Vibrations An optics table has four legs supporting a heavy table-top Between ea

Question

Vibrations

An optics table has four legs supporting a heavy table-top Between each leg and the floor is a hard spring, with spring constant k Between each leg and the table-top is a softer spring, with spring constant k2. (a) What is the total spring constant of the eight springs? Draw diagrams showing each type of spring addition you must make. (b) If the table-top has a mass 50 kg, ki 4 k and the ground vibrates at a frequency of f 1000 Hz, what is the worst possible value for k (Hint: avoid resonance

Explanation / Answer

>> Part (a).

As, between each leg and floor, a spring, k1 is present,

and,

between each leg and table top, a spring, k2 is present,

>> SO, both spring are in series.

So, on a leg, total spring constant = (k1k2)/(k1 + k2) = K

>> Now, there are four legs, and as all are parallel

So, now, we have 4 springs, having spring constant: K, in parallel

=> Net Spring Constant = 4K = 4(k1k2)/(k1 + k2)   .........ANSWER.........PART (a)....

Part (b)...

m = 50 kg

k1 = 4k2

So, Net Spring Constant, Ke = 4(k1k2)/(k1 + k2) = (16/5)k2 = 3.2 k2

>> As, fn = Natural Frequency = 1000 Hz

>> And, for system, natural frequency = (1/2)*(Ke/m)1/2

For avoiding resonance,

(1/2)*(Ke/m)1/2 > 1000

=> (1/2)*(3.2k2/50)1/2 > 1000

Solving,

So, k2 > 616.85 MN/m

So, Worst Value of k2 = 616.85 MN/m            ..........ANSWER...........

Related Questions

Navigate

• Browse (All)
• Browse V
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.