AP Physics 2: Algebra-Based FRQ Practice
1. Fluid Mechanics
Question:
A cylindrical tank of water has a height of 10 meters and a radius of 2 meters. The tank is open to the atmosphere, and the water is flowing out of a small hole at the bottom of the tank.
(a) Calculate the speed of the water exiting the hole using Torricelli’s Law. Assume the water level is 10 meters above the hole.
(b) If the hole is circular with a radius of 0.05 meters, calculate the flow rate of the water (in cubic meters per second).
(c) How long would it take for the water level in the tank to decrease by 1 meter?
Response Guidelines:
- Part (a):
Use Torricelli’s Law to find the velocity of the water at the hole:
v = √(2gh)
Where:
g = 9.8 m/s² (acceleration due to gravity)
h = 10 m (height of water above the hole)
v = √(2 × 9.8 × 10) ≈ 14.0 m/s - Part (b):
The flow rate Q is given by the formula:
Q = A v
Where:
A = πr² is the area of the hole
r = 0.05 m (radius of the hole)
A = π(0.05)² ≈ 7.85 × 10⁻³ m²
Now calculate the flow rate:
Q = 7.85 × 10⁻³ × 14.0 ≈ 0.11 m³/s - Part (c):
To find the time it takes for the water level to decrease by 1 meter, use the formula for volume change:
Volume of water initially:
V_initial = πr²h = π(2)²(10) = 40π m³
Volume of water after 1 meter drop:
V_final = πr²(h - 1) = π(2)²(9) = 36π m³
The volume of water removed is:
ΔV = 40π - 36π = 4π m³
Now calculate the time it takes for this volume to flow out:
t = ΔV / Q = 4π / 0.11 ≈ 114.6 seconds
2. Thermodynamics
Question:
A gas is contained in a piston with an initial volume of 2.0 m³ at a pressure of 1.0 × 10⁵ Pa and a temperature of 300 K. The gas undergoes an isothermal expansion to a final volume of 4.0 m³.
(a) Calculate the work done by the gas during this isothermal expansion.
(b) If the gas is ideal, calculate the final pressure of the gas after expansion.
(c) Calculate the heat absorbed by the gas during this process.
Response Guidelines:
- Part (a):
For an isothermal expansion, the work done by the gas is given by:
W = nRT ln(V_f / V_i)
Where:
n is the number of moles
R = 8.31 J/mol·K (universal gas constant)
T = 300 K
V_i = 2.0 m³
V_f = 4.0 m³
First, calculate the number of moles n using the ideal gas law:
PV = nRT
n = (PV) / (RT) = (1.0 × 10⁵ × 2.0) / (8.31 × 300) ≈ 8.05 mol
Now calculate the work:
W = 8.05 × 8.31 × 300 × ln(4.0 / 2.0) ≈ 5.64 × 10³ J - Part (b):
For an ideal gas undergoing an isothermal process, the pressure and volume are related by:
P_i V_i = P_f V_f
Solving for P_f:
P_f = (P_i V_i) / V_f = (1.0 × 10⁵ × 2.0) / 4.0 = 5.0 × 10⁴ Pa - Part (c):
The heat absorbed by the gas during an isothermal process is equal to the work done by the gas:
Q = W = 5.64 × 10³ J
3. Electromagnetism
Question:
Two point charges, q₁ = 3.0 μC and q₂ = -3.0 μC, are placed 0.5 meters apart in a vacuum.
(a) Calculate the electric force between the two charges.
(b) If the distance between the charges is doubled, calculate the new force between them.
(c) What is the potential energy of the system of charges?
Response Guidelines:
- Part (a):
The electric force between two charges is given by Coulomb's Law:
F = kₑ |q₁ q₂| / r²
Where:
kₑ = 8.99 × 10⁹ N·m²/C² (Coulomb's constant)
r = 0.5 m
F = (8.99 × 10⁹ × 3.0 × 10⁻⁶ × 3.0 × 10⁻⁶) / (0.5)² ≈ 3.24 N - Part (b):
If the distance is doubled, r' = 1.0 m, the new force is:
F' = kₑ |q₁ q₂| / r'²
F' = (8.99 × 10⁹ × 3.0 × 10⁻⁶ × 3.0 × 10⁻⁶) / (1.0)² ≈ 0.81 N - Part (c):
The potential energy of the system of charges is given by:
U = kₑ q₁ q₂ / r
U = (8.99 × 10⁹ × 3.0 × 10⁻⁶ × -3.0 × 10⁻⁶) / 0.5 ≈ -1.62 J
Grading Rubric for AP Physics 2: Algebra-Based FRQs
1. Fluid Mechanics
- Part (a):
0-1 points: Incorrect or incomplete calculation of water velocity.
2 points: Correct calculation of water velocity using Torricelli's Law with proper units. - Part (b):
0-1 points: Incorrect or incomplete calculation of flow rate.
2 points: Correct calculation of flow rate with proper formula and units. - Part (c):
0-1 points: Incorrect or incomplete calculation of time for water level decrease.
2 points: Correct calculation of time with proper formula and units.
2. Thermodynamics
- Part (a):
0-1 points: Incorrect or incomplete calculation of work done.
2 points: Correct calculation of work with proper formula and units. - Part (b):
0-1 points: Incorrect or incomplete calculation of final pressure.
2 points: Correct calculation of final pressure with proper formula and units. - Part (c):
0-1 points: Incorrect or incomplete calculation of heat absorbed.
2 points: Correct calculation of heat absorbed with proper units.
3. Electromagnetism
- Part (a):
0-1 points: Incorrect or incomplete calculation of electric force.
2 points: Correct calculation of electric force with proper formula and units. - Part (b):
0-1 points: Incorrect or incomplete calculation of the new force.
2 points: Correct calculation of the new force with proper formula and units. - Part (c):
0-1 points: Incorrect or incomplete calculation of potential energy.
2 points: Correct calculation of potential energy with proper formula and units.
Total Possible Points: 18 points