Question

A flat raft with a $6\mathrm{ft}\times 6\mathrm{ft}$ surface floats in a fresh water lake. Determine the deflection of the raft if a 190 lb man stands at the geometric center. Fig. P1.7

Short answer

When a 190 lbs man stands at the geometric center of a 6 ft by 6 ft raft, the deflection of the raft is approximately 0.043 ft.

Step-by-step solution

Calculate Initial Buoyancy Force

First, we need to calculate the buoyancy force acting on the raft at its initial position. The buoyancy force can be calculated using the following formula: $F_B=V\times {\rho }_w\times g$ where $F_B$ is the buoyancy force, $V$ is the volume of submerged part of the raft, ${\rho }_w$ is the density of fresh water (approximately $1000\mathrm{kg}/{\mathrm{m}}^3$), and $g$ is the acceleration due to gravity (approximately $9.81\mathrm{m}/{\mathrm{s}}^2$). We don't know $V$ directly, so we need to find the height of the submerged part of the raft, which we'll call $h$. Assume that the initial position of the raft means it is floating without any deflection (all its weight is supported by the buoyancy force). From this assumption, we can calculate $h$: $h\times 6\mathrm{ft}\times 6\mathrm{ft}=190\mathrm{lb}$.

Convert Units

To proceed, we need to convert all the units in SI (i.e., meters and newtons). 1 $\mathrm{ft}=0.3048\mathrm{m}$, and 1 $\mathrm{lb}=4.44822\mathrm{N}$. So, $6\mathrm{ft}=1.8288\mathrm{m}$ and $190\mathrm{lb}=846.16\mathrm{N}$.

Calculate Height of Submerged Raft

Now we can solve the equation for $h$: $h\times 1.8288\mathrm{m}\times 1.8288\mathrm{m}=846.16\mathrm{N}$. $h=\frac{846.16\mathrm{N}}{(1.8288\mathrm{m}{)}^2}\approx 0.253\mathrm{m}$.

Calculate Additional Force Due to the Man's Weight

When the man stands on the raft, his weight will cause additional force on the raft and hence increases the submerged part of the raft. To calculate the additional submerged part of the raft, we can use the additional force due to man's weight and the formula for buoyancy force. $\mathrm{\Delta }{F}_B=\mathrm{\Delta }V\times {\rho }_w\times g$ $\frac{\mathrm{\Delta }{F}_B}{{\rho }_w\times g}=\mathrm{\Delta }V=h_{new}\times 1.8288\mathrm{m}\times 1.8288\mathrm{m}-h\times 1.8288\mathrm{m}\times 1.8288\mathrm{m}$, where $h_{new}$ is the height of the submerged part of the raft when the man is standing on it. Since the additional force is due to the man's weight, we have $\mathrm{\Delta }{F}_B=846.16\mathrm{N}$.

Calculate the New Submerged Height of the Raft

We can now solve the above equation for $h_{new}$. $h_{new}\times 1.8288\mathrm{m}\times 1.8288\mathrm{m}-h\times 1.8288\mathrm{m}\times 1.8288\mathrm{m}=\frac{846.16\mathrm{N}}{1000\mathrm{kg}/{\mathrm{m}}^3\times 9.81\mathrm{m}/{\mathrm{s}}^2}$ $h_{new}\approx 0.253\mathrm{m}+\frac{846.16\mathrm{N}}{1000\mathrm{kg}/{\mathrm{m}}^3\times 9.81\mathrm{m}/{\mathrm{s}}^2\times {1.8288}^2{\mathrm{m}}^2}\approx 0.266\mathrm{m}$.

Calculate the Deflection of the Raft

The deflection of the raft is the difference between the new submerged height of the raft and the initial submerged height of the raft: Deflection $\mathrm{\Delta }h=h_{new}-h\approx 0.266\mathrm{m}-0.253\mathrm{m}\approx 0.013\mathrm{m}$. Now we convert the deflection back to feet: Deflection $\mathrm{\Delta }h=0.013\mathrm{m}\times \frac{1\mathrm{ft}}{0.3048\mathrm{m}}\approx 0.043\mathrm{ft}$. So, the deflection of the raft when the man stands at the geometric center is approximately 0.043 ft.

Key concepts

Buoyancy Force Calculation

Understanding buoyancy force is key when analyzing floating objects, such as rafts. The buoyancy force is the upward force exerted by a fluid on an object submerged in it. This force allows the object to float. For a raft, buoyancy force can be calculated using the formula: $$F_B=V\times {\rho }_w\times g$$where:

• $F_B$ is the buoyancy force,
• $V$ is the volume of the submerged part of the object,
• ${\rho }_w$ is the density of the fluid (approximately 1000 kg/m³ for fresh water),
• $g$ is the acceleration due to gravity (approximately 9.81 m/s²).

The submerged volume here depends on the weight of the object and its distribution across the surface. Initially, the raft supports its own weight entirely using buoyancy without submerging deeply.

Unit Conversion in Engineering

Unit conversion is crucial in engineering calculations to ensure all measurements are in a consistent system, usually the SI system. This exercise requires converting feet to meters and pounds to newtons:

• 1 foot = 0.3048 meters,
• 1 pound = 4.44822 newtons.

For example, in this exercise, the dimensions of the raft and the weight it supports are given in feet and pounds respectively. Converting these to the metric system:

• The size of the raft, originally $6\text{ ft}\times 6\text{ ft}$, becomes $1.8288\text{ m}\times 1.8288\text{ m}$.
• The man's weight of 190 lb converts to 846.16 N.

Once you perform these conversions accurately, you enhance the precision of calculations related to submerged height and deflection.

Submerged Height Determination

Determining the submerged height of the raft is essential for calculating buoyancy. Initially, you calculate how deep the raft is submerged without any additional load. Using the equation:$$h\times l\times w=F_0$$you solve for $h$, the submerged height. Here:

• $h$ is the height you want to find,
• $l$ and $w$ are the length and width of the raft,
• $F_0$ is the initial force supported by the raft's buoyancy, initially equal to its own weight in newtons.

In practical terms, you sum up the forces and volumes to find how much of the raft is below the surface of the water. This step sets the stage for understanding raft behavior when additional weight is applied.

Raft Deflection Analysis

Raft deflection occurs when additional weight, like a person, is placed on the raft, increasing its submerged depth. The difference between the new and initial submerged heights reflects this deflection. Calculate the additional submerged volume needed to support extra weight using:$$\mathrm{\Delta }{F}_B=\mathrm{\Delta }V\times {\rho }_w\times g$$The extra volume is found by rearranging to solve for the new submerged height $h_{new}$. The deflection $\mathrm{\Delta }h$, which is $h_{new}-h$, gives the measure of how much deeper the raft goes due to the weight added. In this exercise, the deflection is initially found in meters and needs conversion back to feet to suit the context of the problem, providing a complete understanding of structural impacts in practical terms.