a solid aluminum ingot weighs 89N in the air. what is its volume? (b) the ingot is suspended from a rope and totally immersed in water. What is the tension in the rope?
VolumeinAir=Mass/density
mass=weight/g
Now in the water, it has the same mass, and the same volume. But it has a force of bouyancy equal to the weight of the displaced water.
bouyancy=densitywater*volume*g
weight in water=89N-bouyancyforce
To find the volume of the solid aluminum ingot, you can use the principle of buoyancy. The buoyant force acting on the ingot when it is fully submerged in water is equal to the weight of the water displaced by the ingot. Since the ingot is made of aluminum, we can assume that it is completely solid without any hollow spaces.
(a) Finding the volume of the ingot:
1. Determine the density of aluminum: The density of aluminum is typically around 2,700 kg/m³.
2. Convert the weight of the ingot to kilograms: Divide the weight (89N) by the acceleration due to gravity (9.8 m/s²) to get the mass in kilograms: 89N / 9.8 m/s² = 9.08 kg.
3. Use the formula for density: Density = Mass / Volume. Rearrange the formula to solve for volume: Volume = Mass / Density.
Volume = 9.08 kg / 2700 kg/m³ ≈ 0.00337 m³.
Therefore, the volume of the solid aluminum ingot is approximately 0.00337 cubic meters.
(b) To determine the tension in the rope when the ingot is immersed in water, we need to consider the forces acting on the ingot:
1. Weight of the ingot: The weight of the ingot remains the same (89N) regardless of whether it is in the air or immersed in water.
2. Buoyant force: The buoyant force acting on the ingot is equal to the weight of the water displaced by the ingot. This buoyant force counteracts the weight of the ingot, reducing the tension in the rope.
3. Tension in the rope: The tension in the rope is the difference between the weight of the ingot and the buoyant force.
Since the buoyant force is equal to the weight of the water displaced by the ingot, we need to find the volume of the ingot (which we already solved in part a) and use it to determine the weight of the displaced water.
4. Find the weight of the displaced water: The weight of the displaced water is equal to the buoyant force. We can calculate it using the formula: Weight of water displaced = density of water × volume of the ingot × acceleration due to gravity.
Density of water ≈ 1000 kg/m³.
Weight of water displaced = 1000 kg/m³ × 0.00337 m³ × 9.8 m/s² ≈ 32.85 N.
5. Calculate the tension in the rope: Tension = Weight of the ingot - Buoyant force.
Tension = 89 N - 32.85 N ≈ 56.15 N.
Therefore, the tension in the rope when the aluminum ingot is totally immersed in water is approximately 56.15 Newtons.