Question

\( \mathrm{~A}\) 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 (c) \(55.0 \mathrm{~kg} ?\)

Short answer

Answer: The net force required to lift the box with a mass of (a) 1000 kg is 9319.5 N, (b) 100 kg is 490.5 N, and (c) 55.0 kg is 49.05 N.

Step-by-step solution

1. Calculate the volume of water displaced by the box

In this case, the volume of water displaced is the same as the volume of the box, that is \(V = 0.0500 m^3\).

2. Compute the mass of the water displaced

To compute the mass of the water displaced, we use the formula \(m_{water} = V \cdot \rho_{water}\), where \(\rho_{water}\) is the density of the water. So, \(m_{water} = 0.0500 m^3 \cdot 1.00 \cdot 10^{3} kg/m^3 = 50.0 kg\).

3. Determine the buoyant force acting on the box

The buoyant force is equal to the weight of the water displaced, which can be calculated using the formula \(F_{buoyancy} = m_{water} \cdot g\), where \(g\) is the acceleration due to gravity (\(9.81 m/s^2\)). So, \(F_{buoyancy} = 50.0 kg \cdot 9.81 m/s^2 = 490.5 N\).

4. Calculate the gravitational force (weight) for each case

For each mass scenario (a) 1000 kg, (b) 100 kg, and (c) 55.0 kg, we need to calculate the gravitational force. We will use the formula \(F_{gravity} = m_{box} \cdot g\). (a) For \(m_{box} = 1000 kg\), \(F_{gravity} = 1000 kg \cdot 9.81 m/s^2 = 9810 N\). (b) For \(m_{box} = 100 kg\), \(F_{gravity} = 100 kg \cdot 9.81 m/s^2 = 981 N\). (c) For \(m_{box} = 55.0 kg\), \(F_{gravity} = 55.0 kg \cdot 9.81 m/s^2 = 539.55 N\).

5. Compute the net force required to lift the box for each case

Finally, we need to calculate the net force required to lift the box by subtracting the buoyant force from the gravitational force for each mass scenario: (a) For \(1000 kg\), \(F_{net} = F_{gravity} - F_{buoyancy} = 9810 N - 490.5 N = 9319.5 N\). (b) For \(100 kg\), \(F_{net} = F_{gravity} - F_{buoyancy} = 981 N - 490.5 N = 490.5 N\). (c) For \(55.0 kg\), \(F_{net} = F_{gravity} - F_{buoyancy} = 539.55 N - 490.5 N = 49.05 N\). So the force required to lift the box with a mass of (a) 1000 kg is 9319.5 N, (b) 100 kg is 490.5 N and (c) 55.0 kg is 49.05 N.