A rectangular block of wood is 40cm wide, 60cm long and 30cm deep. If its relative density is 0.7 and it floats with its 30cm side verticle, determine: (a)the length of the block above the water surface and (b)the minimum force required to completely submerge the block.
Please show your working so i can understand
To determine the length of the block above the water surface, we can use the concept of buoyancy. The buoyant force acting on the block is equal to the weight of the water displaced by the block.
The volume of water displaced by the block can be calculated by multiplying the length, width, and height of the submerged portion of the block:
Volume of water displaced = width x depth x submerged height
= 40 cm x 30 cm x (60 cm - 30 cm)
= 40 cm x 30 cm x 30 cm
= 36000 cm^3
Next, we can calculate the weight of the water displaced by the block:
Weight of water displaced = density of water x volume of water displaced
= 1 g/cm^3 x 36000 cm^3
= 36000 g
= 36 kg
Since the block is floating, the weight of the water displaced is equal to the weight of the block:
Weight of block = relative density x volume of block x density of water
= 0.7 x 40 cm x 60 cm x 30 cm x 1 g/cm^3
= 50400 g
= 50.4 kg
Now we can determine the length of the block above the water surface. Let's assume the length above the water surface is x cm:
Weight of block = weight of water displaced
50.4 kg = 36 kg
x can be calculated using the following equation:
[length above water / total length] = [weight of water displaced / weight of block]
x / 60 cm = 36 kg / 50.4 kg
Cross multiplying the equation:
x = 60 cm x (36 kg / 50.4 kg)
x = 60 cm x 0.7143
x ≈ 42.857 cm
Therefore, the length of the block above the water surface is approximately 42.857 cm.
To determine the minimum force required to completely submerge the block, we need to calculate the weight of the portion of the block that is above the water surface. This weight can be calculated using the relative density of the block:
Weight of block above water = (1 - relative density) x volume of block x density of water
= (1 - 0.7) x 40 cm x 60 cm x 43 cm x 1 g/cm^3
= 0.3 x 40 cm x 60 cm x 43 cm
= 1.548 kg
Therefore, the minimum force required to completely submerge the block is approximately 1.548 kg or 15.48 N (since 1 kg ≈ 9.81 N).
To solve this problem, we need to consider the concept of buoyancy.
The buoyant force is the upward force exerted on an object immersed in a fluid (in this case, water). It is equal to the weight of the fluid displaced by the object. If the buoyant force is greater than or equal to the weight of the object, it will float; otherwise, it will sink.
Let's start with part (a) - determining the length of the block above the water surface.
1. Calculate the volume of the block:
Volume = length × width × depth
Volume = 60 cm × 40 cm × 30 cm
Volume = 72,000 cm³
2. Determine the volume of water displaced by the submerged part of the block:
Volume of water displaced = width × height × depth
Volume of water displaced = 40 cm × height (unknown) × 30 cm
Volume of water displaced = 1,200 cm³ × height (unknown)
3. The relative density of the wood block is given as 0.7, which means it is 0.7 times denser than water.
Hence, the total volume of the block above the water surface is 0.7 times the volume of the entire block:
Total volume above water surface = 0.7 × Volume
Total volume above water surface = 0.7 × 72,000 cm³
Total volume above water surface = 50,400 cm³
4. Set up an equation to find the height (length) of the block above the water surface:
Volume of water displaced = Total volume above water surface
1,200 cm³ × height = 50,400 cm³
height = 50,400 cm³ / 1,200 cm³
height = 42 cm
Therefore, the length of the block above the water surface is 42 cm.
Now let's move on to part (b) - determining the minimum force required to completely submerge the block.
To fully submerge the block, the buoyant force must be equal to or greater than the weight of the block.
1. Calculate the weight of the block:
Weight = mass × gravitational acceleration
Weight = density × volume × gravitational acceleration
Weight = 0.7 × Volume × gravitational acceleration
2. Calculate the buoyant force:
Buoyant force = weight of the water displaced by the block
Buoyant force = density of water × volume of water displaced × gravitational acceleration
Buoyant force = 1 g/cm³ (density of water) × 1,200 cm³ (volume of water displaced) × gravitational acceleration
3. Set up an equation to find the minimum force required:
Buoyant force = Weight
1 g/cm³ × 1,200 cm³ × gravitational acceleration = 0.7 × Volume × gravitational acceleration
Simplify the equation by canceling out the gravitational acceleration:
1,200 cm³ = 0.7 × Volume
Volume = 1,200 cm³ / 0.7
Volume = 1,714.29 cm³
4. Since the block is rectangular, we can express the volume as follows:
Volume = length × width × depth
Now, substitute the given dimensions and solve for the length:
1,714.29 cm³ = length × 40 cm × 30 cm
length = 1,714.29 cm³ / (40 cm × 30 cm)
length ≈ 1.43 cm
Therefore, the minimum force required to completely submerge the block is approximately 1.43 cm.