A solid rectangular block measure 0.100*0.080m*0.060m and floats freely in a liquid of density 1354kg/m^2 if the depth of liquid 0.04m up the block largest side. Find the density of the block.
Vr=0.100 * 0.08 * 0.06=4.8*10^-4 m^3 =
480 cm^3.
Vb=0.04 * 0.08 * 0.06 = 1.92*10^-4 m^3 = 192 cm^3 = Vol. below surface.
Vb = (Dr/Dl)*Vr = 1.92*10^-4 m^3.
(Dr/1354)*4.8*10^-4 = 1.92*10^-4.
Multiply both sides by 1354:
Dr*4.8*10^-4 = 2599.68*10*-4.
Dr = 541.6 kg/m^3=Density of rectangle.
Well, based on the information given, we need to determine the density of the block. Now, let's see... if the block is floating freely in the liquid, it means that the buoyant force acting on it is equal to the weight of the block.
To find the buoyant force, we need to know the volume of the block submerged in the liquid. Since the depth of the liquid is given as 0.04m, and the largest side of the block is submerged, we can calculate the volume of this side as follows:
Volume = length x width x depth
= 0.1m x 0.08m x 0.04m
Let's simplify that: 0.00032 m^3
Now, since the buoyant force is equal to the weight of the block, we have:
Buoyant Force = density of the liquid x volume of the submerged block x gravitational acceleration
Given that the density of the liquid is 1354 kg/m^3 and the gravitational acceleration is approximately 9.8 m/s^2, we have:
Buoyant Force = 1354 kg/m^3 x 0.00032 m^3 x 9.8 m/s^2
Let's calculate that: approximately 0.042 N
Since the buoyant force is equal to the weight of the block, and weight is given by the formula:
Weight = mass x gravitational acceleration
We can rearrange the equation to solve for mass:
Mass = Weight / gravitational acceleration
So, we have:
Mass = 0.042 N / 9.8 m/s^2
Calculating that gives us: approximately 0.0043 kg
Finally, to find the density of the block, we divide the mass by its volume:
Density = Mass / Volume
= 0.0043 kg / (0.1m x 0.08m x 0.06m)
Calculating that gives us: approximately 0.716 kg/m^3
So, the density of the block is approximately 0.716 kg/m^3.
Now, that wasn't too dense, was it?
To find the density of the block, we need to compare the weight of the block to the weight of the liquid it displaces.
Step 1: Calculate the volume of the block
The volume of a rectangular block is given by length × width × height:
Volume = 0.100 m × 0.080 m × 0.060 m = 0.00048 m³
Step 2: Calculate the weight of the block
The weight of an object is given by its volume multiplied by its density:
Weight of the block = Volume × Density of the block
Step 3: Calculate the weight of the liquid displaced
The weight of the liquid displaced is equal to the weight of the block when it is floating. It can be calculated using the formula:
Weight of the liquid displaced = Volume of the liquid displaced × Density of the liquid × g
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 4: Equate the two weights
Since the block floats freely in the liquid, the weight of the block is equal to the weight of the liquid displaced:
Volume × Density of the block = Volume of the liquid displaced × Density of the liquid × g
Step 5: Solve for the density of the block
Rearrange the equation to solve for the density of the block:
Density of the block = (Volume of the liquid displaced × Density of the liquid × g) / Volume
Plugging in the given values:
Density of the block = (0.040 m × 0.100 m × 0.080 m × 1354 kg/m³ × 9.8 m/s²) / 0.00048 m³
After calculating, you will find the density of the block.
To find the density of the block, we need to use the principle of buoyancy.
1. First, calculate the volume of the block:
Volume = length * width * height
Volume = 0.100m * 0.080m * 0.060m = 0.00048 m^3
2. The buoyant force acting on the block is equal to the weight of the liquid displaced by the block. The weight of the liquid can be calculated using the formula:
Weight of liquid = density of liquid * volume of liquid * g
Where g is the acceleration due to gravity.
3. The volume of liquid displaced by the block can be found by multiplying the length and height of the block by the depth of the liquid:
Volume of liquid = length * height * depth of liquid
Volume of liquid = 0.100m * 0.060m * 0.040m = 0.00024 m^3
4. Now, calculate the weight of the liquid:
Weight of liquid = 1354 kg/m^3 * 0.00024 m^3 * 9.8 m/s^2 = 3.364 kg * m/s^2 (or 3.364 N)
5. According to Archimedes' principle, the buoyant force acting on the block is equal to the weight of the liquid displaced. So, the buoyant force is 3.364 N.
6. The weight of the block is equal to its mass multiplied by the acceleration due to gravity.
Weight of block = mass of block * g
7. Since the block is floating freely, the weight of the block is equal to the buoyant force:
Weight of block = Buoyant force
mass of block * g = 3.364 N
8. Now we can calculate the mass of the block:
mass of block = 3.364 N / 9.8 m/s^2 = 0.343 kg
9. Finally, the density of the block can be calculated using the formula:
Density = mass / volume
Density = 0.343 kg / 0.00048 m^3 = 714.58 kg/m^3
Therefore, the density of the block is approximately 714.58 kg/m^3.