A sphere of copper is subjected to 200 MPa of pressure. Copper has a bulk modulus of 130 GPa. By what fraction does the volume of the sphere change?
dV=V*dp/Modulus
dV/V= 200e6/130e9 *200e3=200/130 *1e-3
To find the change in volume of a sphere under pressure, we can use the concept of bulk modulus.
The bulk modulus (K) relates the pressure applied (ΔP) to the resulting fractional change in volume (ΔV/V) of a material, and it is given by the equation:
K = -V(ΔP/ΔV)
where:
K = bulk modulus
ΔP = change in pressure
ΔV = change in volume
V = original volume
We can rearrange this equation to solve for ΔV/V:
ΔV/V = - (ΔP/K)
Now, let's substitute the given values:
ΔP = 200 MPa = 200 × 10^6 N/m²
K = 130 GPa = 130 × 10^9 N/m²
ΔV/V = - (200 × 10^6 N/m²) / (130 × 10^9 N/m²)
Simplifying the equation:
ΔV/V = - 1.54 × 10^-3
The negative sign indicates a decrease in volume.
Therefore, the volume of the copper sphere decreases by approximately 1.54 × 10^-3, or 0.00154, or 0.154%.
To find the change in volume of the copper sphere, we can use the formula:
Change in volume = -V * (Change in pressure / Bulk modulus)
In this case, we are given:
Pressure (Change in pressure) = 200 MPa = 200 * 10^6 Pa
Bulk modulus (K) = 130 GPa = 130 * 10^9 Pa
We need to convert the pressure and bulk modulus into SI units (Pascals) for accurate calculations.
1 GPa = 1 * 10^9 Pa
So, 130 GPa = 130 * 10^9 Pa
Now, we can substitute the values into the formula:
Change in volume = -V * (200 * 10^6 Pa / 130 * 10^9 Pa)
We know that the change in volume is given by the negative of the initial volume times the change in pressure divided by the bulk modulus.
Next, we need to find the fraction change in volume, which can be calculated using the formula:
Fraction change in volume = Change in volume / Initial volume
To find the fraction change in volume, we need to know the initial volume of the copper sphere. If the initial volume is provided, we can substitute the values into the fraction change in volume formula to get the result.