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Q.55

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 386

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Short Answer

A plastic "boat" with a 25cm2 square cross section floats in a liquid. One by one, you place 50g masses inside the boat and measure how far the boat extends below the surface. Your data are as follows:

Draw an appropriate graph of the data and, from the slope and intercept of the best-fit line, determine the mass of the boat and the density of the liquid.

Therefore, the length of bottle above the water is 5.2 cm

See the step by step solution

Step by Step Solution

Step :1 Introduction 

The expression for Buoyancy force acting on a submerged object in a liquid is,

FB=ρfVdisg

Here, ρf is density of the fluid, Vdis is displaced volume of the liquid due to object, and g is acceleration due to gravity.

Step :2 Explanation 

Convert the radius of the soda bottle from centimeter to meter.

r=6.2cm2

=3.1cm1.0m100cm

=3.1×102m

Convert the volume of soda from mile-liters to liters.

V=355mL106m31.0mL

=355×106m3

Step :3 Archimedes principle 

Using Archimedes' Principle, the floating force equals to the weight of the liquid displaced. A half full soda can is floating because it displaces at least half can volume of water plus a bit more to support the can's own weight.

The volume of the soda can V is,

V=πr2l

Here, / is the length of the soda can.

If we change the volume to Sl unit, 355mL is 0.355L. One liter is 103m3, the volume of soda can is 0.355×103m3, so we have calculated the length of the can from the above equation:

l=0.355×103m3π3.1×102m2

=0.118m

Step :4 Density of water

As the soda can is half full with water, it is immersed half-length / to support the half full of water and plus a bit more Δl as a result of the weight of the can itself.

Δ/π3.1×1022ρWg=20×103g

Here, ρW the mass density of water and g is the gravitational acceleration. Solve the equation for Δl.

Δl=20×103π3.1×10221000m

=6.6×103m

The length of the can above water is

l2Δl=0.118m26.6×103m

=0.052m

=0.052m100cm1m

=5.2cm

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