Mechanical Engineering Principles

234 MECHANICAL ENGINEERING PRINCIPLES

Hence,the relative density of the body

=

3305 kg/m^3

1000 kg/m^3

= 3. 305

Problem 9. A rectangular watertight box is

560 mm long, 420 mm wide and 210 mm

deep. It weighs 223 N.

(a) If it floats with its sides and ends

vertical in water of density 1030 kg/m^3 ,

what depth of the box will be

submerged?

(b) If the box is held completely

submerged in water of density

1030 kg/m^3 , by a vertical chain

attached to the underside of the box,

what is the force in the chain?

(a) The apparent weight of a floating body is zero.
That is, the weight of the body is equal to
the weight of liquid displaced. This is given
by: Vρgwhere V is the volume of liquid
displaced, andρis the density of the liquid.
Here,

223 N=V×1030 kg/m^3 × 9 .81 m/s^2

=V× 10 .104 kN/m^3

Hence,

V=

223 N

10 .104 kN/m^3

= 22. 07 × 10 −^3 m^3

This volume is also given by Lbd, where

L=length of box,b=breadth of box, and

d=depth of box submerged, i.e.

22. 07 × 10 −^3 m^3 = 0 .56 m× 0 .42 m×d

Hence,depth submerged,

d=

22. 07 × 10 −^3

0. 56 × 0. 42

= 0 .09384 m= 93 .84 mm

(b) The volume of water displaced is the total
volume of the box. The upthrust or buoyancy
of the water, i.e. the ‘apparent loss of weight’,
is greater than the weight of the box. The force
in the chain accounts for the difference.

Volume of water displaced,

V= 0 .56 m× 0 .42 m× 0 .21 m

= 4. 9392 × 10 −^2 m^3

Weight of water displaced

=Vρg= 4. 9392 × 10 −^2 m^3 ×1030 kg/m^3

× 9 .81 m/s^2

= 499 .1N

Theforce in the chain

=weight of water displaced−weight of box

= 499 .1N−223 N= 276 .1N

Now try the following exercise

Exercise 109 Further problems on

Archimedes’ principle

Take the gravitational acceleration as 9.8 m/s^2 ,

the density of water as 1000 kg/m^3 and the

density of mercury as 13600 kg/m^3.

1. A body of volume 0.124 m^3 is completely
   immersed in water of density 1000 kg/m^3.
   What is the apparent loss of weight of the
   body? [1.215 kN]


2. A body of weight 27.4 N and volume
   1240 cm^3 is completely immersed in
   water of specific weight 9.81 kN/m^3.
   What is its apparent weight?
   [15.24 N]


3. A body weighs 512.6 N in air and
   256.8 N when completely immersed in oil
   of density 810 kg/m^3. What is the volume
   of the body?
   [32.22 dm^3 = 0 .03222 m^3 ]


4. A body weighs 243 N in air and 125 N
   when completely immersed in water.
   What will it weigh when completely
   immersed in oil of relative density 0.8?
   [148.6 N]


5. A watertight rectangular box, 1.2 m long
   and 0.75 m wide, floats with its sides
   and ends vertical in water of density