PHSX 220 Physics I (w/ Calculus) · Unit 2: Applying Newton, statics, fluids, energy · on Exam 2
Ch 6.1 Applying Newton’s laws
Study notes for Applying Newton’s laws in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Multi-object systems (Atwood machines, blocks connected by ropes, stacked blocks): draw one free-body diagram per object, use the same $a$ magnitude for objects tied together, and add the equations to eliminate the tension.
- Frictionless incline: $a = g\sin\theta$ down the slope, independent of mass. Add friction later as an extra term.
- Elevator problems: the scale reads the normal force, $N = m(g + a)$ with $a$ positive upward. In free fall $N = 0$ ("weightless").
- Blocks pushed together: treat the whole system to get $a = F/(m_1+m_2)$, then isolate one block to get the contact force $m_2 a$.
- Sanity checks: heavier side of an Atwood machine accelerates down; $a$ must be less than $g$; tension must lie between the two weights.
Key formulas
Atwood machine
$$a = \frac{(m_2 - m_1)g}{m_1 + m_2},\qquad T = \frac{2m_1m_2}{m_1+m_2}\,g$$
Block on table pulled by hanging mass
$$a = \frac{m_2 g}{m_1 + m_2},\qquad T = m_1 a\ \text{(frictionless)}$$
Frictionless incline
$$a = g\sin\theta,\qquad N = mg\cos\theta$$
Worked example
Masses 3.0 kg and 5.0 kg hang from an ideal pulley. Find the acceleration and the tension.
Show the solution
$a = \frac{(5-3)(9.8)}{8} = 2.45\ \text{m/s}^2$ (5 kg side down). From the 3 kg block: $T - 3g = 3a \Rightarrow T = 3(9.8 + 2.45) = 36.8$ N. Check: between $29.4$ N and $49$ N.
Common mistakes
- Using a different acceleration for two blocks joined by a taut rope.
- Assuming the tension equals the hanging weight when the system is accelerating.
- Forgetting that a pulley with two rope segments pulling up on it exerts 2T.
Exam tipFor connected objects, write one Newton equation per object in the direction of motion, then add them: the internal tensions cancel and a pops out.
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