Oscillations

d) For a damped harmonic oscillator, the equation of motion is m d²x/dt² + γ dx/dt + kx = 0 with m = 0.50 kg, γ = 0.70 kg/s and k = 70 N/m. Calculate (i) the period of motion, (ii) number of oscillations in which its amplitude will become half of its initial value, (iii) the number of oscillations in which its mechanical energy will drop to half of its initial value, (iv) its relaxation time, and (v) quality factor.

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Introduction This is a damped harmonic oscillator problem where we need to calculate multiple physical parameters based on the values of mass, damping constant, and spring constant. The motion is governed by the differential equation: m d²x/dt² + γ dx/dt + kx = 0 Given: m = 0.50 kg γ = 0.70 kg/s k = […]

d) For a damped harmonic oscillator, the equation of motion is m d²x/dt² + γ dx/dt + kx = 0 with m = 0.50 kg, γ = 0.70 kg/s and k = 70 N/m. Calculate (i) the period of motion, (ii) number of oscillations in which its amplitude will become half of its initial value, (iii) the number of oscillations in which its mechanical energy will drop to half of its initial value, (iv) its relaxation time, and (v) quality factor. Read More »

b) The time period of a simple pendulum, called ‘seconds pendulum’, is 2 s. Calculate the length, angular frequency and frequency of the pendulum. What is the difference between a simple pendulum and a compound pendulum?

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Introduction A seconds pendulum is defined as a simple pendulum whose time period is exactly 2 seconds. In this question, we calculate its length, angular frequency, and frequency. Additionally, we differentiate between a simple and compound pendulum. Given Time period (T) = 2 s Acceleration due to gravity (g) = 9.8 m/s² Step 1: Calculate

b) The time period of a simple pendulum, called ‘seconds pendulum’, is 2 s. Calculate the length, angular frequency and frequency of the pendulum. What is the difference between a simple pendulum and a compound pendulum? Read More »

a) A simple harmonic motion is represented by x(t) = a cos(ωt). Obtain expressions for velocity and acceleration of the oscillator. Also, plot the time variation of displacement, velocity and acceleration of the oscillator.

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Introduction Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force is directly proportional to the displacement and acts in the opposite direction. The motion is sinusoidal in time and exhibits characteristics such as constant amplitude and periodicity. Displacement Equation Given the equation of SHM: x(t) = a cos(ωt) where:

a) A simple harmonic motion is represented by x(t) = a cos(ωt). Obtain expressions for velocity and acceleration of the oscillator. Also, plot the time variation of displacement, velocity and acceleration of the oscillator. Read More »

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