Problem Statement
Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A banked circular track has angle 20° and radius 100 m, $\mu_s = 0.4$. Find the maximum safe speed. ($g = 10$ m/s², $\tan20°=0.364$) $v_{max} = \sqrt{rg(\tan\theta+\mu_s)/(1-\mu_s\tan\theta)}$ Step 1: $v_{max}^2 = 100\times10\times\dfrac{0.364+0.4}{1-0.4\
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
Full substitution shown in the steps above.
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.
Leave a Reply