F=ma bridge · 35 min
A bridge set for pulleys, system boundaries, friction direction, and accelerating frames.
Write the system boundary for every multi-body problem.
Write the constraint relation before substituting numbers.
For static friction, decide the slipping tendency first.
Question 1
Masses 3 kg and 5 kg hang over an ideal pulley. Find the acceleration magnitude.
Add the two Newton's second law equations.
5g - T = 5a and T - 3g = 3a. Add: 2g = 8a, so a = 2.45 m/s².
Trap tested: Missing the equal-magnitude string constraint.
Question 2
Three carts of masses 1, 2, and 3 kg are pulled by 18 N on a frictionless track. Find the tension pulling the 3 kg cart.
Find the common acceleration, then isolate the 3 kg cart.
a = 18/6 = 3 m/s². The tension on the 3 kg cart is 3(3) = 9 N.
Trap tested: Solving internal forces before the shared acceleration.
Question 3
A car moves slower than the no-friction design speed on a banked curve. Which way does static friction point along the bank?
Ask how the car would slip without friction.
At low speed the car tends to slide down the bank, so static friction points up the bank.
Trap tested: Assuming friction always points opposite the car's velocity.
Question 4
A truck accelerates forward at 3.0 m/s². A pendulum inside reaches a steady angle. Find tan(θ), where θ is from vertical.
In the truck frame, balance gravity with a backward pseudo-force.
tan(θ) = a/g = 3.0/9.8 = 0.306. The bob leans backward.
Trap tested: Treating the tilted string as mysterious instead of force balance in an accelerating frame.
Question 5
A package is dropped onto a conveyor belt moving right. Initially the package has zero horizontal speed. Which way is friction on the package?
Look at relative slipping at the contact.
Friction on the package points right, because the belt surface slips rightward relative to the package.
Trap tested: Friction opposes relative slipping, not absolute motion.
Question 6
Masses m_2 > m_1 are connected over a pulley of radius R and moment of inertia I. The string does not slip. Derive the acceleration magnitude and state when the two tensions become equal.
Use two mass equations, one pulley torque equation, and α = a/R.
Use m_2g - T_2 = m_2a, T_1 - m_1g = m_1a, and (T_2 - T_1)R = I(a/R). Combining gives (m_2 - m_1)g = (m_1 + m_2 + I/R²)a, so a = (m_2 - m_1)g/(m_1 + m_2 + I/R²). The tensions are equal only in the ideal limit I = 0 or when α = 0.
Trap tested: Applying the ideal massless-pulley shortcut to a massive pulley.
Rubric
score What it means
Start mixing timed ladder rungs R8-R14 and R37-R38.
Redo the missed constraint or system-boundary lesson before retesting.
Pause timing and rebuild from Atwood signs, combined systems, and friction direction.
after Turn the mock into data
If you missed a question, do not just reread the solution. Open the linked lesson, redo one drill, then come back to the mock.