Higher Engineering Mathematics, Sixth Edition

36 Higher Engineering Mathematics

0

2

4

6

5.71

8

0.5

0.555

i (A)

1.0 1.5 t(s)

i 5 8.0 (1 2 e^2 t/CR)

Figure 4.6

Problem 20. The temperatureθ 2 of a winding

which is being heated electrically at timetis given

by:θ 2 =θ 1 ( 1 −e

−t

τ)whereθ 1 is the temperature (in

degrees Celsius) at timet=0andτis a constant.

Calculate,

(a) θ 1 , correct to the nearest degree, whenθ 2 is

50 ◦C,tis 30s andτis 60s

(b) the timet, correct to 1 decimal place, forθ 2 to

be half the value ofθ 1.

(a) Transposing the formula to makeθ 1 the subject

gives:

θ 1 =

θ 2

( 1 −e

−t

T)

=

50

1 −e

− 30

60

=

50

1 −e−^0.^5

=

50

0. 393469 ...

i.e. θ 1 = 127 ◦C, correct to the nearest degree.

(b) Transposing to maketthe subject of the formula

gives:

θ 2

θ 1

= 1 −e

−t

τ

from which, e

−t

τ = 1 −

θ 2

θ 1

Hence −

t

τ

=ln

(

1 −

θ 2

θ 1

)

i.e. t=−τln

(

1 −

θ 2

θ 1

)

Since θ 2 =

1

2

θ 1

t=−60ln

(

1 −

1

2

)

=−60ln0. 5 = 41 .59s

Hence the time for the temperatureθ 2 to be

one half of the value ofθ 1 is 41.6s, correct to 1

decimal place.

Now try the following exercise

Exercise 18 Further problems on the laws

of growth and decay

1. The temperature,T◦C, of a cooling object
   varies with time,tminutes, according to the
   equation:T=150e−^0.^04 t. Determine the tem-
   perature when (a)t=0, (b)t=10 minutes.
   [(a) 150◦C (b) 100. 5 ◦C]


2. The pressureppascals at heighth metres
   above ground level is given by p=p 0 e

−h

C,

where p 0 is the pressure at ground level

andCis a constant. Find pressurepwhen

p 0 = 1. 012 × 105 Pa, heighth=1420m, and

C=71500. [99210]

3. The voltage drop,vvolts, across an induc-
   tor L henrys at time t seconds is given
   by v=200e

−Rt

L ,whereR= 150  and

L= 12. 5 × 10 −^3 H. Determine (a) the voltage

whent= 160 × 10 −^6 s, and (b) thetime for the

voltage to reach 85V.

[(a) 29.32volts (b) 71. 31 × 10 −^6 s]

4. The lengthlmetres of a metal bar at tem-
   peraturet◦Cisgivenbyl=l 0 eαt,where
   l 0 and αare constants. Determine (a) the
   value of αwhenl= 1 .993m,l 0 = 1 .894m
   andt= 250 ◦C, and (b) the value ofl 0 when
   l= 2 .416,t= 310 ◦Candα= 1. 682 × 10 −^4.
   [(a) 2. 038 × 10 −^4 (b) 2.293m]


5. The temperatureθ 2 ◦C of an electrical conduc-
   tor at timetseconds is given by:
   θ 2 =θ 1 ( 1 −e−t/T),whereθ 1 is the initial
   temperature andT seconds is a constant.
   Determine:
   (a) θ 2 when θ 1 = 159. 9 ◦C,t=30s and
   T=80s, and
   (b) the timetforθ 2 to fall to half the value
   ofθ 1 ifTremains at 80s.
   [(a) 50◦C (b) 55.45s ]


6. A belt is in contact with a pulley for a
   sector ofθ= 1 .12radians and the coefficient