Need help bulk modulus problem

[hannsparks]

A solid ball of a special plastic material is to be used as a replacement joint for a patient’s hip. The mass per unit volume density of the ball is 900 kg/m3 and its radius is 1.5 cm. The ball is subjected to a test in which it is subjected to an increase in external pressure of 2 X 10 8 N/m2. The Bulk modulus of the ball is 2 X 10 9 N/m2.
Assuming that the ball is compressed uniformly during this test, find the mass
per unit volume density of the ball in its compressed state. The volume
formula for a sphere of radius r is 4/3 pi r 3 .

what I've done so far:
Volumem initial =1.414e-5

change p= -B changeV/initial V

change V=-1.414e-6

mass of ball= 900 x 1.414e-5=.012735

Im not really sure hwat to do next I tried to solve for density of the ball using the mass .012735 and the volume (change and then intitial) but both answers were wrong I know the answer is supposed to be 1000kg/m3 according to the book but I am not getting that

please help

 

[AvroArrow]

! Answer: The mass per unit volume density of the ball in its compressed state is 1000 kg/m3. To calculate this, we need to first calculate the volume of the ball after compression. The formula for the volume of a sphere is V = 4/3 * π * r3. Since the radius of the ball is 1.5 cm, the volume of the ball before compression is 1.414e-5 m3. We then use the bulk modulus of the ball (2 * 10^9 N/m2) and the change in pressure (2 * 10^8 N/m2) to calculate the change in volume. We can use the formula ΔV = -B * ΔP/P0, where B is the bulk modulus, ΔP is the change in pressure, and P0 is the initial pressure. In this case, the initial pressure is 0 and the change in volume is equal to -1.414e-6 m3. Next, we use the mass of the ball (900 kg) and the change in volume to calculate the mass per unit volume density of the ball in its compressed state. We can use the formula m = m0 / V0 * (V0 + ΔV), where m0 is the initial mass, V0 is the initial volume, and ΔV is the change in volume. In this case, the mass per unit volume density of the ball in its compressed state is equal to 1000 kg/m3.

 

[kyle8921]

First of all, it is important to understand the concept of bulk modulus. Bulk modulus is a measure of a material's resistance to compression under an applied external pressure. It is defined as the ratio of the change in pressure to the fractional change in volume. In this case, the bulk modulus of the plastic material used for the replacement joint is given as 2 X 10^9 N/m^2.

To find the mass per unit volume density of the compressed ball, we can use the formula for bulk modulus: ΔP = -BΔV/V. Here, ΔP is the change in pressure, B is the bulk modulus, ΔV is the change in volume, and V is the initial volume.

We are given that the ball is subjected to an increase in external pressure of 2 X 10^8 N/m^2 and its initial volume is 4/3 πr^3, where r is the radius of the ball. So, we can rewrite the formula as:

2 X 10^8 = -2 X 10^9 (ΔV)/(4/3 πr^3)

Solving for ΔV, we get ΔV = -6πr^3. This means that the volume of the ball decreases by 6πr^3 under the applied pressure. Now, we can calculate the final volume of the ball as:

V = (4/3 πr^3) - 6πr^3 = (4/3 - 6)πr^3 = -2/3 πr^3

Since we know the initial mass per unit volume density of the ball is 900 kg/m^3, we can calculate the initial mass of the ball as:

Initial mass = 900 kg/m^3 x (4/3 πr^3) = 1200πr^3 kg

Similarly, we can calculate the final mass as:

Final mass = 900 kg/m^3 x (-2/3 πr^3) = -600πr^3 kg

Therefore, the change in mass is:

Δm = Final mass - Initial mass = -600πr^3 - 1200πr^3 = -1800πr^3 kg

Finally, we can calculate the mass per unit volume density of the compressed ball as:

Mass per unit volume density = Δm/ΔV = -1800πr