically damped, or underdamped? Justify with the appropriate computations. (You do not need to find a solution; Please just classify the system).
Assume a 4 lb weight stretches a spring 3 inches. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass). Suppose we also that when the mass is moving \(2 ft/sec\) the medium in which the mass moves imparts of force of 6 lbs. Find a function \(u(t)\) that measures the position of the spring if it is initially moved 2 inches from its equilibrium position and released with no initial velocity.
Assume a 8 lb weight stretches a spring 6 inches. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass). Suppose we also consider the effect of dampening, where when the mass is moving \(2 ft/sec\) the medium in which the mass moves imparts of force of 8 lbs. Is this system overdamped, critically damped, or underdamped? Justify with the appropriate computations. (You do not need to find a solution; Please just classify the system).
Assume a 8 lb weight stretches a spring 6 inches. Recall that the gravity constant for standard units is \(32 ft/s^2\) (also recall that a pound is a unit of force (weight) not mass). Suppose we also consider the effect of dampening, where when the mass is moving \(2 ft/sec\) the medium in which the mass moves imparts of force of 8 lbs. Find a function \(u(t)\) that measures the position of the spring if it moved from its equilibrium position with an initial velocity of 2 ft/sec.
Assume a 4 kg mass stretches a spring 19.6 cm, and that when the mass is moving with a velocity of 5 cm/s the medium in which the mass is moving exerts a viscous force of 3 N. Is this system overdamped, critically damped, or underdamped? Justify with the appropriate computations. (You do not need to find a solution; Please just classify the system).
Assume a 4 kg mass stretches a spring 19.6 cm, and that when the mass is moving with a velocity of 5 cm/s the medium in which the mass is moving exerts a viscous force of 3 N. Find an expression which represents the position of the mass with respect to its equilibrium position if there are no external forces acting on the mass, and it is released from its equilibrium position with an initial velocity of 5 cm/s.
Suppose a 2 kg mass stretches a spring 4.9 meters. Suppose we also consider the effect of dampening and that when the mass is moving 1 m/s the the medium in which the mass moves imparts of force of 4 N. Is this system overdamped, critically damped, or underdamped? Justify with the appropriate computations. (You do not need to find a solution; Please just classify the system).
Suppose a 2 kg mass stretches a spring 4.9 meters. Suppose we also consider the effect of dampening and that when the mass is moving 1 m/s the the medium in which the mass moves imparts of force of 4 N. Suppose this mass is stretched 50 cm from its equilibrium position and pushed back with an initial velocity of 1 m/s (by pushed back we mean that the initial velocity is actually -1m/s for this system). Find a function which models the position of the mass \(t\) seconds after the mass is released.
Suppose that a 2 slug mass stretches a spring 1/2 ft. Also suppose that when the mass is moving at a speed of 0.5 ft/sec the medium in which the mass is moving exerts a viscous force of 16 lbs.
(a) Is this system underdamped, critically damped or overdamped?
(b) Given some set of initial conditions, let \(u(t)\) describe the position of the mass with respect to its equilibrium position when no external forces act on the mass, and \(U(t)\) describe the position of the mass when an external force of \(2\cos(t)\) ft-lbs is exerted. Compare the limiting behavior of \(u(t)\) and \(U(t)\). (You need not find either \(u(t)\) or \(U(t)\), but you should say what happens to each of them as \(t \to \infty\)).
When a mass of 4 grams is hung vertically from a spring, at rest it stretches the spring 39.2 cm. (Gravitational acceleration is \(g = 980 cm/sec^2\).) Assume the effect of dampening is negligible and the mass is acted on continuously by the force \(44\cos(6t)\) dynes where the mass is initially at rest in its equilibrium position. Solve the initial value problem. Also use the formulas \(\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\) to write your answer in the form \(A\sin(\frac{\omega_0 +\omega}{2})\sin(\frac{\omega_0 -\omega}{2})\) and sketch your answer for \(0 < t< 6\pi\).
When a mass of 5 grams is hung vertically from a spring, at rest it stretches the spring 2.45 cm. (Gravitational acceleration is \(g = 980 cm/sec^2\).) Assume the effect of dampening is negligible and that the mass is acted on continuously by the force \(256\cos(12t)\) dynes where the mass is initially at rest in its equilibrium position. Solve the initial value problem. Also use the formulas \(\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)\) to write your answer in the form \(A\sin(\frac{\omega_0 +\omega}{2})\sin(\frac{\omega_0 -\omega}{2})\) and sketch your answer for \(0 < t< \pi/2\).
