A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and rad/s.

Question

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force  and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k =  5 MN/m, F0 = 450 kN and   rad/s.

BlackTom AI · Accepted Answer

The question asks for the amplitude of the response X0 of a mass-spring system under harmonic excitation with negligible damping. To proceed, we must restate the given data and identify the missing piece needed for calculation.

Restating the problem:
- Mass (m) = 150 kg
- Spring constant (k) = 5 MN/m = 5,000,000 N/m
- Forcing amplitude (F0) = 450 kN = 450,000 N
- Damping is neglected (c = 0)
- The driving angular frequency is given as ω (rad/s), but the numerical value of ω is not provided in the text (it shows as 'and   rad/s' with the number missing).

What we need to compute X0:
For a single degree-of-freedom system with equation m x'' + k x = F0 cos(ωt) and zero damping, the steady-state displacement amplitude is:
X0 = F0 / |k − m ω^2|
This formula directly uses F0, k, m, and ω. Since damping is neglected, the denominator is the magnitude of (k − m ω^2).

Why the missing ω prevents a numeric answer:
- Without ω, we cannot evaluate (k − m ω^2). The value of k − m ω^2 could be positive, negative, or zero; its magnitude determines X0 directly via the above expression.
- If k − m ω^2 were zero, the amplitude would theoretically go to infinity (resonance in the ideal undamped case), which is another reason the exact ω matters for a meaningful result here.

Given answer option(s):
- The provided data includes an answer option '15.8 cm', but there are no explicit answer choices listed in the question (the answer_options array is empty). With no ω value and no alternative options to compare, we cannot confirm whether 15.8 cm is correct or not.

What you can do to complete the calculation:
1) Obtain the missing angular frequency ω (in rad/s).
2) Compute the denominator: D = |k − m ω^2| = |5,000,000 − 150 × ω^2|.
3) Compute X0 = 450,000 / D (units in meters). If desired in centimeters, multiply by 100.

Example structure if ω were known:
- Suppose ω is given as ω0, then D = |5,000,000 − 150 × ω0^2| (N/m).
- Then X0 = 450,000 / D meters.
- Convert to centimeters: X0_cm = X0 × 100.

Summary of the issue:
- The calculation is incomplete because the driving frequency ω is missing.
- The final numeric value for X0 depends critically on ω, due to the term (k − m ω^2).
- Without ω, we cannot verify or derive whether 15.8 cm is correct, nor select among multiple options (which are not provided).

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and rad/s.

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A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force F=F0cos(ωt) and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and ω=120 rad/s.

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and rad/s.

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