Category: Part 1: Mechanics

Problem Statement Solve the rotational mechanics problem: At a given instant, different points of a rigid body have velocities $\vec v_1$ and $\vec v_2$ at positions $\vec r_1$ and $\vec r_2$. Find the angular velocity $\vec\omega$ of the body. For a rigid body: $\vec v_i = \vec\omega \times \vec r_i + \vec v_0$ The difference…

Problem Statement Solve the kinematics problem: A wheel of radius $R = 0.5\,\text{m}$ rolls without slipping at $v_{cm} = 10\,\text{m/s}$. Find: (a) angular velocity; (b) speed of the topmost point; (c) speed of the point of contact with ground. Rolling constraint: $v_{cm} = \omega R$ (a) Angular velocity: $$\omega = \frac{v_{cm}}{R} = \frac{10}{ Given Information…

Problem Statement Solve the kinematics problem: A solid body rotates with angular velocity $\vec\omega$. A point on it is at position $\vec r$ from the rotation axis. Express its velocity. The velocity of a point at $\vec r$ from a fixed point on the axis: $$\vec v = \vec\omega \times \vec r$$ where $\vec\omega$ points…

Problem Statement Solve the kinematics problem: Find the angular velocities of (a) the hour hand, (b) the minute hand, and (c) the second hand of a clock. Find the angular velocity of the second hand relative to the minute hand. (a) Hour hand: $T_H = 12\,\text{h} = 43200\,\text{s}$ $$\omega_H = \frac{2\pi}{43200} \approx 1.45\times10^{-4}\,\text{ Given Information…

Problem Statement Solve the Newton’s Laws / mechanics problem: A point accelerates uniformly from rest along a circle of radius $R$. At the moment when its linear velocity equals $v$, find the angle $\varphi$ the total acceleration vector makes with the radius (centripetal direction). At speed $v$: $w_n = v^2/R$, $w_\tau = \text{const}$ Using $v^2…

Problem Statement Solve the kinematics problem: A disk rotates at $n = 1800\,\text{rpm}$. Express the angular velocity $\omega$ in rad/s and find the linear speed of a point at $r = 15\,\text{cm}$ from the axis. Angular velocity: $$\omega = 2\pi n = 2\pi\times\frac{1800}{60} = 60\pi \approx 188\,\text{rad/s}$$ Linear speed: $$v = \omega r = 188\t…

Problem Statement Solve the Newton’s Laws / mechanics problem: A car starts from rest on a circular track of radius $R = 500\,\text{m}$ with tangential acceleration $w_\tau = 0.80\,\text{m/s}^2$. At what time will the normal acceleration equal the tangential? How far has the car gone? Speed at time $t$: $v = w_\tau t$ Normal acceleration:…

Problem Statement Solve the Newton’s Laws / mechanics problem: Find the centripetal acceleration of a point on the Earth’s equator due to Earth’s rotation. Compare with $g$. Data: $R_E = 6.37\times10^6\,\text{m}$, $T = 86400\,\text{s}$ Angular velocity: $\omega = 2\pi/T = 2\pi/86400 = 7.27\times10^{-5}\,\text{rad/s}$ Centripetal acceleration: $$a_c = \omega^2 R Given Information $T = 86400\,\text{s}$ Physical…

Problem Statement Solve the kinematics problem: A point moves on a circle of radius $R = 1.00\,\text{m}$ with angular velocity $\omega = 2.00 + 3.00t\,\text{rad/s}$. At $t = 0$ find: (a) $\beta$; (b) $w_n$ and $w_\tau$; (c) total acceleration; (d) angle between $\vec w$ and $\vec v$. (a) Angular acceleration: $\beta = d\omega/dt = 3.00\,\text{rad…

Problem Statement Solve the kinematics problem: A particle is thrown horizontally from height $h$ above the ground. Find the horizontal distance covered when it hits the ground. Horizontal throw: $v_{0y}=0$, $v_{0x}=v_0$ Time to fall height $h$: $$h = \frac12 gt^2 \implies t = \sqrt{\frac{2h}{g}}$$ Horizontal distance: $$x = v_0 t = v_0\sqrt{\fra Given Information See…