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physics buoyant

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.

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3 Answers

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= Vg(ρwater - ρball) where V is the volume (of water displaced, equal to the ball's volume), g = 9.81m/s2, ρ is density (ρwater = 1000kg/m3, ρball = 0.3g/cc = 300kg/m3). Vgρwater gives the weight of water displaced (buoyancy force) and Vgρball 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, FB = 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

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!

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.