A spring with k=1400N/m is firmly anchored to the bottom of a swimming pool. A diver in full SCUBA gear with total mass of 100kg and density of 0.65g/cm3 holds on to the top of the spring. The system reaches a new static equilibrium. After the print is elongated by a length del L. How do I find del L please?
Ah, no way a diver would be that poorly weighted but anyway
.65 g/cm^3 = 650 kg/m^3
volume of diver = 100 kg/650 kg/m^3
= .154 m^3
buoyancy force on diver = .154m^3 * 1000 kg/m^3 * 9.81 = 1509 Newtons up
diver weight down = 100*9.81 = 981 N
so
force up on spring = 1509-981 = 528 Newtons
528 = 1400 * delta L
so delta L = .377 meters
To find the elongation length (ΔL), we need to consider the forces acting on the system and use Hooke's law.
1. Determine the weight of the diver:
The weight (W) of an object is given by the formula W = mass × gravity, where the mass is measured in kilograms (kg) and gravity is approximately 9.8 m/s².
In this case, the mass of the diver is given as 100 kg. Therefore, W = 100 kg × 9.8 m/s² = 980 N.
2. Determine the buoyant force on the diver:
The buoyant force (F_b) acting on an object immersed in a fluid is given by the formula F_b = density × gravity × volume, where the density is measured in kilograms per cubic meter (kg/m³), gravity is approximately 9.8 m/s², and volume is measured in cubic meters (m³).
In this case, the density of the diver is given as 0.65 g/cm³. Converting this to kg/m³ gives 650 kg/m³. Let's assume the volume of the diver is V m³.
3. Equate the forces acting on the system:
In static equilibrium, the weight of the diver is balanced by the buoyant force acting in the opposite direction, and the spring force (F_s) balances the difference.
Thus, we can set up the equation: W - F_b = F_s.
4. Calculate the volume of the diver:
Since we know the density (ρ) and mass (m) of the diver, we can use the formula: density = mass / volume (ρ = m / V) to find the volume (V) of the diver.
Rearranging the formula, we have: V = m / ρ.
Plugging in the values, V = 100 kg / 650 kg/m³ = 0.154 m³.
5. Determine the spring force:
According to Hooke's Law, the spring force (F_s) is given by the formula F_s = -k × ΔL, where k is the spring constant.
In this case, the spring constant (k) is given as 1400 N/m, and we are trying to find the elongation of the spring (ΔL).
6. Substitute the values into the equation:
Plugging in the known values, our equation becomes: 980 N - (650 kg/m³ × 9.8 m/s² × 0.154 m³) = -1400 N/m × ΔL.
7. Solve for ΔL:
Rearranging the equation to solve for ΔL, we have: ΔL = (980 - 650 × 9.8 × 0.154) / 1400.
Evaluating the right-hand side of the equation, we find: ΔL ≈ 0.305 m.
Therefore, the elongation of the spring (ΔL) is approximately 0.305 meters.