When placed deep in water, an empty plastic box of mass m accelerates upward at a rate of g/6 . Determine the mass of sand that must be added to the box so that it accelerates downward at a rate of g/6 .

Step-by-step solution

Let the buoyant force on the completely submerged box be B . When the empty box of mass m accelerates upward at g 6 , the equation of motion is: B - mg = m ( g 6 ) B = mg + mg 6 = 7mg 6 Let M be the mass of the sand added. The total mass becomes m + M . Since the volume of the box remains the same, the buoyant force B remains unchanged. For the box to accelerate downward at g 6 , the equation of motion is: (m + M)g - B = (m + M) ( g 6 ) Substituting the value of B into the equation: (m + M)g - 7mg 6 = (m + M) ( g 6