A wooden block of volume 5.24 3 1024 m3 floats in water, and a small steel objec

Question

A wooden block of volume 5.24 3 1024 m3 floats in water, and a small steel object of mass m is placed on top of the block. When m 5 0.310 kg, the system is in equilibrium, and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by a second steel object with a mass less than 0.310 kg? What happens to the block when the steel object is replaced by yet another steel object with a mass greater than 0.310 kg?

Explanation / Answer

# Given

Volume of the wooden block Vwood = 5.243 x 10-4 m3

Mass of the steel object msteel = 0.310 Kg

# Known Values

Density of water water= 1 x 103 kg/m3

# Solution

a) Finding Density of the wood

The buoyancy force due to the displaced water = weight of the wooden block + weight of the steel object

Fb = (mwood) g + (msteel)g

Since buoyancy force is equal to the weight of the displaced water

Fb = (mwater)g = (mwood) g + (msteel)g

since density() = mass(m) / volume(V)

mass = density x volume

m = V

so the above equation becomes

(water Vwater)g = (woodVwood) g + (msteel)g

water Vwater = woodVwood + msteel

since the volume of the water displaced is equal to the volume of the block submerged

Vwater = Vwood

so the above equation will become

water Vwood = woodVwood + msteel

1 x 103 x 5.243 x 10-4= wood x 5.243 x 10-4 + 0.310

0.5243 = wood x 5.243 x 10-4 + 0.310

wood = 0.5243 – 0.310 / 5.243 x 10-4

wood = 0.2143 / 5.243 x 10-4

wood = 408.74 kg/m3

So thedensity of the wood is 408.74 kg/m3

B) Analyzing the effect of change in mass

The buoyancy force due to the displaced water = weight of the wooden block + weight of the steel object.

An object with lesser mass will have lesser weight so when the steel object is replaced by a second steel object with a mass less than 0.310 kg, the buoyancy force will be less than the original buoyancy force and hence the wooden block will not be completely submerged under the water

C) Analyzing the effect of change in mass

The buoyancy force due to the displaced water = weight of the wooden block + weight of the steel object.

An object with greater mass will have greater weight so when the steel object is replaced by yet another steel object with a mass greater than 0.310 kg, the buoyancy force will be greater than the original buoyancy force and hence the wooden block sink along with the steel object.

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