Dynamics 2: Mathematical modeling & analysis of rigid bodies

Solve a 3d disk-rod-collar problem by four equations and a dot product constraint to compute VB and omega BC; VB points negative y, omega BC has x and z positive.

Fix a rod-attached rotating frame, define omega and omega dot, and derive rotating-frame derivatives and velocity terms, including I dot = omega J and J dot = - omega I.

Use the cross product to describe rotating frames, derive i dot and j dot from omega, extend to three-dimensional, and split time derivatives into rotating frame and frame-rotation terms.

Derive the two-dimensional rotation matrix to transform vectors between body and inertial frames using theta, cosine, and sine, and explain the inverse as the transpose of an orthonormal matrix.

Transform the collar’s velocity and acceleration from the inertial frame to the body frame via the inverse rotation matrix, then combine B’s motion with C’s to obtain v_c and a_c.

Compute the velocity and acceleration of the signal horn for a 3d satellite dish using angular velocity and acceleration about the z and x axes at theta 25 degrees.

Compute velocity of point A in a 3d satellite dish via cross product of total angular velocity with its position vector, as the mechanism rotates about the inertial z axis.

Derive the velocity of point A using a rotating body frame: (d r_A/dt)_body + Ω × r_A, then compute its acceleration as α × r_A + Ω × v_A.

Demonstrates how three-dimensional rotations depend on order, showing angular displacements are not vectors except for infinitesimal rotations; derives that angular velocity and acceleration are vectors.

Establish a rotating frame about the z axis, compute velocity of point A as omega cross r_A in the inertial frame, and include omega dot in the acceleration.

Map the angular velocity from the inertial frame to the purple frame, combining omega1 about z double prime with omega2 about y, especially at theta=0.

Compute color c’s velocity and acceleration in the inertial frame by combining relative motion in the purple body frame with the rotation of that frame, using omega and cross products.

Compute the mass moment of inertia about an axis by integrating r squared over the mass; I = ∫ r^2 dm = ∫ r^2 ρ dV.

Compute mass moment of inertia from radius of gyration K and mass M, then apply parallel axis theorem to each component and sum mass moments about O.

Compute the mass moment of inertia of a composite body about point O by applying the parallel axis theorem to a rod and a sphere, then sum the results.

Apply the parallel axis theorem to compute the mass moment of inertia about point A for a thin wheel with spokes, summing ring and spoke contributions to 23/3 kg m^2.

Analyze the bicycle braking problem: a 40 kg bike with a 60 kg rider decelerating at 3 m/s^2 to determine front and rear wheel normal reactions.

Analyze the 460 kg pipe on a truck bed; determine maximum acceleration causing loss of contact at A with pivot about B, and compute board B's force on the pipe.

Determine the maximum acceleration a_G at which the pipe pivots about point B and the corresponding force F_B via moment and force analyses.

Compute the hydraulic cylinder's compressive force to support an 8000 kg payload and 2000 kg boom in a pulley system, with 2 m/s^2 cable acceleration.

Solve the compressive force in the CD hydraulic system of a crane using two parallel cables and pulley kinematics, finding C equals D at about 88,870 newtons.

Analyze the curvilinear translation of a 20 kg platform under 50 N and compute AB and CD tensile forces and the bars' acceleration after chord failure, neglecting link masses.

Analyze a curvilinear translation problem to find forces in links A, B, C, D and the acceleration of point G using tangential and normal components.

Compute the pin reaction force for a pendulum made of a uniform rod and a sphere, given torque, angular velocity, and angle, using their mass moments of inertia.

Compute tangential gravity components for the rod and sphere, apply moments about point O to solve for angular acceleration, then determine tangential and pin forces and their magnitude.

Use rigid-body dynamics to find angular velocities of gears A and B at t = 3 s from rest, given torque on gear A, and masses and radii of gyration.

Compute angular velocities in two-gear problem: gear A about C and gear B about D at contact E, giving omega C ≈ 38.31 rad/s and omega D ≈ 57.47 rad/s.

Explore a two-block pulley system with kinetic friction mu_k on block A and a thin disk pulley with inertia; determine A's acceleration when B of mass M is released.

Model blocks a and b and a rotating pulley with kinetic friction and its mass moment of inertia; apply free body diagrams and no slipping to derive the common acceleration.

