Question

A large plastic cylinder with mass 30.0 kg and density 395 kg/m3 is in the water of a lake. A light vertical cable runs between the bottom of the cylinder and the bottom of the lake and holds the cylinder so that 30.0% of its volume is above the surface of the water. What is the tension in the cable?

Answer 1

The tension in the cable can be calculated using Archimedes' principle and equilibrium conditions. Use Archimedes' principle to calculate the volume of water displaced by the submerged portion of the cylinder, and then use the density of water and the acceleration due to gravity to calculate the weight of the submerged portion. This weight is equal to the tension in the cable.

Step-by-step explanation:

The tension in the cable can be calculated using Archimedes' principle and equilibrium conditions. Archimedes' principle 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. In this case, the buoyant force is equal to the weight of the volume of water displaced by the submerged portion of the cylinder.

Since 30.0% of the cylinder's volume is above the surface of the water, 70.0% of the volume is submerged. The density of water is 1000 kg/m^3. We can calculate the volume of water displaced by the submerged portion of the cylinder using the mass of the cylinder and its density.

The tension in the cable is equal to the weight of the submerged portion of the cylinder, which can be calculated by multiplying the volume of water displaced by the density of water and the acceleration due to gravity.

In summary, use Archimedes' principle to calculate the volume of water displaced by the submerged portion of the cylinder, and then use the density of water and the acceleration due to gravity to calculate the weight of the submerged portion. This weight is equal to the tension in the cable.

Answer 2

Tension = 227.6 N

Step-by-step explanation:

Applying the equilibrium condition on the cylinder with the downward direction taken as positive:

`Tension + Weight\ of\ Cylinder - Bouyant\ Force = 0\\Tension = \rho Vg - Weight\ of\ Cylinder`

where,

ρ = density of water = 1000 kg/m³

g = acceleration due to gravity = 9.81 m/s²

V = Volume of water displaced = 70% of Volume of Cylinder

V =
`0.7((Mass\ of\ Cylinder)/(Density\ of\ Cylinder) ) = 0.7((30\ kg)/(395\ kg/m^3))` = 0.0532 m³

Weight of Cylinder = (mass)(g) = (30 kg)(9.81 m/s²) = 294.3 N

Therefore,

`Tension = (1000\ kg/m^3)(0.0532\ m^3)(9.81\ m/s^2) - 294.3\ N\\`

Tension = 227.6 N