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 ?
To solve this problem, we will break it down into several steps:
(a) Draw the free-body diagram for the box:
In order to draw the free-body diagram, we need to identify all the forces acting on the box. The forces acting on the box are:
1. Weight force (mg) acting downward, where m is the mass of the box and g is the acceleration due to gravity (9.8 m/s^2).
2. Buoyant force (Fb) acting upward, which opposes the weight force and is equal in magnitude to the weight of the water displaced by the box.
3. Water resistance force (Fr) acting upward, which opposes the motion of the box underwater.
(b) Determine the magnitudes of the forces on the box:
1. Weight force (mg) can be calculated by multiplying the mass of the box by the acceleration due to gravity. Since density (ρ) is mass per unit volume, the volume of the box can be calculated by multiplying its dimensions.
2. Buoyant force (Fb) is equal to the weight of the water displaced by the box. This can be calculated using Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
3. Water resistance force (Fr) can be assumed to be equal to the drag force, which is given by the equation Fr = 1/2 * ρf * Av^2, where ρf is the density of the fluid (seawater), A is the surface area of the box facing the fluid, and v is the velocity of the box.
(c) Calculate the work done by each force and the net work done on the box:
Work done by a force is defined as the product of the magnitude of the force and the distance over which it acts. Since both the weight force and the buoyant force act vertically, and the box has a constant vertical velocity, the work done by these forces is zero. The water resistance force acts opposite to the motion of the box, so the work done by this force can be calculated by multiplying the magnitude of the force by the distance traveled by the box. The net work done on the box can be determined by summing up the work done by all the forces.
(d) Calculate the change in the box’s gravitational potential energy:
The change in the box's gravitational potential energy is equal to the negative of the work done by the weight force. This can be calculated using the formula ΔPE = mgh, where ΔPE is the change in potential energy, m is the mass of the box, g is the acceleration due to gravity, and h is the vertical displacement of the box.
(e) Calculate the change in the box’s total energy:
The change in the box's total energy is equal to the net work done on the box, according to the work-energy theorem. This can be calculated by summing up the work done by all the forces acting on the box.
With these steps, we should be able to answer all the questions related to the forces, work, and energy of the box in this scenario.