Category: Part 1: Mechanics

Problem Statement Solve the Newton’s Laws / mechanics problem: A block barely moves down a rough incline of angle $\alpha = 30°$. Find the kinetic friction coefficient and the deceleration if it is given a push down. Barely moves’ means constant velocity (or at the transition): $$\mu_s = \tan\alpha = \tan30° = \frac{1}{\sqrt3} \approx 0.577$$…

Problem Statement Solve the momentum/collision problem: A ball of mass $m = 0.14\,\text{kg}$ traveling at $v_1 = 40\,\text{m/s}$ is struck by a bat and returns at $v_2 = 50\,\text{m/s}$ in the opposite direction. The contact lasts $\Delta t = 1.5\,\text{ms}$. Find the average force. Impulse-momentum theorem: $$F_{avg}\Delta t = \Delta p = m(v_2-(-v_1)) = Given…

Problem Statement Solve the Newton’s Laws / mechanics problem: A ball of mass $m$ on a string of length $l$ moves in a horizontal circle (conical pendulum). String makes angle $\theta$ with vertical. Find: (a) angular velocity; (b) tension; (c) period. Equations ($r = l\sin\theta$ = radius of circle): $$T\cos\theta = mg \quad\text{(vertical)}$$ $$T\sin\theta =…

Problem Statement Solve the Newton’s Laws / mechanics problem: A car moves around a banked curve of radius $R = 500\,\text{m}$ and bank angle $\theta = 10°$. Find the speed for which no friction is needed. On the banked curve, the normal force $N$ provides both the vertical support and the centripetal force. Vertical: $N\cos\theta…

Problem Statement A ball is threaded on a circular loop of radius $R$. What minimum speed must it have at the top to maintain contact? Given Information See problem statement for all given quantities. Physical Concepts & Formulas Circular motion requires a centripetal force directed toward the centre, providing the centripetal acceleration $a_c = v^2/r…

Problem Statement Solve the Newton’s Laws / mechanics problem: Two blocks of mass $m_1 = 3\,\text{kg}$ and $m_2 = 5\,\text{kg}$ are in contact on a frictionless surface. Force $F = 16\,\text{N}$ pushes $m_1$ into $m_2$. Find (a) acceleration; (b) contact force. System acceleration: $$a = \frac{F}{m_1+m_2} = \frac{16}{3+5} = \frac{16}{8} = 2.0\,\text{m/s}^2$$ Co Given Information…

Problem Statement Solve the Newton’s Laws / mechanics problem: A ball of mass $m$ on a string of length $l$ moves in a vertical circle at constant speed $v$. Find the tension at the top and bottom. At the bottom (centripetal acceleration directed upward = toward center): $$T_{bot} – mg = \frac{mv^2}{l} \implies T_{bot} =…

Problem Statement Solve the Newton’s Laws / mechanics problem: A block of mass $m$ rests on a smooth incline of angle $\alpha$. The incline is accelerated horizontally at $a_0$. Find the normal force and whether the block slides. In the non-inertial frame of the incline, a pseudo-force $ma_0$ acts horizontally on the block (opposite to…

Problem Statement Solve the Newton’s Laws / mechanics problem: A person of mass $m = 70\,\text{kg}$ climbs a vertical rope. Find the minimum force they must exert on the rope to: (a) climb at constant speed; (b) accelerate upward at $a = 1.0\,\text{m/s}^2$. By Newton’s 3rd law, if the person pulls down on rope with…

Problem Statement Solve the Newton’s Laws / mechanics problem: A block starts sliding down an incline at angle $\alpha$. The kinetic friction coefficient is $\mu_k$. Find the acceleration. Forces along the incline (positive down-slope): $$mg\sin\alpha – \mu_k mg\cos\alpha = ma$$ $$a = g(\sin\alpha – \mu_k\cos\alpha)$$ Condition for sliding: $\tan\alpha > \mu_s$ Given Information See problem…