A solid aluminum ingot weighs 89 N in air. What is its volume? The ingot is suspended from a rope and totally immersed in water. What is the tension in the rope?
density of Al = 2.7*10^3 kg/m^3
weight = m g = 89 N
so
m = 89 N /9.8 m/s^2 = 9.08 kg
mass = density*volume
so
volume = mass/(2.7 *10^3)
so
volume = 9.08 /(2.7*10^3) = 3.36*10^-3 m^3
density of water = 10^3 kg/m^3
buoyancy = volume * density of water * g
= 3.36*10^-3*10^3 * 9.8 = 33 Newtons
weight - buoyancy = tension = 89-33
Why did the ingot go to the water park? Because it wanted to make a splash! Now, let's dive into your questions. To find the volume of the solid aluminum ingot, we need to use Archimedes' principle. Since it's completely immersed in water, the buoyant force acting on it is equal to the weight of the water it displaces.
Given that the weight of the ingot in air is 89 N, we can use this information to find its weight in water. Since the buoyant force equals the weight of the water displaced, the weight in water is equal to 89 N (since the ingot is fully submerged).
Now, the volume of the water displaced is the same as the volume of the ingot. We know that the density of water is approximately 1000 kg/m³ (it's pretty dense, you know). We can make use of this information to find the volume of the ingot using the formula:
Density = Mass / Volume
Rearranging the formula, we get:
Volume = Mass / Density
Since weight is equal to mass multiplied by the acceleration due to gravity (approximately 9.8 m/s² on Earth), we can convert the weight of the ingot in water to its mass using:
Weight = Mass x Gravity
Therefore, we have:
Mass = Weight in water / Gravity
But here comes the punchline, since the weight in water is the same as the weight in air, we can substitute the weight in air:
Mass = Weight in air / Gravity
Now that we have the mass, we can calculate the volume using:
Volume = Mass / Density
Once we know the volume of the ingot, we can move on to the second part of your question, the tension in the rope when the ingot is suspended in water. But wait, maybe the ingot wants to take a break from its weighing adventures, and float in the water just like a rubber duck. Just kidding, let's get back to the question.
When the ingot floats submerged in water, the tension in the rope is equal to the weight of the ingot. So, the tension in the rope is the same as the weight of the ingot, which we already know is 89 N.
I hope this answer weighed in with the right amount of humor and information. Let me know if you have any more questions, and I'll be here to add a touch of laughter to your day!
To find the volume of the aluminum ingot, we can use Archimedes' principle, which states that the buoyant force acting on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
1. First, find the buoyant force acting on the aluminum ingot when it is fully submerged in water. The buoyant force can be calculated using the following formula:
Buoyant force = weight of the water displaced
The weight of the water displaced is equal to the weight of the ingot in air minus the weight of the ingot in water.
The weight of the ingot in water can be found using the formula:
Weight in water = weight in air - buoyant force
2. Calculate the volume of water displaced by the ingot. Since aluminum has a density of 2700 kg/m^3, we can use the formula:
Volume = Weight in water / Density of water
3. Finally, calculate the tension in the rope when the ingot is fully immersed in water. The tension in the rope is equal to the weight of the ingot minus the buoyant force.
Let's do the calculations step by step:
Step 1: Finding the buoyant force
Weight of the ingot in air: 89 N
Step 2: Calculating the volume of water displaced
Density of aluminum: 2700 kg/m^3
Density of water: 1000 kg/m^3 (approximately)
Step 3: Calculating the tension in the rope
Weight of the ingot: 89 N
Following these steps, we can find the volume of the ingot and the tension in the rope.
To find the volume of the aluminum ingot, we need to use Archimedes' principle. According to this principle, when a solid is immersed in a fluid (in this case, water), it experiences an upward buoyant force equal to the weight of the fluid it displaces.
Step 1: Determine the weight of water displaced
The weight of water displaced is equal to the weight of the aluminum ingot when it is totally immersed in water. We can calculate this as follows:
Weight of water displaced = Weight of aluminum ingot in air
Since the weight of the aluminum ingot in air is given as 89 N, the weight of water displaced is also 89 N.
Step 2: Calculate the volume of the aluminum ingot
To find the volume of the ingot, we divide the weight of water displaced by the density of water:
Volume = weight of water displaced / density of water
The density of water is typically taken to be 1000 kg/m^3 or 1000 N/m^3, since 1 kg weighs 9.8 N. However, it's important to note that the density of water can vary slightly depending on temperature and impurities. For simplicity, let's use 1000 N/m^3.
Volume = 89 N / 1000 N/m^3
By canceling the units, we find that the volume of the aluminum ingot is 0.089 m^3.
Now, let's move on to finding the tension in the rope.
Step 3: Calculate the buoyant force
The buoyant force acting on the aluminum ingot when it is immersed in water is equal to the weight of the water displaced.
Buoyant force = weight of water displaced = 89 N
Step 4: Calculate the tension in the rope
When the ingot is suspended from the rope, there are two forces acting on it: the tension in the rope and the buoyant force.
Since the system is in equilibrium, the tension in the rope must be equal and opposite to the buoyant force.
Tension in the rope = Buoyant force = 89 N
Therefore, the tension in the rope is 89 N.