i) An object is executing simple harmonic motion. Obtain expressions for its kinetic and potential energies.
ii) A spring mass system is characterized by and
The system is oscillating with amplitude of 0.40 m. Obtain an expression for the velocity of the block as a function of displacement and calculate its value a
### (i) Expressions for Kinetic and Potential Energy in SHM For a simple harmonic motion (SHM) system, the displacement of an object is given by: \[ x(t) = A \cos(\omega t + \phi) \] where: - \( A \) = amplitude, - \( \omega \) = angular frequency (\(\omega = \sqrt{k/m}\) for a spring-mass system), - \( \phi \) = phase constant. The velocity is obtained by differentiating displacement: \[ v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \] The kinetic energy (\( KE \)) of the object is: \[ KE = \frac{1}{2} m v^2 __________ _____ _____ ___ ________ _____ ___ ________ ________ _________ _______.
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d) A body of mass 0.2 kg is suspended from a spring of force constant 80 A
damping force is acting on the system for which γ = 4 Nsm−1. Write down the equation of motion of the system and calculate the period of its oscillations. Now a harmonic force F =10cos10t is applied. Calculate a and θ when the steady state
response is given by a cos(ωt − θ).
e) i) Show that only odd harmonics can be generated in a closed-end organ pipe.
ii) Determine the fundamental frequency and the first 3 overtones of an organ pipe
of length 1.7 m and closed at one end. Take the speed of sound to be 340
a) A Transverse waves propagating on a stretched string encounter another string of different characteristic impedance. (i) Write down the equations of particle displacement due to the incident, reflected and transmitted waves. (ii) Specify the boundary conditions and (iii) use these to obtain expressions for reflection and
transmission amplitude coefficients.
b) Consider a particle undergoing simple harmonic motion. The velocity of the particle at position is
and velocity of the particle at position
is
Show that the ratio of time period (T) and amplitude (A) is:
i) An object is executing simple harmonic motion. Obtain expressions for its kinetic and potential energies.
ii) A spring mass system is characterized by and
The system is oscillating with amplitude of 0.40 m. Obtain an expression for the velocity of the block as a function of displacement and calculate its value a
Consider N identical masses connected through identical springs of force constant k. The free ends of the coupled system are rigidly fixed at and
The masses are made to execute longitudinal oscillations on a frictionless table.
i) Depict the equilibrium as well as instantaneous configurations.
ii) Write down their equations of motion, decouple them and obtain frequencies of normal modes.
i) A sound wave of frequency 400 Hz travels in air at a speed of 320 Calculate the phase difference between two points on the wave separated by a distance of 0.2 m along the direction of travel of the wave.