In the following exercises assume that there’s no damping.
Q6.1.1
1. An object stretches a spring \(4\) inches in equilibrium. Find and graph its displacement for \(t>0\) if it is initially displaced \(36\) inches above equilibrium and given a downward velocity of \(2\) ft/s.
2. An object stretches a string \(1.2\) inches in equilibrium. Find its displacement for \(t>0\) if it is initially displaced \(3\) inches below equilibrium and given a downward velocity of \(2\) ft/s.
3. A spring with natural length \(.5\) m has length \(50.5\) cm with a mass of \(2\) gm suspended from it. The mass is initially displaced \(1.5\) cm below equilibrium and released with zero velocity. Find its displacement for \(t>0\).
4. An object stretches a spring \(6\) inches in equilibrium. Find its displacement for \(t>0\) if it is initially displaced \(3\) inches above equilibrium and given a downward velocity of \(6\) inches/s. Find the frequency, period, amplitude and phase angle of the motion.
5. An object stretches a spring \(5\) cm in equilibrium. It is initially displaced \(10\) cm above equilibrium and given an upward velocity of \(.25\) m/s. Find and graph its displacement for \(t>0\). Find the frequency, period, amplitude, and phase angle of the motion.
6. A \(10\) kg mass stretches a spring \(70\) cm in equilibrium. Suppose a \(2\) kg mass is attached to the spring, initially displaced \(25\) cm below equilibrium, and given an upward velocity of \(2\) m/s. Find its displacement for \(t>0\). Find the frequency, period, amplitude, and phase angle of the motion.
7. A weight stretches a spring \(1.5\) inches in equilibrium. The weight is initially displaced \(8\) inches above equilibrium and given a downward velocity of \(4\) ft/s. Find its displacement for \(t > 0\).
8. A weight stretches a spring \(6\) inches in equilibrium. The weight is initially displaced \(6\) inches above equilibrium and given a downward velocity of \(3\) ft/s. Find its displacement for \(t>0\).
9. A spring–mass system has natural frequency \(7\sqrt{10}\) rad/s. The natural length of the spring is \(.7\) m. What is the length of the spring when the mass is in equilibrium?
10. A \(64\) lb weight is attached to a spring with constant \(k=8\) lb/ft and subjected to an external force \(F(t)=2\sin t\). The weight is initially displaced \(6\) inches above equilibrium and given an upward velocity of \(2\) ft/s. Find its displacement for \(t>0\).
11. A unit mass hangs in equilibrium from a spring with constant \(k=1/16\). Starting at \(t=0\), a force \(F(t)=3\sin t\) is applied to the mass. Find its displacement for \(t>0\).
12. A \(4\) lb weight stretches a spring \(1\) ft in equilibrium. An external force \(F(t)=.25\sin8 t\) lb is applied to the weight, which is initially displaced \(4\) inches above equilibrium and given a downward velocity of \(1\) ft/s. Find and graph its displacement for \(t>0\).
13. A \(2\) lb weight stretches a spring \(6\) inches in equilibrium. An external force \(F(t)=\sin8t\) lb is applied to the weight, which is released from rest \(2\) inches below equilibrium. Find its displacement for \(t>0\).
14. A \(10\) gm mass suspended on a spring moves in simple harmonic motion with period \(4\) s. Find the period of the simple harmonic motion of a \(20\) gm mass suspended from the same spring.
15. A \(6\) lb weight stretches a spring \(6\) inches in equilibrium. Suppose an external force \(F(t)={3\over16}\sin\omega t+{3\over8}\cos\omega t\) lb is applied to the weight. For what value of \(\omega\) will the displacement be unbounded? Find the displacement if \(\omega\) has this value. Assume that the motion starts from equilibrium with zero initial velocity.
16. A \(6\) lb weight stretches a spring \(4\) inches in equilibrium. Suppose an external force \(F(t)=4\sin\omega t-6\cos\omega t\) lb is applied to the weight. For what value of \(\omega\) will the displacement be unbounded? Find and graph the displacement if \(\omega\) has this value. Assume that the motion starts from equilibrium with zero initial velocity.
17. A mass of one kg is attached to a spring with constant \(k=4\) N/m. An external force \(F(t)=-\cos\omega t-2\sin\omega t\) n is applied to the mass. Find the displacement \(y\) for \(t>0\) if \(\omega\) equals the natural frequency of the spring–mass system. Assume that the mass is initially displaced \(3\) m above equilibrium and given an upward velocity of \(450\) cm/s.
18. An object is in simple harmonic motion with frequency \(\omega_0\), with \(y(0)=y_0\) and \(y'(0)=v_0\). Find its displacement for \(t>0\). Also, find the amplitude of the oscillation and give formulas for the sine and cosine of the initial phase angle.
19. Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?
20. Two objects suspended from identical springs are set into motion. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?
21. Two identical objects suspended from different springs are set into motion. The period of one motion is \(3\) times the period of the other. How are the two spring constants related?
