Question
Figure (a) shows a spring of force constant k fixed at one end and carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. figure (b) shows the same spring with both ends free and a mass m fixed at each free end. Each of the spring is stretched by the same force F. the mass is case (a) and the masses in case (b) are then released.
Which of the following statements is/are true?

While oscillating, the maximum extension of the spring is more in case (a) than in case (b).

The time period of oscillation is the same in both cases.

None of these

The maximum extension of the spring is the same in both cases. & The time period of oscillation in case (a) is times that in case (b).
medium
Solution
The maximum extension of the spring is the same in both cases. & The time period of oscillation in case (a) is times that in case (b).
The maximum extension x produced in the spring in fig (a) is given by
F = kx
or x = F/k
The time period of oscillation is
In case (a) one end A of the spring is fixed to the wall. When a force F is applied to the free and B in the direction shown in fig (a) the spring is stretched exerting a force on the wall which in turn exerts an equal and opposite reaction force on the spring, as a result of which every coil of the spring is elongated producing a total extension x. In case (b) shown in fig (b), both ends of the spring are free. Therefore, the reaction force is absent, as a result of which every coil of the spring is not elongated when force F is applied at each in opposite directions. The coil at point O in the middle of the spring is not elongated. This situation can be visualized as two springs each of length l/2 (where l is the length of the complete spring) are joined to each other at point O. since extension is proportional to the length of the spring, the force F applied at end B produces an extensionx/2 in the part OB of the spring and the force F applied at A produces an extension x/2 in the part OA. The total extension in the spring is
Thus, the maximum extension produced in the spring in cases (a) and (b) is the same.
Now, the force constant of half the spring is twice that of the complete spring. In case (b) the force constant = 2k.hence the time period of oscillation will be
SIMILAR QUESTIONS
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? ^{ }
Two springs of force constants k_{1} and k_{2} 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 k_{1} = k_{2}?
Two springs of force constants k_{1} and k_{2} 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 k_{1} = k_{2}?
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
A vertical Utube of uniform crosssectional 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
A cylindrical piece of cork of height h and density ρ_{c} floats vertically in a liquid of density ρ_{l}. The cork is depressed slightly an released. If viscous effects are neglected, the time period of vertical oscillations of the cylinder is given by
An air chamber of volume V has a neck of crosssectional area a into which a light ball of mass m can move without friction. The diameter of the ball is equal to that of the neck of the chamber. The ball is pressed down a little and released. If the bulk modulus of air in B, the time period of the resulting oscillation of the ball is given by
A simple pendulum of length l and bob mass m is displaced from its equilibrium position O to a position P so that the height of P above O is h. It is then released. What is the tension in the string when the bob passes through the equilibrium position O? Neglect friction. V is the velocity of the bob at O.
A trolley of mass m is connected to two identical springs, each of force constant k, as shown in fig. The trolley is displaced from its equilibrium position by a distance x and released. The trolley executes simple harmonic motion of period T. After some time it comes to rest due to friction. The total energy dissipated as heat is (assume the damping force to be weak)
A simple pendulum of bob mass m is oscillating with an angular amplitudeα_{m} (in radius). The maximum tension in the string is