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Aspring mass system is characterized by k=16Nm/sec and m=1.0kg .the system is oscillating with an amplitude of 0.20m then i) calculate the angular velocity

of oscillation.ii)obtain an expression for the velocity of the block as a function of displacement and calculate its value at X=0.1m .iii)calculate energy of the spring mass system

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A spring mass system oscillates in a simple harmonic motion with an angular speed ω. The restoration force F due to the spring on the mass m is given by:

F = m d²x/dt² = - k x d²x /dt² = - (k/m) x = - ω² x ω = √(k/m) = √16/1 = 4 rad/sec as k = 16 N m/sec and m = 1.0 kg

Amplitude = A = 0.20 meters ============ The displacement x from its mean position at any point of time t of the mass m can be given as: x = A Sin (ω t) = 0.20 Sin 4 t

velocity instantaneous of the mass = v = dx/dt = v = A ω Cos ωt obtained by differentiating x wrt t. = ω √[A² - x²]

So v = 4 * √[ 0.2² - 0.1²] = 0.4 * √3 m/sec =======================

Energy of the spring mass system is the total potential energy when the displacement is maximum ie., displacement = amplitude or the total KE when the displacement is 0.

Total energy = 1/2 k A² = 1/2 * 16 * 0.2² = 0.32 Joules.

we can also calculate this as follows: The velocity of the mass when x = 0 is: v = ω A So total energy = 1/2 * m * v² = 1/2 * 1 * (4 * 0.2)² = 0.32 Joules