Finding Time Intervals for a Capacitor
If a charged capacitor is connected to a coil by closing a switch, the energy
transferred to the coil and then back to the capacitor exhibits an oscillatory
motion. The voltage V (in volts) across the capacitor at time t (in seconds)
is given by: V(t) = e – t/6 cos(
3π
5
t)
Find the first time interval when will the voltage V be between – 0.3 and 0.3
volts? (round your answer to two decimal places).
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On your graphing calculator,
hit the Y = key
and enter in the following:
y1 = e – t/6 cos(
y2 = 0.3
y3 = – 0.3
3π
5
t)
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Next, hit the Window key and enter
in the following:
Xmin = 0
Xmax = 10
Xscl = 1
Ymin = – 1
Ymax = 1
Yscl = 1
Xres = 1
Be sure your calculator is in radian mode. Now, hit the graph key. You
should get a picture like this.
A
B
The first time interval that the voltage is between – 0.3 & 0.3 volts is here.
Finding Time Intervals for a Capacitor
To find the lower endpoint (A), we
will need to find the intersection of
y1 and y2. To do so, hit the CALC
key (2ND Trace). Arrow down to
option 5: intersect and hit enter.
This will take you back to the
graph and you will notice that Y1
is selected. Your picture should
look like the one on the below.
If you hit enter, you should then
see that Y2 is selected (see picture
to the right). If you hit the up
or down arrow keys, you will notice
that the selected curve changes.
With Y2 curve selected, hit the
enter key. Now, the calculator will
ask you to make a guess for the
x-value of the intersection. Since
the intersection is close to x = 0,
type 0 and hit enter.
You should get the following:
To ensure the voltage is below 0.3, we will need to round x up to 0.66
Finding Time Intervals for a Capacitor
If you repeat the same steps but
with Y1 and Y3, you should get
the upper endpoint B. To ensure that
the voltage is above – 0.3, we will
need to round x down to 1.02.
Thus, the first interval where the
voltage is between – 0.3 and 0.3
is 0.66 sec < t < 1.02 sec
If we need to find the second interval
where the voltage is between – 0.3
and 0.3, we do exactly the same thing
except that we will change our guess.
Since the intersection of Y1 and Y3 is
close to x = 2, we will use 2 as our
guess and since the intersection of
Y1 and Y2 is close to x = 3, we will
use 3 as our guess.
3
2
Making those calculations, you should get the following answers:
The first value, we will need to round up to 2.26 and the second value we
will need to round down to 2.76. to ensure that the voltage is between – 0.3
and 0.3 volts. Thus, the second interval where the voltage is between – 0.3
and 0.3 is 2.26 sec < t < 2.76 sec. You can repeat the process to find the
other intervals. You should get:
3rd interval
4th interval
5th interval
3.85 sec < t < 4.53 sec
5.40 sec < t < 6.39 sec
t > 6.86 sec