CP A 950 -kg, cylindrical can buoy floats vertically in salt...

CP A 950 -kg, cylindrical can buoy floats vertically in salt water. The diameter of the buoy is 0.900 $\mathrm{m}$ . (a) Calculate the additional distance the buoy will sink when a 70.0 -kg man stands on top of it. (Use the expression derived in part (b) of Problem 14.76.) (b) Calculate the period of the resulting vertical SHM when the man dives off. (Use the expression derived in part (c) of Problem 14.76 , and as in that problem, you can ignore the damping due to fluid friction.)

CP A 950 -kg, cylindrical can buoy floats vertically in salt water. The diameter of the buoy is 0.900 $\mathrm{m}$ . (a) Calculate the additional distance the buoy will sink when a 70.0 -kg man stands
on top of it. (Use the expression derived in part (b) of Problem 14.76.) (b) Calculate the period of the resulting vertical SHM when the man dives off. (Use the expression derived in part (c) of Problem 14.76 , and as in that problem, you can ignore the damping due to fluid friction.)

Key Concepts

Simple Harmonic Motion (SHM) in Buoyant Systems

When a floating object is disturbed from its stable equilibrium position in a fluid, it undergoes oscillations that can be modeled as simple harmonic motion, provided the restoring force is proportional to the displacement. In a buoyant system, the effective stiffness (or spring constant) can be derived from the change in buoyant force with displacement, which in turn is used to calculate the period of oscillation.

Buoyant Equilibrium

Buoyant equilibrium occurs when the upward buoyant force exactly balances the downward gravitational force, allowing an object to float at a constant level. This concept is essential to understand how floating objects adjust their submerged volume in response to added weights or changes in density.

Archimedes' Principle

This concept states that a body wholly or partially submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the body. It is fundamental for understanding why objects float, sink, or remain neutrally buoyant and is used to calculate the effects of changes in displacement when extra weight is added to a floating object.

Additional Displacement Due to Added Load

When extra mass is placed on a floating object, it sinks further into the fluid until the new volume of displaced fluid produces a buoyant force equal to the increased weight. The calculation of this additional displacement relies on the geometric properties of the object and the principle of buoyancy.

Transcript

00:01 High in the given problem mass of the person standing on the cylinder is given as 70 kilogram so when he'll stand on the cylinder the cylinder will dip a little more in the water in the salt water let it be h means the cylinder is dipped into the salt water by a height of h meter further so the as per archimedes principle, weight of the more water displaced, more water displaced will be equal to the weight of the person.

00:55 Weight of the person.

01:00 As diameter of the cylinder is given as 0 .90 meters, so its radius is 0 .9000 meters.

01:09 Meter by 2 means 0 .450 meter.

01:15 This is the radius.

01:16 Hence we can say the weight of the person means m g will be equal to weight of the more water displaced means v row g.

01:27 So this g will be cancelled for m this is 70 kilogram for volume it will be area of the cross -sectional area of the cylinder pi r square into h, the more height of the cylinder dipped into the salt water.

01:46 And as for a density of the salt water is concerned, this is 1039 kilogram per meter cube.

01:54 Hence, the expression for height will come out to be 70 by 70 divided by pi r square, pi r square into 1033.

02:11 For the density of the salt water.

02:15 Hence here this is 70 divided by 3 .14 divided by 0 .450 to the whole square and 1039.

02:28 So finally when we calculate it this height comes out to be 1 .106 meter.

02:36 So this is the answer for the first part of the problem in which we have to find the height of the cylinder which will be dipped.

02:49 This is the height of the cylinder.

02:53 The additional distance, the boy will sink when this 70 kilogram man stands on top of it.

03:01 Now, in the second part of the problem, they are saying, if this person dives off the boy, then what will be the term? of the time period of oscillation of this boy in the water.

03:20 So when this person jumps off, then the cylinder will experience a restoring force and that restoring force will be given by this is the restoring force...

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