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Question 1 A 2kg object is attached to a spring and will stretch the spring 612.5 mm by itself as shown in Figure Q1 below. Unstretched spring L u = 0 u(t): object displacement at any time t $mu'' + ku = F(t)$ $k = \frac{mg}{L}$ u(0): initial displacement u'(0): initial velocity System at rest Initial state u(0)=20cm Figure 01: Spring mass system (Source: Google Image) Assume that there is no damping in the system and the force $F(t) = 5 \cos 4t$ is applied to the object and the system will experience resonance. The object is initially displaced 20 cm downward from its equilibrium position and given a velocity of 10 cm/sec upward. (a) Show that the IVP for this system is $2u'' + 32u = 5 \cos 4t$, $u(0) = 0.2$ and $u'(0) = -0.1$ (b) Hence find and sketch the object displacement at anytime t. (c) Explain what will happen to the displacement as $t \to \infty$.

          Question 1
A 2kg object is attached to a spring and will stretch the spring 612.5 mm by itself as shown in
Figure Q1 below.
Unstretched spring
L
u = 0
u(t): object displacement at any time t
$mu'' + ku = F(t)$
$k = \frac{mg}{L}$
u(0): initial displacement
u'(0): initial velocity
System at rest
Initial state
u(0)=20cm
Figure 01: Spring mass system (Source: Google Image)
Assume that there is no damping in the system and the force $F(t) = 5 \cos 4t$ is applied to the
object and the system will experience resonance. The object is initially displaced 20 cm
downward from its equilibrium position and given a velocity of 10 cm/sec upward.
(a) Show that the IVP for this system is
$2u'' + 32u = 5 \cos 4t$, $u(0) = 0.2$ and $u'(0) = -0.1$
(b) Hence find and sketch the object displacement at anytime t.
(c) Explain what will happen to the displacement as $t \to \infty$.

Question 1
A 2kg object is attached to a spring and will stretch the spring 612.5 mm by itself as shown in
Figure Q1 below.
Unstretched spring
L
u = 0
u(t): object displacement at any time t
mu” + ku = F(t)
k = (mg)/(L)
u(0): initial displacement
u'(0): initial velocity
System at rest
Initial state
u(0)=20cm
Figure 01: Spring mass system (Source: Google Image)
Assume that there is no damping in the system and the force F(t) = 5 cos 4t is applied to the
object and the system will experience resonance. The object is initially displaced 20 cm
downward from its equilibrium position and given a velocity of 10 cm/sec upward.
(a) Show that the IVP for this system is
2u” + 32u = 5 cos 4t, u(0) = 0.2 and u'(0) = -0.1
(b) Hence find and sketch the object displacement at anytime t.
(c) Explain what will happen to the displacement as t →∞.

Added by Brenda B.

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