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Potential energy of a particle with mass m is U=kx3 , where k is a positive constant. The particle is oscillating about the origin on x-axis. If the amplitude of oscillation is a, then its time period, T is
  • a)
    Proportional to a2
  • b)
    Independent of a
  • c)
    Proportional to √a
  • d)
    Proportional to 1/√a
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Potential energy of a particle with mass m is U=kx3 , where k is a pos...
P.E.=kx^2
F=-du/dx=-3kx^2
Maximum force=-3ka^2=-mw^2a\
w^2=3ka/m
and w^2 also equals to 4π^2/T^2
Comparing both, we get  T is  directly proportional to 1/√a.
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Potential energy of a particle with mass m is U=kx3 , where k is a pos...
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Potential energy of a particle with mass m is U=kx3 , where k is a pos...
According to the given question, the potential energy of a particle with mass m is given by U = kx^3, where k is a positive constant.

To analyze the time period of oscillation, we need to consider the motion of the particle.

The motion of the particle can be described by the equation of motion, which is derived from the potential energy function. The equation of motion for this system is given by:

m(d^2x/dt^2) = -dU/dx

Differentiating the potential energy function with respect to x, we get:

dU/dx = 3kx^2

Substituting this value into the equation of motion, we have:

m(d^2x/dt^2) = -3kx^2

Rearranging the equation, we get:

(d^2x/dt^2) = -(3k/m)x^2

This is a non-linear differential equation, and its solution gives the equation of motion for the particle.

To find the time period of oscillation, we can use the small angle approximation, where the amplitude of oscillation (a) is much smaller than the distance between the two extreme points of oscillation.

In this case, we can assume that x < a,="" which="" simplifies="" the="" equation="" of="" motion="" />

(d^2x/dt^2) = -(3k/m)a^2

This is a linear differential equation with a simple harmonic motion solution. The general solution to this equation is given by:

x(t) = A*cos(ωt + φ)

Where A is the amplitude of oscillation, ω is the angular frequency, t is time, and φ is the phase constant.

The time period of oscillation (T) is given by the reciprocal of the angular frequency:

T = 2π/ω

From the equation of motion, we can see that the angular frequency ω is proportional to the square root of the constant (3k/m)a^2.

Therefore, the time period of oscillation (T) is inversely proportional to the amplitude of oscillation (a).

Hence, the correct answer is option D: Proportional to 1/a.
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Attempt All sub parts from each question.Damping: When an analog instrument is used to measure a physical parameter, a deflecting torque is applied to the moving system which is deflected from its initial position and should move steadily to the deflected position. But due to inertia, the moving system keeps on oscillating about equilibrium. To remove the oscillation of the moving system a damping torque is required. The damping torque should be of such that the pointer quickly comes to its final steady position, without overshooting. If the instrument is underdamped, the moving system will oscillate about the final steady position with a decreasing amplitude and will take some time before it comes to rest. When the moving system moves rapidly but smoothly to its final steady position, the instrument is said to be critically damped or deadbeat. If the damping torque is more than what is required for critical damping, the instrument is said to be overdamped. In an overdamped instrument, the moving system moves slowly to its final steady position in a lethargic fashion.Methods of producing damping torque:(i) Air friction damping(ii) Fluid friction damping(iii) Eddy current dampingAir Friction Damping: A light piston is attached to the moving system. This piston moves in an air chamber closed at one end. When there is an oscillation, the piston moves in and out of the chamber. When the piston moves into the chamber, the air inside is compressed and an air pressure is built up which opposes the motion of the piston and thus the moving system faces a damping torque which ultimately reduces the oscillation. Fluid Friction Damping: In this type of damping oil is used in place of air. Viscosity of the oil being greater, the damping torque is also more. A disc is attached to the moving system which is completely dipped into the oil. When the moving system oscillates, the disc moves in oil and a frictional drag is produced. This frictional drag opposes the oscillation. Eddy Current Damping: The moving system is connected to an aluminium disc which rotates in a magnetic field. Rotation in magnetic field induces an emf in it and if the path is closed, a current (known as eddy current) flows. This current interacts with the magnetic field to produce an electromagnetic torque which opposes the motion. This torque is proportional to the oscillation of the moving system. This electromagnetic torque ultimately reduces the oscillation. Air friction damping provides a very simple and cheap method of damping. The disadvantages of fluid friction damping are that it can be used only for instruments which are in vertical position. Eddy current damping is the most efficient form of damping.Q. In Fluid Friction Damping the amount of damping torque

Attempt All sub parts from each question.Damping: When an analog instrument is used to measure a physical parameter, a deflecting torque is applied to the moving system which is deflected from its initial position and should move steadily to the deflected position. But due to inertia, the moving system keeps on oscillating about equilibrium. To remove the oscillation of the moving system a damping torque is required. The damping torque should be of such that the pointer quickly comes to its final steady position, without overshooting. If the instrument is underdamped, the moving system will oscillate about the final steady position with a decreasing amplitude and will take some time before it comes to rest. When the moving system moves rapidly but smoothly to its final steady position, the instrument is said to be critically damped or deadbeat. If the damping torque is more than what is required for critical damping, the instrument is said to be overdamped. In an overdamped instrument, the moving system moves slowly to its final steady position in a lethargic fashion.Methods of producing damping torque:(i) Air friction damping(ii) Fluid friction damping(iii) Eddy current dampingAir Friction Damping: A light piston is attached to the moving system. This piston moves in an air chamber closed at one end. When there is an oscillation, the piston moves in and out of the chamber. When the piston moves into the chamber, the air inside is compressed and an air pressure is built up which opposes the motion of the piston and thus the moving system faces a damping torque which ultimately reduces the oscillation. Fluid Friction Damping: In this type of damping oil is used in place of air. Viscosity of the oil being greater, the damping torque is also more. A disc is attached to the moving system which is completely dipped into the oil. When the moving system oscillates, the disc moves in oil and a frictional drag is produced. This frictional drag opposes the oscillation. Eddy Current Damping: The moving system is connected to an aluminium disc which rotates in a magnetic field. Rotation in magnetic field induces an emf in it and if the path is closed, a current (known as eddy current) flows. This current interacts with the magnetic field to produce an electromagnetic torque which opposes the motion. This torque is proportional to the oscillation of the moving system. This electromagnetic torque ultimately reduces the oscillation. Air friction damping provides a very simple and cheap method of damping. The disadvantages of fluid friction damping are that it can be used only for instruments which are in vertical position. Eddy current damping is the most efficient form of damping.Q. The most efficient form of damping is

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Potential energy of a particle with mass m is U=kx3 , where k is a positive constant. The particle is oscillating about the origin on x-axis. If the amplitude of oscillation is a, then its time period, T isa)Proportional to a2b)Independent of ac)Proportional to √ad)Proportional to 1/√aCorrect answer is option 'D'. Can you explain this answer?
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