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Answer :
The amplitude of the mass-spring system with mechanical energy of 0.4 J, force constant of 186 N/m, and mass of 0.25 kg can be found by equating the mechanical energy to the potential energy of the spring. After the calculation, the amplitude is found to be approximately 6.56 cm.
To determine the amplitude of the oscillatory motion of the mass-spring system with mechanical energy of 0.4 J and a force constant of 186 N/m, one can use the relationship between mechanical energy, force constant, and amplitude. The mechanical energy (E) in a mass-spring system is the sum of potential and kinetic energy. However, at the points of maximum displacement (amplitude), the velocity is zero, and thus all the mechanical energy is potential energy stored in the spring.
The formula for potential energy (U) stored in a spring is:
U = 1/2kx²
Where k is the force constant of the spring, and x is the displacement from the equilibrium position, which at maximum displacement is the amplitude (A). By setting U equal to the total mechanical energy (E), we can solve for x:
0.4 J = 1/2(186 N/m)A²
A² = (2 0.4 J)/(186 N/m)
A = √[(2 0.4 J)/(186 N/m)]
After calculating A, to convert the amplitude to centimeters (cm), we multiply by 100 since 1 m = 100 cm.
Now performing the calculation:
A = √[(2*0.4 J)/(186 N/m)]
A = √[(0.8 J)/(186 N/m)]
A ≈ √(0.00430107527 m²)
A ≈ 0.06558241407 m
A ≈ 6.56 cm
Therefore, the amplitude of the oscillatory motion is approximately 6.56 cm.
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