Simple oscillations

Problem:

A particle that hangs from a spring oscillates with an angular frequency of 2 rad/s.  The spring is suspended from the ceiling of an elevator car and hangs motionless (relative to the car) as the car descends at a constant speed of 1.5 m/s.  The car then suddenly stops.  Neglect the mass of the spring.
(a) With what amplitude does the particle oscillate?
(b) What is the equation of motion for the particle?  (Choose the upward direction to be positive.)

Solution:

Problem:

A mass of 250 g hangs on a spring and oscillates vertically with a period of 1.1 s.  What mass must be added to double the period?

Solution:

Problem:

Two equal masses, each of mass m, are connected by a spring having a spring constant k.  If the equilibrium separation is L0 and the spring rests on a frictionless horizontal surface, what is the frequency of vibrations w in terms of k, m and L0?

Solution:

Problem:

A certain oscillator satisfies the equation d2x/dt2 + 4x = 0.  Initially the particle is at the point x = √3 when it is projected towards the origin with speed u = 2.
(a)  Find x(t).
(b)  How long does it take for the particle to first reach the origin?

Solution:

Problem:

A simple pendulum is attached to the ceiling of a boxcar which accelerates at a constant rate a.  Find the equilibrium angle of the pendulum, and also the frequency of small oscillations.

Solution:

Problem:

Calculate the frequencies of oscillations of the mass m for the two spring configurations shown in the figures.  The springs have elastic constants k1 and k2.

 image

Solution:

Problem:

When the system shown in the diagram is in equilibrium, the right spring is stretched by x1.  The coefficient of static friction between the blocks is µs.  There is no friction between the bottom block and the supporting surface.  The force constants of the springs are k and 3k (see the diagram).  The blocks have equal mass m.  Find the maximum amplitude of the oscillations of the system shown in the diagram that does not allow the top block to slide on the bottom.

image

Solution:

Problem:

Three small identical coins of mass m each are connected by two light non-conducting strings of length d each.  Each coin carries an unknown charge q.  The coins are placed on a horizontal frictionless non-conducting surface as shown (the angle between the strings is very close to 180°).  After the coins are released, they are observed to vibrate with period T.  Find the charge q on each of the coins in terms of m, d, and T.

image 

Solution:

Problem:

Consider the motion of a point of mass m subjected to a potential energy function of the form
U(x) = U0[1 - cos(x/R0)] for πR0/2 < x < πR0/2, 
where x denotes distance, and U0 and R0 are positive constants with dimensions of energy and length, respectively.   
(a)  Find the position of stable equilibrium for the mass.
(b)  Show that the motion of the mass in proximity of the stable equilibrium position is SHM.
(c)  Find the period of the small oscillations. 
(d)  Find the period of the small oscillations for the same mass in the potential
U(x) = -U0/[1 + (x/R0)2].

Solution:

Problem:

A flexible “U-tube” with cross sectional area A is filled with a liquid of density ρ that has no viscosity.  The total length of the fluid column is L.  If one side of the U tube is suddenly lowered so that the fluid level on that side is lowered by an amount h with respect to its initial position, when will the heights of the two columns again be equal?  (You may assume that the fluid flow is laminar but with no sticking to the inner surface of the U tube so that all particles in the fluid move at the same speed.  Also, neglect all friction, such as any effects of air resistance in the tube above the liquid.)

image

Solution:

More oscillation problems