Category: Part 1: Mechanics

Problem Statement Solve the optics problem: This problem involves optics/kinematics — an object moves toward a spherical mirror. Find the velocity of the image using the mirror equation. Mirror equation: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ (standard sign convention) Differentiating with respect to time (with $u$ changing): $$-\frac{1}{v Given Information See problem statement for all given…

Problem Statement Solve the kinematics problem: A particle moves along a spiral $r = b\varphi$ with constant angular velocity $\omega$. Find the tangential and normal accelerations as functions of $\varphi$. Position: $r = b\varphi$, $\dot\varphi = \omega = $ const $\dot r = b\dot\varphi = b\omega$, $\ddot r = 0$ Speed: $v = \sqrt{\dot r^2…

Problem Statement A ball is dropped from height $h = 100\,\text{m}$ at latitude $\lambda = 45°N$. Find the eastward Coriolis deflection. Given Information $h = 100\,\text{m}$ Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically…

Problem Statement Solve the kinematics problem: A rigid body rotates about a fixed axis with angular acceleration $\beta(t)$. Express the linear acceleration of a point at radius $r$ from the axis. A point at perpendicular distance $r$ from the axis has: Tangential acceleration (along the circle): $$w_\tau = \beta r$$ Centripetal (normal) acceler Given Information…

Problem Statement A train travels east at $v = 100\,\text{km/h}$ at latitude $\lambda = 60°N$. Find the force on the rails from Earth’s rotation (Coriolis effect). Train mass $m = 1000\,\text{t}$. Given Information $v = 100\,\text{km/h}$ $m = 1000\,\text{t}$ Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution…

Problem Statement Solve the gravitation problem: Find the orbital speed and period of a satellite at height $h = 200\,\text{km}$ above the Earth’s surface. ($R_E = 6370\,\text{km}$, $g = 9.8\,\text{m/s}^2$) Orbital radius: $r = R_E+h = 6570\,\text{km} = 6.57\times10^6\,\text{m}$ Gravitational acceleration at altitude: $$g(r) = g\frac{R_E^2}{r^2} = Given Information $h = 200\,\text{km}$ $R_E = 6370\,\text{km}$…

Problem Statement Find the radius $r_0$ of a geostationary orbit (period 24 hours). Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to…

Problem Statement Solve the Newton’s Laws / mechanics problem: The Moon orbits Earth at $R = 3.84\times10^5\,\text{km}$ with period $T = 27.3\,\text{days}$. Find the Moon’s centripetal acceleration and compare with $g$ at Earth’s surface. Angular velocity: $$\omega = \frac{2\pi}{T} = \frac{2\pi}{27.3\times86400} = 2.66\times10^{-6}\,\text{rad/s}$$ Centripetal a Given Information $T = 27.3\,\text{days}$ Physical Concepts & Formulas Circular…

Problem Statement Solve the rotational mechanics problem: A disk of radius $r$ rolls without slipping on the outside of a cylinder of radius $R$. Find the angular velocity of the disk if the axis of the cylinder rotates at $\omega_0$. Speed of disk center: $v_c = \omega_0 R$ (center of disk orbits cylinder axis at…

Problem Statement Solve the kinematics problem: A small disk of radius $r$ rolls without slipping inside a large cylinder of radius $R$. Find the angular velocity of the disk and the velocity of its center if the center moves at $v$. Rolling constraint (no slip at contact): $$v = \omega_{disk}\cdot(R-r) \quad\text{(center orbits at radius }R-r\te…