(a) Explain what a principal axis system for a rigid body is. (b) Write down the definition of the moment of inertia of a rigid body with respect to a fixed axis ñ. (c) If the principal moments of inertia of a rigid body are all equal, I₁ = I₂ = I3, does that imply that the mass-distribution p(x) in the rigid body is spherically symmetric? Give a short explanation. (d) State what a normal mode is.

(a) Explain what a principal axis system for a rigid body is. (b) Write down the definition of the moment of inertia of a rigid body with respect to a fixed axis ñ. (c) If the principal moments of inertia of a rigid body are all equal, I₁ = I₂ = I3, does that imply that the mass-distribution p(x) in the rigid body is spherically symmetric? Give a short explanation. (d) State what a normal mode is.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

A massless cable is used to connect the center of mass of the cylinder A to the box B hanging over the pulley C. Take the coefficient of static friction as µs = 0.25 and that of kinetic friction as µk = 0.2. Assume that there is no slipping at the contact point D and solve the problem using Newton's second law to find: a) angular accelerations of the cylinder A and pulley C (aA and ac), b) tension forces in the cable on both sides of the pulley C (cylinder side is TA and box side is TB), and c) normal and rolling forces at D (Np and FD). d) Is the assumption of no slipping at the contact point D correct? Why or why not? Notes: 1) We are asuming that tension forces on two sides of the pulley C are not the same, but acceleration of the cable from A to B stays the same. 2) For a cylinder or disk of mass m and radius r, moment of inertia about its center is given as I = mr2/2. 3)Do not forget to sketch the FBD’s for the cylinder A, box B and pulley C. m, = 12 kg mc = 4 kg rc =0.2 m TA=0.5…
Using the data for a falling mass of 0.0183 kg, calculate the uncertainty on the moment of inertia (in g·m2) for the single disc (disc 1), taking into consideration the given uncertainty on the angular acceleration and torque. Round your answer to 4 decimal places. Please answer the second part using the 0.0183kg.
(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m/v|d. Show that if r is the position of the particle then the vector J =r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = w X ri and that the total angular momentum J is J = Σm₁ [r}w - (r; · w)r;]. (ii) Show further that the component of J along the axis of rotation can be written as Iw, where I, the moment of inertia of the collection about the axis or rotation, is given by 1 = Σm₁p². Interpret pi geometrically. (iii) Prove that the total kinetic energy of the particles is 1².
Engineering Dynamics need help from 4,5,6,7 thank you A ball of mass m is moving along a vertical semi-cylinder of radius R as it is guided by the arm OA. The arm moves in a clockwise direction with a constant angular velocity ω. Assume 0° ≤ Φ ≤ 90°. Neglect any friction. Neglect also the size of the ball and the thickness of the arm. Find the relationship between r, R and θ where r is the distance between O and the ball. