Problem 41 - University Physics with Modern Physics

Question

A box with a volume \(V=0.0500 \mathrm{~m}^{3}\) lies at the bottom of a lake whose water has a density of \(1.00 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). How much force is required to lift the box, if the mass of the box is (a) \(1000 . \mathrm{kg},\) (b) \(100 . \mathrm{kg},\) and \((\mathrm{c}) 55.0 \mathrm{~kg} ?\)

Step-by-Step Solution

Answer

Answer: The net forces required to lift the box for each case are: (a) 9319.5 N (b) 490.5 N (c) 49.05 N

Step 1: Calculate the mass of water displaced by the box

We can find the mass of the water displaced by the box using the formula: mass = volume × density For the given volume (V) of the box and the density of water, the mass of the water displaced (m_water) is: m_water = V × (1.00 × 10^3 kg/m³) m_water = 0.0500 m³ × (1.00 × 10^3 kg/m³) m_water = 50 kg

Step 2: Calculate the buoyant force

We can calculate the buoyant force (F_buoyant) using the formula: F_buoyant = m_water × g where g is the gravitational acceleration (9.81 m/s²). So, F_buoyant = 50 kg × 9.81 m/s² F_buoyant = 490.5 N

Step 3: Calculate the weight of the box for each case

We can calculate the weight (F_gravity) of the box for each case using the formula: F_gravity = m_box × g where m_box is the mass of the box. For each case: (a) m_box = 1000 kg F_gravity(a) = 1000 kg × 9.81 m/s² = 9810 N (b) m_box = 100 kg F_gravity(b) = 100 kg × 9.81 m/s² = 981 N (c) m_box = 55.0 kg F_gravity(c) = 55.0 kg × 9.81 m/s² = 539.55 N

Step 4: Calculate the net force required to lift the box for each case

Now, we can calculate the net force required to lift the box (F_net) for each case by subtracting the buoyant force from the weight: F_net = F_gravity - F_buoyant For each case: (a) F_net(a) = 9810 N - 490.5 N = 9319.5 N (b) F_net(b) = 981 N - 490.5 N = 490.5 N (c) F_net(c) = 539.55 N - 490.5 N = 49.05 N Thus, the forces required to lift the box for each case are: (a) 9319.5 N (b) 490.5 N (c) 49.05 N

Key Concepts

Fluid Mechanics

Fluid mechanics is a branch of physics concerned with the behavior of fluids (liquids, gases, and plasmas) and the forces on them. It has a variety of applications, including calculating forces involved in fluid flow and understanding the buoyancy effect on objects submerged in fluids.

In our exercise, we're essentially dealing with a static fluid problem where an object (a box) is submerged in water. One key concept here is understanding how the pressure exerted by the fluid acts on the box in all directions. Despite this pressure being exerted on the box, the box experiences a net upward force, known as the buoyant force. This force is crucial in determining whether an object sinks, floats, or rises in a fluid.

Archimedes' Principle

Archimedes' principle states that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. It is a fundamental principle in fluid mechanics and plays a central role in the study of buoyancy.

In relation to our example, the principle helps us calculate the buoyant force acting on the submerged box. The weight of the water displaced by the box is essentially what the buoyant force counteracts. Once we know the volume of the box and the density of the water, we can easily calculate the mass of the water displaced, and hence the buoyant force, using the formula \(F_{buoyant} = m_{water} \times g\), where \(g\) is the gravitational acceleration.

Density and Buoyancy

Density is defined as mass per unit volume and is a measure of how tightly matter is packed within a substance. The concept of density is vital in determining an object's buoyancy—the ability of an object to float or sink in a fluid.

When addressing the box's behavior in water for our exercise, we compare the density of the box to that of the water. If the box’s density is greater than that of the water, it will tend to sink, and if it's less, it will float. The buoyant force is directly related to the density of the fluid; the greater the fluid's density, the greater the buoyant force. This is reflected in the step by step solution where the mass of the displaced water, which contributes to buoyancy, is found using the water's density.

Gravitational Force

Gravitational force is a force of attraction that exists between any two masses, the earth being a massive body exerting gravitational pull on objects near its surface. This force governs the motion of objects and is responsible for the weight of an object, which is the force due to gravity acting on an object's mass.

The weight of the box in our problem is calculated taking into account the gravitational force, using the formula \(F_{gravity} = m_{box} \times g\). In the context of buoyancy, gravitational force is what the buoyant force has to overcome for an object to rise in a fluid. The net force to lift the box from the lakebed is thus the difference between the box's weight and the buoyant force exerted by the water, as shown in the step by step solution.