periodic motion – the energy/power in any periodic phenomenon is proportional to the
sum of the squares of the amplitudes of the sinusoidal components. If this reminds you
of Pythagoras’ theorem, well...
17.13 Answers to reinforcement exercises
- (i)
2 s^2 +s+ 2
s^3
(ii)
s^2 + 6 s+ 21
(s+ 2 )(s^2 + 9 )
(iii)
s− 1
s^2 − 2 s+ 5
(iv)
2
s^2 + 4
- (i)
t^2
6
(t+ 24 ) (ii) t+
5
2
t^2 +
7
6
t^3
(iii) e^4 t (iv) e−^4 t
(v)
1
2
e^3 t/^4 (vi)
c
a
e−bt/a
(vii) cos 3t (viii) 2 sin 3t
(ix) 5 cos 3t+
4
3
sin 3t (x) acosct+
b
c
sinct
- (i)
3
2
−
1
2
e−^2 t (ii)
1
4
(e^3 t−e−t)
(iii)
3
2
e−t−
1
2
cost+
1
2
sint (iv)
1
16
e−^4 t−
1
16
+
1
4
t
(v)
1
9
et−
1
9
e−^2 t−
1
3
te−^2 t (vi)
1
10
sin 2t−
1
15
sin 3t
- (i)
3
2
−
1
2
e−^2 t (ii)
1
4
(e^3 t−e−t)
(iii) 3e−t−cost+sint (iv)
1
16
e−^4 t−
1
16
+
1
4
t
(v)
1
9
et−
1
9
e−^2 t−
1
3
te−^2 t (vi)
3
5
sin 2t−
2
5
sin 3t
- (i) Period
π
2
(ii) Period
4
3
(iii) Not periodic
(iv) Period 2π (v) Not periodic (vi) Periodπ
(vii) PeriodL (viii) Period
π
2 ω
(ix) Periodπ
(x) Not periodic
2 π
ω
