Question

A.Suppose you attach an object to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length. If the spring has a force constant of 8.5 N/m and a 0.19 kg-mass object is set in motion as described, find the amplitude of the oscillations.

B.One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. Calculate the force constant of its plunger’s spring if you must compress it 0.149 m to drive the 0.046 kg plunger to a top speed of 23.6 m/s.

C. What is the period of a 1.333 m pendulum?

D.Two parakeets sit on a swing with their combined center of mass 12 cm below the pivot. At what frequency do they swing?

E. Find the frequency of a tuning fork that takes 2.14×10⁻³ s to complete one oscillation.

Answer 1

(A) Force constant of the spring = k = 8.5 N / m.

Mass attached to the spring = m = 0.19 kg.

So, vertical force applied on the spring = F = Weight of the mass = mg = ( 0.19 kg x 9.8 m / s² ) = 1.862 N.

Hence, maximum displacement of the spring from its equilibrium position = y = F / k

or, y = 1.862 N / 8.5 N / m ~ 0.219 m.

Hence, amplitude of the oscillations = 2y = 2 x 0.219 m ~ 0.438 m, or, 43.8 cm.

(B) Mass of plunger = m = 0.046 kg.

Compression of the spring = d = 0.149 m.

Maximum speed of the plunger = v = 23.6 m / s.

Let, the force constant of the spring be k, in N / m.

Hence, from principle of conservation of energy,

Maximum potential energy gained by the spring = Maximum kinetic energy acquired by the plunger

or, kd² / 2 = mv² / 2

or, k = mv² / d² = 0.046 x 23.6² / 0.149²

or, k ~ 1154.

Hence, force constant of the plunger's spring is : 1154 N / m.