Question

A vertical U-tube of uniform cross-sectional area A contains a liquid of density ρ. The total length of the liquid column in the tube is L. The liquid column is disturbed by gently blowing into the tube. If viscous effects are neglected, the time period of the resulting oscillation of the liquid column is given by 

Solution

Correct option is

 

Let the liquid in the left arm be depressed by, say, x. The liquid level in the right arm is raised by an equal amount x. when left, the liquid levels will oscillate in each arm about their respective equilibrium position A, A’(see fig). If viscous effects are neglected, the liquid will never come too rest.

Since the column in the right arm is higher by 2x than the column in the left arm, the mass of this column of liquid is m = 2Aρx. The gravitational forcemg of this column provides the restoring force F. Thus 

                               

If L is the total length of the liquid in the U-tube, the total mass M of the oscillating liquid is  

                                

Hence the acceleration of the liquid column is given by

   

                                 

                                 

                                   

Hence the motion is simple harmonic. The time period of the motion is given by 

                     

Note: The period of oscillation is

independent of the density ρ of the

 liquid and the cross-sectional area

A of the U-tube.        

SIMILAR QUESTIONS

Q1

Two identical strings of a stringed  musical instrument are in unison when stretched with the same tension. When the tension in one string is increased by 1%, the musician hears 4 beats per second. What was the frequency of the note when the string were in unison?

Q2

A horizontal platform is executing simple harmonic motion in the vertical direction of frequency v. A block of mass m is placed on the platform. What is the maximum amplitude of the platform so that the block is not detached from it?

Q3

A spring stretches by 0.05 m when a mass of 0.5 kg is hung from it. A body of mass 1.0 kg is attached to one of its ends, the other end being fixed to the wall. The body is pulled 0.01 m along a horizontal frictionless surface and released. What is the total energy of the oscillator. Assume the string to have negligible mass and take g = 10 ms–2.  

Q4

A spring of negligible mass having a force constant k extends by an amount y when a mass m is hung from it. The mass is pulled down a little and released. The system begins to execute simple harmonic motion of amplitude A and angular frequency ω. The total energy of the mass-spring system will be

Q5

A spring has a natural length of 50 cm and a force constant of  A body of mass 10 kg is suspended from it and the spring is stretched. If the body is pulled down further stretching the spring to a length of 58 cm and released, it executes simple harmonic motion. What is the net force on the body when it is at its lowermost position of its oscillation? Take g = 10 ms –2.

Q6

A small trolley of mass 2 kg resting on a horizontal frictionless turntable is connected by a light spring to the centre of the table. The relaxed length of the spring is 35 cm. When the turntable is rotated an angular frequency of 10 rad s–1, the length of the spring becomes 40 cm. what is the force constant of the spring?  

Q7

Two springs of force constants k1 and k2 are connected to a mass mplaced on a horizontal frictionless surface as shown in fig (a) and (b). What is the ratio of the time periods of horizontal oscillation in cases (a) and (b) if k1 = k2?

                                                      

Q8

Two springs of force constants k1 and k2 are connected to a mass m as shown in fig (a) and (b). What is the ratio of the time periods of vertical oscillation in cases (a) and (b) if k1 = k2?

                                        

Q9

A tray of mass M = 10 kg is supported on two identical springs, each of spring constant k, as shown in fig. When the tray is depressed a little and released, it executes simple harmonic motion of period 1.5 s. When a block of mass m is placed on the tray, the period of oscillation becomes 3.0 s. The value of m is  

                                                                 

Q10

A cylindrical piece of cork of height h and density ρc floats vertically in a liquid of density ρl. The cork is depressed slightly an released. If viscous effects are neglected, the time period of vertical oscillations of the cylinder is given by