physics buoyant

Question

a styrofoam ball is attached to the bottom of a swimming pool with a nylon cord.  the cord is 1.20 m long and the pool is 3.3 m.  diameter of ball is 46cm. density of styroforam is 0.3 g/cc. find the tension in the cord.  350N is answer.

Answer 1

Since the forces on the ball will be balanced (ie. we know the buoyant force will act upwards, but this must be balanced by the tension acting downwards, or else the cord would snap and the ball would float up) this means the tension in the rope is equal to the buoyancy force. The buoyant force acting on the object is equal to the weight of water displaced - note that the buoyancy force is only related to weight of water displaced, not depth, so the pool depth and cord length are irrelevant.

 

The net buoyancy force will be equal to the weight of water displaced minus the weight of the ball.

We use the formula: F_B = Vg(ρ_water - ρ_ball) where V is the volume (of water displaced, equal to the ball's volume), g = 9.81m/s², ρ is density (ρ_water = 1000kg/m³, ρ_ball = 0.3g/cc = 300kg/m³). Vgρ_water gives the weight of water displaced (buoyancy force) and Vgρ_ball gives the weight of the ball.

 

Volume of the ball/water displaced is the volume of a sphere, and the radius is 1/2 diameter, so 23cm or r = 0.23m

V = 4/3 * pi * r³ = 4/3 * pi * 0.23³ = 0.050965 m³

Now from the formula for net buoyancy force, F_B = 0.050965 * 9.81 * (1000 - 300) = 349.98N 

Since the tension must equal the net buoyancy force for the ball to remain stationary, T = 350N

Answer 2

There are three forces acting on the ball:

1. Fg - The weight of the ball (force of Earth's gravity) pulls it down,

2. Fb - The buoyant force of the displaced water pushes it up, and

3. Ft - The tension force in the nylon cord also pulls it down.

Since the ball is in a state of static equilibrium (it is not moving or accelerating) the net force from those three must equal zero. Since the tension force is down, weight is down and buoyant force is up, we can write the equation:

Fb - Fg - Ft = 0 N. Or: Ft = Fb - Fg.

So we have to find Fg and Fb. Fg = m g.

The mass of the ball can be found from m = ρ V.

The density ρ = 0.3 g/cc = 300 kg/m³.

The volume of the ball (a sphere) is V = 4/3 π (0.23m)³ = 0.051 m³.

So m = 300 kg/m³ * 0.051 m³ = 15.3 kg.

So Fg = mg = 15.3 kg * 9.8 N/kg = 150N

The buoyant force is simply the weight of the displaced water, so it has the same volume as the ball, but a different density: ρ = 1000 kg/m³:

Fb = m g = ρVg = 1000 kg/m³ * 0.051 m³ * 9.8 N/kg = 500 N

Now we can solve the problem!

Ft = Fb - Fg = 500 N - 150 N = 350 N.

You're welcome.

Answer 3

Love it that it's stated to be a styroforam ball. Didn't know those little fellas could do tests (yes, that's what foraminifera skeletons are called!) of polystyrene. Thought they were limited to chitin or calcim carbonate.

Though -- come to think of it, I have seen model dinosaur skeletons that distinctly had polystyrene characteristics! They'd better watch out, if a recycling truck catches them, T. rex could end up as Trex!

-- Cheers, --Mr. d.

Answer 4

Note that the length of the cord (1.2m) and the depth of the pool (3.3m) a both irrelevant.

Of course, this ignores the slight variation in gravity as the ball gets closer to the center of the Earth (but we don't even know how far the pool is above sea level  ;^>  ), and it assumes the nylon cord is inextensible (and has no spring-constant) and that the styrofoam is incompressible (and non-porus).  However, I assume this is only High School physics (or intro. college physics), so those exotic factors are ignored -- especially when the data is given with only two places of precision!

Answer 5

Draw a free body diagram, in which the up force is the buoyancy force Fb, and down forces are the weight W and the tension force T.

Apply Newton's second law in vertical direction: Fb - W - T = 0

Solve for T,
T = Fb - W = (4/3)π(.23)³ (1000 - 300)g = 350 N

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Attn: Both Fb and W are proportional to the volume of the ball.

Answer 6

Pretty straight forward. The buoyant force F_b of the displaced water is in opposition to the weight of the styrofoam ball W. Since the ball is tethered to the bottom by a cord,the cord's tension T is added to the ball's weight W so the sum of the forces is zero.

F_b = W + T or T = F_b-W F_b = ρVg where V = 4/3 π r³ = 4/3 π (.23 )³ = 0.050965 m³,

F_b = (1000 kg/m³ )(0.050965 m³ )(9.8 m/s² ) = 499.457 N ≈ 500 N,

W = mg = ρVg = (300 kg/m³ )(0.050965 m³ )(9.8 m/s² ) = 149.837 N ≈ 150 N,

T = F_b - W = 500 N - 150 N = 350 N