Category: Part 1: Mechanics

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: Write the complete equation of motion of a body in Earth’s rotating frame, including centrifugal and Coriolis pseudo-forces. Identify each term physically. Earth’s angular velocity vector: $\boldsymbol{\Omega}$ Position: $\mathbf{r}$, velocity in rotating Given Information Mass(es), forces, angles, and coefficients…

Problem Statement Find the effective acceleration of free fall at Earth’s surface as a function of latitude $\lambda$, accounting for the centrifugal pseudo-force due to Earth’s rotation. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on…

Problem Statement Solve the rotational mechanics problem: Solve the rotational mechanics problem: Find the deflection of a plumb line from the true vertical due to Earth’s rotation at latitude $\lambda$. What is the maximum deflection and at what latitude does it occur? Earth’s angular velocity: $\Omega$, latitude: $\lambda$, Earth’s radius: $R_E$ In Earth Given Information…

Problem Statement A body is projected horizontally northward with speed $v_0$ at latitude $\lambda$. Find the magnitude and direction of deflection due to the Coriolis force after traveling distance $l$. (Take Earth’s rotation into account.) Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section)…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A body falls freely from height $h$ above Earth’s surface at latitude $\lambda$. Find the horizontal deflection $\Delta x$ due to the Coriolis force (eastward or westward deflection) assuming $h$ is small compared to Earth’s radius. Height: $h$, latitude:…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A ball is thrown horizontally inside a train moving with acceleration $w_0$ relative to the ground. In the train’s frame, the ball appears to have an extra horizontal deceleration. Find the equation of motion in the train’s frame and…

Problem Statement Solve the kinematics problem: Solve the rotational mechanics problem: A wheel of radius $R$ rolls without slipping along a flat surface with the center moving at speed $v_0$. Find the speed and acceleration of a point on the rim as a function of the angle $\phi$ from the contact point. Wheel radius: $R$,…

Problem Statement A particle moves in a circle of radius $R$ with initial speed $v_0$. Tangential deceleration is constant: $w_\tau = a$. Find the total arc length traveled before the particle stops and the number of revolutions. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see…

Problem Statement Solve the kinematics problem: Solve the kinematics problem: A particle moves along a circle of radius $R$ with speed increasing as $v = v_0 + at$ (tangential acceleration $a = \text{const}$). Find the angle $\phi$ between the total acceleration vector and the radius at time $t$. Radius: $R$, speed: $v = v_0+at$, tangential…

Problem Statement Analyze the circuit: Analyze the circuit: A projectile of mass $m$ is launched at angle $\theta$ with speed $v_0$ against quadratic air drag $F = \beta v^2$. Find the approximate range for small drag ($\beta v_0^2 \ll mg$). Initial speed: $v_0$, angle: $\theta$, drag: $\beta v^2$, small drag approximation For small drag Given…