Category: Part 1: Mechanics

Problem Statement Solve the Newton’s Laws / mechanics problem: A block of mass $m$ is pressed against a vertical wall with horizontal force $F$. The friction coefficient is $\mu$. Find the minimum $F$ to keep the block from sliding down. Forces: Normal $N = F$ (horizontal), friction $f = \mu N = \mu F$ (upward),…

Problem Statement Solve the Newton’s Laws / mechanics problem: In an Atwood machine, masses $m_1$ and $m_2 > m_1$ hang over a massless frictionless pulley. Find: (a) acceleration; (b) tension in the string. Free-body diagram: $m_2$ accelerates down, $m_1$ accelerates up, both with same $|a|$. Newton’s 2nd law for each: $$m_2 g – T =…

Problem Statement Solve the Newton’s Laws / mechanics problem: A particle of mass $m$ moves under force $\vec F = F_0\cos(\omega t)\hat x$. At $t=0$, $v=0$, $x=0$. Find $v(t)$ and $x(t)$. Newton’s 2nd law: $m\dot v = F_0\cos\omega t$ Velocity: $$v = \frac{F_0}{m\omega}\sin\omega t$$ Position: $$x = -\frac{F_0}{m\omega^2}\cos\omega t + \frac{F_0}{m\omega^2} = \f Given Information…

Problem Statement Solve the Newton’s Laws / mechanics problem: A car of mass $m = 1500\,\text{kg}$ decelerates from $v_0 = 100\,\text{km/h}$ to rest under constant friction force $F$. Stopping distance $s = 50\,\text{m}$. Find $F$ and the braking time $t$. Work-energy theorem: $$F\cdot s = \frac12 mv_0^2$$ $$F = \frac{mv_0^2}{2s} = \frac{1500\times(100/3.6)^2}{ Given Information $m…

Problem Statement Solve the kinematics problem: Point $A$ moves on a circle of radius $R_1$ and point $B$ on a circle of radius $R_2$ ($R_2>R_1$), both centered at $O$, at constant angular velocities $\omega_1$ and $\omega_2$. Find the angular velocity of $B$ as seen from $A$. Position vectors: $$\vec r_A = R_1(\cos\omega_1 t, \sin\omega_1 t),\qu…

Problem Statement Solve the kinematics problem: A particle moves with velocity $v$. An observer at a fixed point measures the time for the particle to travel distance $l$. Find the time measured by the particle’s own clock (proper time). This is an introduction to relativistic time dilation. Lab frame time: $t = l/v$ Proper time…

Problem Statement Solve the kinematics problem: A straight spoke of a wheel rotates at constant $\omega$. A bead slides outward along it at constant speed $u$ relative to the spoke. Find the velocity and acceleration of the bead in the lab frame. In polar coordinates, $r = r_0 + ut$ (moving out at speed $u$…

Problem Statement Solve the kinematics problem: A uniform rod of length $l$ slides with its ends on two perpendicular walls. One end moves downward at velocity $v_A$. Find the velocity of the other end and the velocity of the midpoint. Let end A be on the vertical wall (at height $y$) and end B on…

Problem Statement Solve the Newton’s Laws / mechanics problem: A rope passes over a fixed pulley of radius $R$ at the top of a vertical wall. One end hangs vertically, the other goes horizontally and is pulled at speed $v_0$. Find the speed at which the hanging end descends. The rope is inextensible. If one…

Problem Statement Solve the kinematics problem: In a frame rotating at constant angular velocity $\omega$ about the $z$-axis, a particle moves with velocity $\vec v’$. Find its velocity and acceleration in the lab frame. Velocity transformation: $$\vec v = \vec v’ + \vec\omega\times\vec r$$ Acceleration transformation (from the rotating frame equ Given Information See problem…