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Answer :
Final answer:
Resonance in a spring mass system occurs when the frequency of the external force matches the system's natural frequency. The natural frequency is represented by ω, which is derived by setting it equal to the square root of the ratio of the spring constant to the mass. Therefore, to cause resonance a value of ω must be selected that makes the frequencies match.
Explanation:
In this physics problem, we are looking at the resonant angular frequency in a spring mass system modeled by the given differential equation. In a spring mass system, resonance occurs when the frequency of the external force matches the natural frequency of the system. The natural frequency, or angular frequency (ω), often symbolizes this in the modelling differential equations of such systems.
Resonance in a spring-mass system can lead to an increase in the amplitude of the system's oscillation. In the given differential equation 4y′′+9y=7cos(ωt), you can derive the natural frequency of the system by setting it equal to the square root of the ratio of the spring constant (k) to the mass (m), or ω = sqrt(k/m). The ω (omega) in the right side of the equation represents the frequency of the external force.
Given the values you would plug in for omega (ω), it's crucial to remember that resonance happens when the natural frequency equals the frequency of the external force being applied. Hence, the value of ω that would lead to resonance would be one that makes these two frequencies equal.
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