Category: Part 1: Mechanics

Problem Statement A gyrocompass consists of a gyroscope free to rotate in a horizontal plane on Earth’s surface at latitude $\lambda$. Explain why it points to geographic north and find the oscillation period about the north-pointing equilibrium. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see…

Problem Statement Solve the rotational mechanics problem: Solve the rotational mechanics problem: A torque-free symmetric body (moments $I_1 = I_2 \neq I_3$) spins with angular velocity components $\omega_3$ along the symmetry axis and $\omega_\perp$ perpendicular to it. Show that the symmetry axis precesses about the angular momentum vector, and find the Given Information Mass $m$,…

Problem Statement A symmetric top (moments of inertia $I_1 = I_2$, $I_3$ for spin axis) spins at rate $n$ (revolutions per second) with its axis at angle $\theta$ to the vertical. Find the two possible precession rates $\Omega$ under gravity (torque $= mgl\sin\theta$). Given Information All quantities, constants, and constraints stated in the problem above…

Problem Statement Solve the magnetic field/force problem: Solve the magnetic field/force problem: A gyroscope precesses uniformly with angular velocity $\Omega_{\rm prec}$ under a constant torque $\tau$. Show that $\Omega_{\rm prec} = \tau/(I\omega)$ and find the angle of the precession cone if the gyroscope also has nutation with small amplitude. Spin: $\ Given Information Current $I$…

Problem Statement Solve the gravitation problem: Solve the rotational mechanics problem: A gyroscope (spinning top) with moment of inertia $I$, spinning at angular velocity $\omega$ about its symmetry axis, is supported at one end. The center of mass is at distance $l$ from the support. Find the precession angular velocity $\Omega_{\rm prec}$. Mom Given Information…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A disk of mass $m$ and radius $R$ rotates on a rough horizontal surface with initial angular velocity $\omega_0$. The coefficient of kinetic friction is $\mu$. Find the time for the disk to stop. Mass: $m$, radius: $R$, initial…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A particle of mass $m$ lies at rest relative to a disk rotating uniformly with angular velocity $\omega$ at distance $r$ from the center. Find the normal force and friction force on the particle from the disk surface. The…

Problem Statement A smooth rod rotates in a horizontal plane with constant angular velocity $\omega$ about one end. A bead of mass $m$ can slide freely along the rod. At $t=0$, the bead is at distance $r_0$ from the axis with radial velocity $\dot{r}_0 = 0$. Find $r(t)$. Given Information All quantities, constants, and constraints…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: A particle moves in a circle of radius $r$ in a frame rotating with angular velocity $\Omega$. What is the Coriolis acceleration experienced by the particle if it moves with speed $v’$ relative to the rotating frame? Speed in…

Problem Statement A Foucault pendulum oscillates at latitude $\lambda$. Find the period of precession of the pendulum’s oscillation plane 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 fundamental physical principles. The…