A hockey puck raft made of lead (density= 11,300kg/m^3) with diameter 13.5cm and

Question

A hockey puck raft made of lead (density= 11,300kg/m^3) with diameter 13.5cm and height 4.0cm. A 4.0cm x 4.0cm x 4.0cm cube of Iridium (density= 22,560 kg/m^3) is suspended by a string from the bottom of lead puck. The puck and cube are then placed in a tank of mercury (density= 13,600 kg/m^3) where they are at rest and a float.

a) what is the tension in the string supporting the Iridium cube?

b) how much of the lead pucks 4.0cm height is submerged in the mercury? (note: don’t forget to account for the connected cube. )

Explanation / Answer

The Density ratio equals the fraction of volume of the object that is under the surface of the liquid.
Density ratio = Density of object / Density of liquid
Density ratio = 11300 / 13600

The fraction that is under the surface of the liquid = 113/136

This is the fraction of volume of the hockey puck that is under mercury. Since the puck is 4.0 cm high, the fraction of the height that is under mercury = 113/136
So, the height of the puck that is under mercury = 113/136 * 4.0

This is also how deep below the surface of the mercury the bottom of the lead puck is!
How deep below the surface of the mercury is the bottom of the lead puck?
113/136 *4.0 cm = 3.323 cm ........Ans.

Net force up = Buoyant force – Weight
The net force is the force that is pushing a floating object up.
If the Buoyant force > Weight, the object will float. As the buoyant force increases, more of the object is above the surface.

Volume of lead = * radius^2 * height
Mass of lead = Volume * Density
Radius = ½ * diameter = ½ * 13.5 cm = 6.75 cm = 0.0675 m
Height = 4.0 cm = 0.04 m
Volume = * 0.0675^2 * 0.04
Mass = ( * 0.0675^2 * 0.04) * 11300
Weight of lead = ( * 0.0675^2 * 0.06) * 11300 * 9.8 = 95.05 N

weight of Iridium = 0.043* 22560 * 9.8 = 14.15 N

total tension = 109.20 N.......Ans.

Buoyant force = volume of puck under surface * Density of Hg * 9.8
Buoyant force = volume of puck under surface * 13,600 * 9.8

As the puck sinks into the mercury, the entire area of the bottom is under and part of the height is under.
Volume under = * 0.0675^2 * h
Buoyant force = * 0.0675^2 * h * 13,600 * 9.8
Buoyant force = weight
* 0.0675^2 * h * 13600 * 9.8 = ( * 0.0675^2 * 0.04) * 11300 * 9.8
divide both sides by ( * 0.0575^2 * 9.8)

h * 13600 = 0.04 * 11300
h = 0.04 * 11300 ÷ 13600
h = 0.04 * 113 ÷ 136 = 3.323 cm ........Ans.

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