Problem Statement
Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A banked curve of radius 50 m is banked at 15°. Find the design speed at which no friction is needed. ($g = 10$ m/s²) $v = \sqrt{rg\tan\theta}$ Step 1: $v = \sqrt{50 \times 10 \times \tan15°} = \sqrt{500 \times 0.268} = \sqrt{134} \approx 11.6$ m/s. $$\bo
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Friction is a contact force opposing relative motion (kinetic friction) or impending motion (static friction). On an inclined plane, the weight component along the slope is $mg\sin\theta$ and the normal force is $N = mg\cos\theta$, giving maximum static friction $f_{s,\max} = \mu_s mg\cos\theta$. The condition for sliding is $\tan\theta > \mu_s$.
- $f = \mu N$ — kinetic friction force
- $N = mg\cos\theta$ — normal force on incline
- $mg\sin\theta – \mu mg\cos\theta = ma$ — Newton’s 2nd law along incline
- $\tan\theta_c = \mu_s$ — critical angle for sliding
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
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Given Information
- Mass(es), forces, angles, and coefficients of friction as given
- $g = 9.8\,\text{m/s}^2$ (acceleration due to gravity)
Physical Concepts & Formulas
Newton’s three laws form the complete foundation of classical mechanics. The second law $\vec{F}_{\text{net}} = m\vec{a}$ is the workhorse: draw a free-body diagram for each object, resolve forces into components along chosen axes, and write $\sum F_x = ma_x$, $\sum F_y = ma_y$. For systems connected by strings over pulleys, the tension is common to both sides of a massless string. Friction force $f = \mu N$ opposes relative sliding and is proportional to the normal force, not the contact area.
- $\vec{F}_{\text{net}} = m\vec{a}$ — Newton’s second law
- $f_k = \mu_k N$ — kinetic friction
- $f_{s,\max} = \mu_s N$ — maximum static friction
- $N = mg\cos\theta$ — normal force on incline of angle $\theta$
- $a = g\sin\theta – \mu_k g\cos\theta$ — acceleration on rough incline
Step-by-Step Solution
Step 1 — Free-body diagram: Draw all forces on each object separately.
Step 2 — Choose coordinate axes: Align one axis along the direction of motion.
Step 3 — Apply Newton’s 2nd Law component by component:
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Step 4 — Constraint equations: For Atwood machine: $a_1 = -a_2 = a$, same tension $T$.
Step 5 — Solve the system of equations for $a$ and $T$.
Worked Calculation
Substituting all values with units:
Atwood machine: $m_1 = 3\,\text{kg}$, $m_2 = 5\,\text{kg}$:
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Answer
$$\boxed{a = \dfrac{(m_2-m_1)g}{m_1+m_2}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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