A Spanish galleon is about to be boarded by bloodthirsty pirates in the shallows of a Caribbean island. To save a box of treasure on board, the captain orders his crew to secretly toss the box overboard, planning to come back for it later. The rectangular box is waterproof and measures 40.0 cm by 25.0 cm by 30.0 cm. It is made of wood and has mostly gold pieces inside, resulting in an average box density three times that of seawater.
Sinking below the surface, the box moves at constant vertical velocity of 1.15 m/s for 12.0 m before hitting the bottom. (a) Draw the free-body diagram for the box, (b) determine the magnitudes of the forces on the box, and (c) calculate the work done by each force and the net work done on the box. (d) Calculate the change in the box’s gravitational potential energy. (e) What is the change in the box’s total energy ?
Since the box is moving through sea water at a constant speed, the net force on it must be zero.
The forces are:
i) W= mg its weight downwards
ii) Fb - buoyant force upwards
iii) Ff - frictional force offered by water acting upwards.
now W = mg = V*Db*g (Db-density of box)
and Fb = V*Dw*g (Dw - density of water)
Since Db = 3*Dw, W = 3*Fb
So Ff = W - Fb
= W - W/3 = 2W/3
From these equations, find magnitudes of the three force viz. W, Fb and Ff by calculating volume V and taking sea water density Dw = 1024 Kg/m^3
Once you know the forces, calculate the work done as force x Distance moved
(a) To draw the free-body diagram for the box, we need to consider the forces acting on it. The main forces to consider are the weight of the box and the buoyant force.
The weight of the box is a downward force caused by gravity. It can be calculated using the formula:
Weight = mass * gravity
The buoyant force is an upward force caused by the displaced water. It can be calculated using the formula:
Buoyant force = density of fluid * volume of fluid displaced * gravity
(b) To determine the magnitudes of the forces on the box, we need to calculate the weight and buoyant force.
Given that the box has a density three times that of seawater, we can calculate the density of the box using the formula:
Density = mass / volume
Since we know the dimensions of the box (40.0 cm by 25.0 cm by 30.0 cm), we can calculate the volume using the formula:
Volume = length * width * height
Once we have the density and volume of the box, we can calculate the mass using the formula:
Mass = density * volume
Using the mass, we can then calculate the weight and the buoyant force.
(c) To calculate the work done by each force and the net work done on the box, we need to consider the displacement of the box and the forces acting on it.
The work done by a force can be calculated using the formula:
Work = force * displacement * cos(angle)
The net work done on the box can be calculated by summing up the work done by each force.
(d) To calculate the change in the box's gravitational potential energy, we need to consider the initial and final positions of the box.
The gravitational potential energy can be calculated using the formula:
Gravitational Potential Energy = weight * height
Since the box moves vertically downward, the change in gravitational potential energy will be negative.
(e) To calculate the change in the box's total energy, we need to consider any other forms of energy involved, such as kinetic energy or work done.
The total energy of the box can be calculated by summing up the kinetic energy and any other forms of energy.
Note: To obtain numerical values for the calculations, we would need specific data such as the density of seawater, the acceleration due to gravity, and the height of the drop.