John and Tau take their toys down to a nearby pond to play. John has a toy boat made from wood. He places the wooden boat on the pond and it floats. Tau, being a cheeky little boy, wants to make John’s wooden boat sink. So he attaches a piece of lead on one end of the string, and ties the other loose end onto John’s boat and gently places the lead into the water. Determine the smallest amount of lead (mass) that will be enough to sink John’s boat, assuming the specific gravity of wood is 0.5 and the density of the pond water is equal to the density of pure water

The smallest amount of lead that needs to be attached to John's boat in order to sink it has to have a mass slightly greater than the mass of the boat.

Explanation:

A body floats in a fluid when its density is less than the density of the fluid.

The body sinks when its density is more than the density of the fluid.

Density of the pond = density of pure water = 1 g/cm³

Specific gravity of an object = (density of object)/(density of water)

0.5 = (density of John's boat)/1

Density of the John's boat = 0.5 g/cm³

And density is given as

Density = (mass/volume)

0.5 = (mass of John's boat)/(volume of John's boat)

Let the volume of John's boat be v and the mass of John's boat be m

0.5 = (m/v)

v = (m/0.5) = 2m

To sink the boat, we need the total mass on the boat to increase the density to a value greater than 1.

Let the minimum mass of lead required for this to be M

The volume of the boat remains the same, but the total mass on the boat is now (m+M)

1 < (m + M)/v

M + m > v

Recall, v = 2m

M + m > 2m

M > 2m - m

M > m

Hence, the smallest amount of lead that needs to be attached to John's boat in order to sink it has to have a mass slightly greater than the mass of the boat.

Hope this Helps

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