A metal block with $5 c m$ edge and RD $9$ is suspended by a thread so as to be completely immersed in a liquid of RD $1.2$ . Find the tension in the thread.

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Solution

Step 1: Given data,
Side of the metal cube = $5 c m$
So, the volume of the metal cube will be = ${(5 c m)}^{3} = 125 c m^{3}$
Relative density of the metal cube = $9$
Relative density of the liquid in which it is immersed = $1.2$
Step 2: Finding the density of the cube,
$R e l a t i v e d e n s i t y o f m e t a l c u b e = \frac{d e n s i t y o f t h e c u b e}{d e n s i t y o f w a t e r a t 4^{\circ} C} S o, d e n s i t y o f t h e c u b e = r e l a t i v e d e n s i t y o f m e t a l c u b e \times d e n s i t y o f w a t e r a t 4^{\circ} C = 9 \times 1 g / c m^{3} = 9 g / c m^{3}$
Step 3: Finding the mass of the cube,
$D e n s i t y = \frac{m a s s}{v o l u m e} S o, m a s s = d e n s i t y \times v o l u m e = 9 g / c m^{3} \times 125 c m^{3} = 1, 125 g$
Step 4: Finding the weight of the cube,
$W e i g h t = m a s s \times a c c e l e r a t i o n d u e t o g r a v i t y = 1, 125 g \times 1, 000 c m / s^{2} = 11, 25, 000 N$ (Considering the value of acceleration due to gravity be $1, 000 c m / s^{2}$ )
Step 5: Finding the density of the liquid,
$R e l a t i v e d e n s i t y o f t h e l i q u i d = \frac{d e n s i t y o f t h e l i q u i d}{d e n s i t y o f w a t e r a t 4^{\circ} C} S o, d e n s i t y o f t h e l i q u i d = r e l a t i v e d e n s i t y o f t h e l i q u i d \times d e n s i t y o f w a t e r a t 4^{\circ} C = 1.2 \times 1 g / c m^{3} = 1.2 g / c m^{3}$
Step 6: Finding the buoyant force acting of the cube when fully immersed,
$B u o y a n t f o r c e (F_{B}) = v o l u m e o f t h e o b j e c t (V) \times t h e d e n s i t y o f t h e f l u i d (ρ) \times a c c e l e r a t i o n d u e t o g r a v i t y (g) F_{B} = 125 c m^{3} \times 1.2 g / c m^{3} \times 1, 000 c m / s^{2} = 1, 50, 000 d y n e$
$S o, T e n s i o n i n t h e t h r e a d = a p p a r e n t w e i g h t = (11, 25, 000 - 1, 50, 000) d y n e = 9, 75, 000 d y n e = 9.75 N$
Hence, the tension in the thread is $9.75 N$ .

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