Two balls of the same volume are released inside a tank filled with water as shown in the figure. The densities of balls A and B are 0.86 g/cm3 and 0.55 g/cm3. (The density of water is 1.00 g/cm3. Take the +y direction to be up. Indicate the direction with the sign of your answer.)

(a) Find the acceleration of ball A.

(b) Find the acceleration of ball B.

If they expect you to neglect hydrodynamic friction (drag), use

a = F/m to get the acceleration a,
where F is the weight minus the buoyancy force.

The accleration is up in both cases.

In the real world, drag will be important in reducing a and will depend upon the diameter and the velocity. Drag can be neglected the instant the ball is released, becasue it is not moving yet.

To find the acceleration of the balls A and B, we can use Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

The buoyant force (Fb) can be calculated using the formula:

Fb = ρ * g * V

where:
- ρ is the density of the fluid (water in this case)
- g is the acceleration due to gravity (assumed to be 9.8 m/s2)
- V is the volume of the fluid displaced by the object

Since the two balls have the same volume, they will displace the same volume of water. Therefore, the buoyant force acting on each ball will be the same.

The weight of the ball (W) can be calculated using the formula:

W = m * g

where:
- m is the mass of the ball
- g is the acceleration due to gravity

The net force acting on each ball can be calculated using the formula:

net force = buoyant force - weight

The acceleration (a) can be calculated using Newton's second law of motion:

net force = m * a

Now, let's solve for the acceleration of each ball:

(a) For ball A:
- density (ρ) = 0.86 g/cm3 = 0.86 * 1000 kg/m3 = 860 kg/m3
- g = 9.8 m/s2
- volume (V) is the same for both balls
- weight (W) = m * g
- net force = buoyant force - weight
- m * a = ρ * g * V - m * g
- a = (ρ * g * V - m * g) / m

(b) For ball B:
- density (ρ) = 0.55 g/cm3 = 0.55 * 1000 kg/m3 = 550 kg/m3
- g = 9.8 m/s2
- volume (V) is the same for both balls
- weight (W) = m * g
- net force = buoyant force - weight
- m * a = ρ * g * V - m * g
- a = (ρ * g * V - m * g) / m

Note: To find the volume (V) of each ball, you can use the formula for the volume of a sphere:

V = (4/3) * π * r3

where:
- r is the radius of the ball (which is the same for both balls)

Plug in the values for the densities, acceleration due to gravity, and volumes in the above formulas to find the accelerations of balls A and B.