Problem Statement
Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: Two masses of 3 kg and 5 kg are connected by a light string over a frictionless pulley. Find the acceleration and tension. ($g = 10$ m/s²) Atwood machine: $a = (m_2-m_1)g/(m_1+m_2)$; $T = 2m_1m_2g/(m_1+m_2)$ Step 1: $a = (5-3)(10)/(5+3) = 20/8 = 2.5$ m/s²
Given Information
- $T = 2m$
- $a = (5-3)(10)/(5+3) = 20/$
Physical Concepts & Formulas
Newton’s second law $\mathbf{F}_\text{net} = m\mathbf{a}$ is the fundamental relation between net force and acceleration. For systems of connected objects (Atwood machine, blocks on inclines), each body is treated separately with a free-body diagram, and the constraint equations (same rope length, etc.) link the accelerations.
- $\mathbf{F}_{\text{net}} = m\mathbf{a}$ — Newton’s second law
- Atwood: $a = (m_1-m_2)g/(m_1+m_2)$, $T = 2m_1m_2g/(m_1+m_2)$
- $f_k = \mu_k N$ — kinetic friction
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\sum F_x = ma_x\quad,\quad \sum F_y = ma_y = 0\quad\text{(if no vertical acceleration)}$$
$$a = \frac{(m_2-m_1)g}{m_1+m_2} = \frac{(5-3)\times9.8}{8} = \frac{19.6}{8} = 2.45\,\text{m/s}^2$$
$$T = \frac{2m_1 m_2 g}{m_1+m_2} = \frac{2\times3\times5\times9.8}{8} = \frac{294}{8} = 36.75\,\text{N}$$
Answer
$$\boxed{a = \dfrac{(m_2-m_1)g}{m_1+m_2}}$$
Physical Interpretation
The Atwood machine amplifies the effect of gravity: a small mass difference $\Delta m$ produces an acceleration much smaller than $g$, making it useful for measuring $g$ precisely in lab settings.
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