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Question
Figure shows a spring of force constant k fixed at one end carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. Figure (ii)show the same spring with both ends free and a mass m fixed at each free end. The same force F stretches each of the spring. The mass in case(i) and the masses in cases (ii) are then released. Which of the following statements is/are true?
Figure shows a spring of force constant k fixed at one end carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. Figure (ii)show the same spring with both ends free and a mass m fixed at each free end. The same force F stretches each of the spring. The mass in case(i) and the masses in cases (ii) are then released. Which of the following statements is/are true?
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Solution
The correct option is D The time period of oscillation in case (i) is √2 time that in case (ii)
The maximum extension x produces in the spring in case (a) is given by
F=kx or x=Fk
The time period of oscillation is
T=2π√massforce constant=2π√mk
In case (a) one end A of the spring is fixed to the wall. When a force F is applied to the free end B in the direction shown in figure, 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 figure (b), both ends of the spring forces are free equal forces are applied at ends. By application of forces both the cases are same.
Thus, the maximum extension produced in the spring in cases (a) and (b) is the same. In case (b) the mid-point of spring will not move, we can say the blocks are connected with the springs whose lengths are half the original length of spring. Now, the force constant of the half spring is twice of complete string. In case (b) the force constant = 2k. Hence, the time period of oscillation will be
T=2π√m2k
T′T=√2
Hence, the correct choices are (b) and (d).
The maximum extension x produces in the spring in case (a) is given by
F=kx or x=Fk
The time period of oscillation is
T=2π√massforce constant=2π√mk
In case (a) one end A of the spring is fixed to the wall. When a force F is applied to the free end B in the direction shown in figure, 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 figure (b), both ends of the spring forces are free equal forces are applied at ends. By application of forces both the cases are same.
Thus, the maximum extension produced in the spring in cases (a) and (b) is the same. In case (b) the mid-point of spring will not move, we can say the blocks are connected with the springs whose lengths are half the original length of spring. Now, the force constant of the half spring is twice of complete string. In case (b) the force constant = 2k. Hence, the time period of oscillation will be
T=2π√m2k
T′T=√2
Hence, the correct choices are (b) and (d).
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