Assume a 10 kg mass stretches a spring 19.6 cm, and that when the mass is moving with a velocity of 4 cm/s the medium in which the mass is moving exerts a viscous force of 4 N.
(a) Find an expression which represents the position of the mass with respect to its equilibrium position if there are no external forces acting on the mass, and it is initially stretching the spring 1 cm beyond the equilibrium position an is released with no initial velocity.
(b) Suppose instead that we impart an external force of \(10 \cos(t/2)\) N where initially the mass is at rest in its equilibrium position. Find an expression which represents the position of the mass with respect to its equilibrium position.
(c) Compare the behavior of your result in part (a) with your result in part (b) as \(t \rightarrow \infty\).
When a mass of 4 grams is hung vertically from a spring, at rest it stretches the spring 39.2 cm. (Gravitational acceleration is \(g = 980 cm/sec^2\).) Assume the effect of dampening is negligible and the mass is acted on continuously by the force \(40\cos(5t)\) dynes where the mass is initially at rest in its equilibrium position. Solve the initial value problem. Discuss the solution as \(t \to \infty\)
The vertical displacement of a mass on a spring is given by \(w(x) = 4e^{-x} \cos(14x) - 3e^{-x}\sin(14x)\), where the positive displacement is in the upward direction.
Express \(w(x)\) in the amplitude-phase form, i.e. in the form \(w(x) = Ae^{-x}\cos(\omega x - \delta)\), where \(A>0\) and \(\delta \in [0, 2\pi)\). Identify the phase of the oscillation (you can express it as an inverse trig function) and the quasiperiod of the oscillation.
A stamping machine applies hammering forces on metal sheets through a die attached to the plunger. It describes an up and down motion through a flywheel spinning at a constant speed. The base on which the metal sheet is located has a mass of \(4000 kg\). The force acting on the base is described by the function \(F(t) = 4000 \sin(10t)\), where \(t\) measures the time in seconds. The base is supported by an elastic foundation with an equivalent spring constant of \(4 \times 10^5 \frac{N}{m}.\) Knowing that the base is initially depressed down by \(0.1 m\) in a resting position, write down and answer the following:
a) the differential equation describing the instantaneous position of the base (e.g. \(x(t)\));
b) Does the load yield a resonant vibration? How can you determine whether resonance occurs?
c) the instantaneous position of the base (e.g. \(x(t)\)) that solves the differential equation you computed in a).
One of the equations below describes a mass-spring system which undergoes resonance. Identify the equation and find its general solution.
a) \(w'' + 9w = 14\cos(9t); \)
b) \(4w'' + 16w = 7\cos(2t);\)
c) \(w'' + 4w' + 4w = 200\sin(2t). \)
\(\bf{Some~Parameter~Analysis}\)
Consider the mass-spring system described by the following differential equation \(2u'' + 3u' + \alpha u = 0\). For what values of \(\alpha\) does the system describe an under, over or critically damped oscillation?
For what values of \(\beta\) does the system described in the following equation undergo resonance?
a) \(3z'' + \beta z = -\pi^2 \cos(x);\)
b) \(8v'' + \beta v = 28 \sin(6x).\)
Consider the mass-spring system described by the following differential
equation \(4v'' + \gamma v' + 36v = 0\). For what values of \(\gamma\) does the system describe an under, over or critically damped oscillation?
A vibrating system satisfies the following differential equation:
\[z'' + \alpha z' + z = 0.\] What is the value of \(\alpha\) for which the quasiperiod of the damped motion is \(50\%\) higher than the period of the undamped corresponding motion?
The motion of an undamped spring-mass system satisfies the following initial-value problem:
\[v'' + 2v = 0, ~~v(0) = 0, ~~v'(0) = 2.\] What is the solution of this initial-value problem? How do \(v(x)\) and \(v'(x)\) compare to one another?
The position of a spring-mass system is described by the initial-value problem:
\[3z'' + \beta z =0, ~~z(0) = 2,~~ z'(0) = \gamma.\] If the period of the resulting motion is \(\pi\) and the amplitude of the same motion is \(3\), determine the values of \(\beta\) and \(\gamma\) such that the conditions are ensured.