Spring-Mass Problem
Two identical particles \(A\) and \(B\) of mass \(m\) are connected by a light spring of natural length \(l_{0}\) and spring constant \(k .\) The system is placed on the ground with \(A\) vertically above \(B\).
(a) Let \(y_{A}\) be the displacement of \(A\) measured from the ground. Determine \(y_{A}\) when the system is in equilibrium.
(b) If \(A\) is projected upward from the equilibrium position with velocity \(u(>2 g \sqrt{m / k})\)
find the values of \(y_{A}\) and \(\dot{y}_{A}\) when \(B\) just leaves the ground.
(c) Let \(y_{B}\) be the displacement of \(B\) after it has left the ground. While the system is still in air above the ground, show that the extension of the spring \(\epsilon\left(=y_{A}-y_{B}-l_{0}\right)\) is described by a simple harmonic motion (SHM). What is the angular frequency of the SHM?
Homework Answers
. If a body oscillates vertically from a spring, the restoring force has magnitude kx. Therefore...
. If a body oscillates vertically from a spring, the restoring force has magnitude kx. Therefore the vertical motion is SHM (b) A body is suspended from the(c) If the body is displaced from spring. It is in equilibrium when the equilibrium, the net force on the body upward force exerted by the stretched is proportional to its displacement. spring equals the body's weight. The oscillations are SHM Al- A hanging spring that obevs Hooke's law ΔΙ img mg
A 288-kg mass, when attached to the end of a spring hanging vertically, stretches the spring...
A 288-kg mass, when attached to the end of a spring hanging vertically, stretches the spring 8 m. The mass is in a medium that exerts a viscous resistance of 576 N when the mass has a velocity of 4 m/sec. Assume the mass is given an initial velocity of 18 m/s from the equilibrium position. a) Determine the spring constant k. Use g = 10 m/sec. k b) Determine the damping coeffient 7. 7= c) If the initial value...
A 189-kg mass, when attached to the end of a spring hanging vertically, stretches the spring...
A 189-kg mass, when attached to the end of a spring hanging vertically, stretches the spring 9 m. The mass is in a medium that exerts a viscous resistance of 3024 N when the mass has a velocity of 4 m/sec. Assume the mass is given an initial velocity of 14 m/s from the equilibrium position. a) Determine the spring constant k. Use g = 10 m/sec. k b) Determine the damping coeffient 7. 7 c) If the initial value...
Pulley Problem
In the figure, a particle \(A\) of mass \(m\) is connected to another particle \(B\) of mass \(2 m\) by an inextensible string which passes over a smooth fixed pulley. A third particle \(C\) of mass \(m\) is connected to \(A\) by an elastic string of natural length \(L\). All the particles are initially held with the elastic string just unstretched and are then released from rest. You can assume that the tension of the elastic string is directly proportional...
Consider a vertical mass-spring system. The spring constant is k = 48N the mass is m...
Consider a vertical mass-spring system. The spring constant is k = 48N the mass is m = 1.0 kg. (a) Find the angular frequency. The mass is pulled from equilibrium 0.10 m down. (b) Write the equation for the displacement as a function of time, with a vertical xaxis pointing upward.
A physics lab demonstrates the principles of simple harmonic motion (SHM) by using a spring affixed...
A physics lab demonstrates the principles of simple harmonic motion (SHM) by using a spring affixed to a horizontal support. The student is asked to find the spring constant, k. After suspending a mass of 295.0 g from the spring, the student notices the spring is displaced by 49.5 cm from equilibrium. With this information, calculate the spring constant. = ___________ N/m The student realizes that the spring demonstrates SHM with the attached mass of 295.0 g. The student is...
One end of a spring with a force constant of k 10.0 N/m is attached to...
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...
8. For a mass spring system in SHM a. The force is directly proportional to displacement...
8. For a mass spring system in SHM a. The force is directly proportional to displacement b. The force is inversely proportional to the displacement c. The force is equal to zero d. The force is in equilibrium e. None of the above apply
A lightweight spring of 50 N/m is suspended vertically on a ceiling and in an initial...
A lightweight spring of 50 N/m is suspended vertically on a ceiling and in an initial equilibrium state. An object with a mass of 0,25 kg is attached to the end of the spring so that the system is in a final equilibrium state. a. Draw the free body diagram of the object in the final equilibrium state. b. What is the length of the spring in the final equilibrium state relative to its length in the initial equilibrium state?...
A spring is suspended vertically from a fixed support. The spring has spring constant k=24 N m −1 k=24 N m−1 . An object...
A spring is suspended vertically from a fixed support. The
spring has spring constant k=24 N m −1 k=24 N m−1 . An object of
mass m= 1 4 kg m=14 kg is attached to the bottom of the spring. The
subject is subject to damping with damping constant β N m −1 s β N
m−1 s . Let y(t) y(t) be the displacement in metres at the end of
the spring below its equilibrium position, at time t...
. If a body oscillates vertically from a spring, the restoring force has magnitude kx. Therefore the vertical motion is SHM (b) A body is suspended from the(c) If the body is displaced from spring. It is in equilibrium when the equilibrium, the net force on the body upward force exerted by the stretched is proportional to its displacement. spring equals the body's weight. The oscillations are SHM Al- A hanging spring that obevs Hooke's law ΔΙ img mg
A 288-kg mass, when attached to the end of a spring hanging vertically, stretches the spring 8 m. The mass is in a medium that exerts a viscous resistance of 576 N when the mass has a velocity of 4 m/sec. Assume the mass is given an initial velocity of 18 m/s from the equilibrium position. a) Determine the spring constant k. Use g = 10 m/sec. k b) Determine the damping coeffient 7. 7= c) If the initial value...
A 189-kg mass, when attached to the end of a spring hanging vertically, stretches the spring 9 m. The mass is in a medium that exerts a viscous resistance of 3024 N when the mass has a velocity of 4 m/sec. Assume the mass is given an initial velocity of 14 m/s from the equilibrium position. a) Determine the spring constant k. Use g = 10 m/sec. k b) Determine the damping coeffient 7. 7 c) If the initial value...
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...
8. For a mass spring system in SHM a. The force is directly proportional to displacement b. The force is inversely proportional to the displacement c. The force is equal to zero d. The force is in equilibrium e. None of the above apply
A lightweight spring of 50 N/m is suspended vertically on a ceiling and in an initial equilibrium state. An object with a mass of 0,25 kg is attached to the end of the spring so that the system is in a final equilibrium state. a. Draw the free body diagram of the object in the final equilibrium state. b. What is the length of the spring in the final equilibrium state relative to its length in the initial equilibrium state?...
A spring is suspended vertically from a fixed support. The
spring has spring constant k=24 N m −1 k=24 N m−1 . An object of
mass m= 1 4 kg m=14 kg is attached to the bottom of the spring. The
subject is subject to damping with damping constant β N m −1 s β N
m−1 s . Let y(t) y(t) be the displacement in metres at the end of
the spring below its equilibrium position, at time t...
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