Explore a two-block and pulley problem to determine acceleration of block A, treating the pulley as a 3 kg disk of radius 0.15 m and neglecting cord mass and slipping.

Solve a two-block and pulley system by modeling each object with free-body diagrams, incorporating gravity, tensions, and the pulley’s moment of inertia without slipping to obtain acceleration and tensions.

Explore the spool on an inclined surface: a 75 kg body with radius of gyration 0.38 m experiences kinetic friction 0.15; find the initial chord tension and angular acceleration.

Solve a wheel friction problem by using a free body diagram and motion equations to compare static friction with its maximum, then apply kinetic friction when slipping occurs.

Relate the truck’s linear acceleration to the culvert’s angular acceleration under no slipping, using its mass, radius, and moment of inertia to model a rolling rigid body.

Solve a rigid-body pipe-on-truck problem using free body and kinetic diagrams, apply moments about point B to determine the angular acceleration alpha and the acceleration of point G.

Solve a disk-and-pulley problem by assuming no slipping and deriving the minimum static friction coefficient. Use free-body diagrams and equations of motion to relate the disk's rotation, translation, and acceleration.

Learn how a rigid body's angular momentum about its center of mass combines differential contributions, and how products of inertia reveal mass symmetry and define principal axes.

Compute the inertia tensor of a cube about blue reference axes using the parallel axis theorem, showing i_xx = i_yy = i_zz = 2/3 m a^2.

This lecture derives products of inertia for a cube, showing zero values on principal axes due to center of mass, and applies the parallel-axis theorem for products of inertia.

Compute the shaft moments on the wind turbine blades as a function of theta in the blade-attached frame, using inertial and blade frames and constant angular velocities omega_s and omega_P.

Analyze the angular velocity in rotating frames, transform to the read frame, derive its time derivative, and compute external moments on wind turbine blades from angular momentum.

Explore how a gyroscope's angular momentum h chases gravity's moment, causing rotation about the z axis, with h dot equals alpha dot cross h linking torque and rotation.

Explore a gyroscope at O with spin omega_s and omega_p = 0.5 rad/s, seeking the omega_s to keep theta at 45 degrees and constant, with radius of gyration.

Describes a fixed blue frame for a precessing gyroscope, maps omega p to z and y axes to keep inertia constant, notes nutation and theta, omega s, omega p equations.

Explain how a circular gyroscope with constant density keeps inertia constant and spins about the z axis. Derive angular momentum and equations of motion in the rotating blue body frame.

Derives the gyroscope's equations of motion about point O and simplifies under constant theta, omega P, and omega s, yielding zero net moments about X, Y, and Z.

Analyze work and energy for rigid bodies, comparing translational and rotational energy about centers of mass and fixed points using the parallel axis theorem.

Solve the rolling spool on a smooth plane using work and energy, with no friction at point A, to find how far center G descends before six radians per second.

Compute sa as two thirds of sg for the rolling spool with friction, and use work energy to find sg ≈ 0.859 m (vs 0.661 m without friction).

Compute how two tugboats applying constant perpendicular forces to the ship's centerline, with mass M and radius of gyration k, produce angular velocity after a 90-degree turn.

Explore the torsion spring exercise to determine the angular velocity at theta = 90 degrees for rod AB (6 kg) and disk C (9 kg) with their inertias.

Solve the torsion spring problem by computing the total moment about point A via parallel axis theorem, then use work-energy to obtain omega ≈ 4.9 rad/s at theta 90 degrees.

Apply the work-energy principle to the climbing rod problem to determine omega just before theta reaches 45 degrees, starting from zero kinetic energy, yielding omega about 4.97 rad/s.

Identify conservative forces and moments, derive gravitational and elastic potential energies, construct a potential function, and apply the conservation of energy and work-energy equations, including non-conservative forces.

Compute the spring stiffness K for a rod and spring using conservation of energy as the rod rotates 90 degrees and momentarily stops.

Model a two-gear and cylinder system to analyze rigid-body dynamics, using given masses and radii of gyration to find the 50 kg cylinder's speed after descending 2 meters from rest.

Apply energy conservation to the two-gear cylinder system, compute final speed after two meters, yielding about 3.675 meters per second.

