Question

A wooden cylinder (0 02 x 0 02 x 0 1m) floats vertically in water with one-third of ts length immersed. a)-Determine the density of wood b)-Determine if this is a stable condition c)-Determine if the plece of wood would float stably in alcohol with density 700 kg/m3

Answer 1

Answer:

a)- the density of wood is 333.33 Kg/m³

b)-unstable condition

c)-unstable condition

Explanation:

Given data

wooden cylinder = 0.02m  x 0.02m x 0.1m

floating = 1/3 × Length

to find out

density of wood,  is it stable condition and wood would float stably in alcohol with density 700 kg/m3

Solution

First we find out density of wood

we know density of water is 1000 kg/m³

and we know wood float 1/3 of length so fraction of density will be

density of wood/ density of water = 1/3

density of wood = 1/3 ×  density of water

density of wood  = 1/3 × 1000 = 333.33 Kg/m³

Now in 2nd part we know for stable condition in partially submerged of body the metacentric height is greater than the centroid of body

we check these condition now,

metacentric height (GM)= I/v  

I = ( 0.02 × 0.02³ / 12 )  

v = ( 0.02 × 0.02 × 0.1 )

(GM)= I/v =  ( 0.02 × 0.02³ / 12 ) / ( 0.02 × 0.02 × 0.1 ) = 0.000333

and we know centroid of body (BM) =  0.05 - 0.033 = 0.017

we know height is 0.1m so G act at 0.05 and B act at (0.1 × 0.3 ) = 0.033

we can see that now metacentric height is less than centroid of body so our body is unstable condition

Now in 3rd part we use alcohol so we calculate ratio of density of wood and density of alcohol i.e. = 333.33 / 700 = 0.48

so now our new G will be 0.05 and B will be (0.1 × 0.48 ) = 0.048

metacentric height (GM)= I/v =  ( 0.02 × 0.02³ / 12 ) / ( 0.02 × 0.02 × 0.1 ) = 0.000333

centroid of body (BM) =  0.05 - 0.048 = 0.002

we can see that now metacentric height is less than centroid of body so our body is unstable condition