Tutorial FDE Applic

8/3/2019 Tutorial FDE Applic

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Problem Sheet - I(Law of natural growth, Decay, Law of cooling, Electric circuits, Motion of a

particle)

1. The number of bacteria in a culture grows at a rate proportional to the

number present at that instant. If initially the number present was 100 and

increased to 332 in one hour, what would its value be after one and half hour?

2. The number N of bacteria in a culture grew at a rate proportional to N. The

value of N was initially 100 and increased to 332 in one hour. What would

be the value of N after 1½ hours?

3. The rate at which bacteria multiply is proportional to the instantaneous

number present. If the original number doubles in 2 hours, in how many

hours will triple?4. In 1770, the population of Great Britain was estimated to be 6.4 billion by

1790, the population grown to 8 million. Estimate k, predict the population

in the year 2010. 5. The populations of a country increases at the rate proportional to the current

population. If the population doubles in 40 years, in how many years will it

treble?6. Radium decomposes at a rate proportional to the amount present. If 50

percent of the original amount disappeared in L years, how much will

remain at the end of 2L years?

7. A radioactive substance disintegrates at a rate proportional to its mass.

When mass is 10 mg, the rate of disintegrates is 0.051 mg per day. How

long will it take for the masses to be reduced from 10 to 5 mg.8. Half-life of a radio active element is the time required for half of the

radioactive nuclei present in a sample to decay.

9. The radioactive lifetime of plutonium-210 is extremely short and so is

measured in days rather than years. The number of atoms remaining after t

days with an initial amount of y0 radioactive atom is given byt xe y y )1095.4(

0

3−−

= . Find the half life of plutonium-210.

10. Scientists where do carbon-14 dating use a value of 5700 years for its half-

life. Find the age of an object that has been excavated and found to have90%.

11. If 30 per cent of a radioactive substance disappeared in 10 days, how longwill it take for 90 per cent of it to disappear?

12. In a chemical reaction, a given substance is being converted into another at

a rate proportional to the amount of the substance unconverted. If one-fifth

of the original amount has been transformed in four minutes, how muchtime will be required to transform one half?



13. Under certain conditions cane sugar in water is converted into dextrose at a

rate which is proportional to the amount unconverted at any time. If, out of

75g which were present initially, 8g are converted during the first 30

minutes, find the amount converted in 90 minutes.

14. If a body cools from 150 C to 100 C in 40 minutes, how long will it take for the body to cool to 80 C? What will be the temperature of the body after one hour? Assume that the air is maintained at 30 C.

15. If the temperature of the air is 20 C and the temperature of a body drops

from 100 C to 75 C in 10 minutes, what will be its temperature after 30

minutes? When will the temperature be 30 C?

16. If the temperature of the air is 30 C and the substance cools from 100 C to

70 C in 15 minutes. Find when the temperature will be 40 C.

17. A body originally at 80 C cools down to 60 C in 20 minutes, the

temperature of the air being 40 C, what will be the temperature of the body

after 40 minutes from the original.

18. If the air is maintained at 30 C and the temperature of the body cools from

80 C to 60 C in 12 minutes, find the temperature of the body after 24

minutes.

19. A hot body cools in air at a rate proportional to the difference between the

temperature of the body and that of the surrounding air. If the air ismaintained at 30 C and the body cools form 80 C to 60 C in 12 minutes,

find the temperature of the body after 24 minutes. When will the

temperature of the body be 40 C?

20. A cup of coffee at temperature 100 C is placed in a room where thetemperature is 15 C and it cools to 75 C in 3 minutes. When will the

temperature of coffee be reduced further by 25 C? What will be the

temperature of coffee after a further interval of 3 minutes?

21. When a resistance R ohms is connected in series with an inductance L

henries with an e.m.f of E volts, the current i amperes at time t is given by

, E Ridt

di L =+ if E=10sint volts and i=0, when t=0. find i as a function of t.



22. Solve the equation ,cos nt E Ridt

di L =+ where L, R, E and n are positive

constants, given that i=0 when t=0. Show that, if t is sufficiently large, t is

approximately periodic with periodn

π 2.

23. Solve the equation ,300cos200 t Ridt di L =+ when R = 100 ohms and L =

0.05 henry, given that i=0 when t=0. In the long run, what is the maximum

value of i ?

24. An electric circuit contains a resistance R and a condenser of capacitance Cin series and an e.m.f E0 cos pt applied to it. Find the current in the circuit,

if i=0 when t=0.

25. An electric circuit contains a resistance R and a condenser of capacitance C

in series and an e.m.f E sinwt applied to it. Find the charge on the

condenser at any time, if its initial value is zero.

26. A body of mass m, falling from rest is subject to the force of gravity and anair resistance proportional to the square of the velocity. If it falls through a

distance x and possesses a velocity v at that instant, prove that

−

=22

2

log2

va

a

m

kx, where mg=ka2.

27. An object is dropped from a height of 500m. When will the object reach

ground level, and with what speed?

28. A particle falls under gravity in a medium whose resistance varies as the

velocity. Show that the distance fallen through in time t is equal to

,

−+−

g

Le

g

Li L L

gt

where L is the limiting velocity in the medium.

29. A particle falls from rest under gravity in a medium whose resistance varies

as the square of the velocity. When its velocity is v, show that it might have

descended through a distance

− 22

22

log2 v L

L

g

L, where L is the terminal velocity

of the medium.

30. L is the terminal velocity in a resisting medium where a resistance varies asthe square of the velocity. If a particle is projected vertically upwards with

a velocity α tan L in such a medium, show that

(i) the greatest height reached = α seclog2

g

L

(ii) the velocity with which it returns to the point of projection =α sin L .

(iii) the total time of flight = { })seclog(tan α α α ++

g

L.

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Tutorial FDE Applic