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
Let L be the relaxed length of the spring and y the extension produced in it due to force mg so that
ky = mg …(i)
The displacement of the mass during oscillation is given by
At the instant when the displacement is x
Taking gravitational PE at the mean position to be zero,
Gravitational PE at x = – mg x … (v)
Adding (iii), (iv) and (v), we get
Total energy of mass-spring system
A body oscillates harmonically with amplitude 0.05 m. At a certain instant of time its displacement is 0.01 m and acceleration is 1.0 ms-2. What is the period of oscillation?
A tuning fork produces 4 beta per second when sounded with sonometer of vibrating length 48cm. It produces 4 beats per second also when the vibrating length is 50cm. What is the frequency of the tuning fork?
The displacement x (in centimetres) of an oscillating particle varies with time t (in seconds) as
The magnitude of the maximum acceleration of the particle is
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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?
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?
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.
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.