Analyze the disk-rod-collar system with rolling without slipping to determine the collar's velocity at theta=30 degrees, released from rest at theta=45 degrees.

Compute final kinetic energy by summing translational and rotational energies of the disk, rod BC, and collar C, using no slipping to relate velocities and energy conservation to find VC.

Analyze a 20 kg sphere rotating about axle at 60 rad/s; apply 50 N·m torque to shaft c and determine angular velocity after 90 degrees, assuming principal axis of inertia.

Calculate the kinetic energy of a 200 kg satellite using its center-of-mass velocity and angular velocities about principal axes, with radii of gyration about x, y, z.

Compute the satellite's kinetic energy by summing translational and rotational parts: 1/2 m v_g^2 plus 1/2 I_x' omega_x^2 plus 1/2 I_y' omega_y^2 plus 1/2 I_z' omega_z^2, yielding about 37 megajoules.

Explore linear and angular momenta of rigid bodies, derive angular momentum about a point using center of mass and mass moment of inertia, and apply the parallel axis theorem.

Explore the space capsule exercise: a 1200 kg capsule with 900 kg m^2 inertia experiences a 400 N thrust for 0.3 s, determining its angular velocity after the jets stop.

compute the capsule's angular velocity after 0.3 seconds using angular impulse and momentum, with zero initial angular momentum and a constant couple moment, yielding 0.386 rad/s.

Solve the airplane yaw problem to determine the angular velocity using angular impulse and the mass moment of inertia about point g; omega is approximately 0.0178 radians per second.

solve this rotating rod and disk exercise under torque 20 t^(3/2) n·m, with initial angular velocity -6 rad/s about the z-axis, and compute ω at t=0 and t=3 s.

Apply parallel axis theorem to find rod and disk inertia about point A; sum to 8.1 kg m^2, then use angular momentum to get omega at 3 s, 9.4 rad/s.

Analyze a double pulley with cylinders a and b released from rest to determine their speeds after four seconds, given no slip, a 60 kg, b 70 kg.

solve the two-cylinder and pulley problem to find the velocities of cylinders a and b after four seconds, using angular impulse and angular momentum about O with no slipping.

apply conservation of angular momentum to a satellite with solar panels; calculate the new angular velocity using parallel axis theorem and moment of inertia, from 0.5 to 3.56 rad/s.

Solve the Hubble telescope rotation problem: internal drive makes solar panels rotate about the y axis at 0.6 rad/s relative to telescope; find the inertial angular velocity, neglecting orbital rotation.

Solve the Hubble telescope rotation problem by using angular momentum conservation, noting internal moments cancel and initial angular momentum is zero in the inertial frame.

Use angular momentum conservation to solve the hammer impact problem, deriving post-impact values: hammer angular velocity -0.93 rad/s and box velocity 1.54 m/s.

Derive the general angular momentum about a random point in 3D using mass moments of inertia and products of inertia, then compare center of mass and fixed-point formulations.

Derives the angular momentum about point a from the center of mass, showing h_a = h_g + rho_g with respect to a cross (M v_g).

Determine the capsule's angular velocity after the rock sticks, using angular momentum conservation for the rock-spacecraft system; final components: omega_x ≈ -0.0625, omega_y ≈ -0.1185, omega_z ≈ 0.106 rad/s.

Compute the angular momentum of the satellite about a random point A using h_A = h_G + r_A G × m v_g.

Calculate the satellite's angular momentum about G and A by evaluating mass moments of inertia about red axis under a principal axis, applying rho vectors and cross products.

Use the Lagrange method to derive equations of motion and natural frequency for conservative-force systems. Define L = T − V and apply Euler–Lagrange, illustrated by a cart with springs.

Analyze a pulley and block vibration system with a 50 kg block and 10 kg pulley (radius of gyration 125 mm) to determine the natural frequency using Newton and Lagrange.

Solve the natural frequency of a pulley–block system by applying Newton and Lagrange methods, accounting for gravity, spring force, and the pulley’s moment of inertia under a tiny displacement.

Apply Lagrange to a pulley and block vibration, deriving kinetic and potential energies with gravity and spring, formulating the Lagrangian, and obtaining the equation of motion and omega_n.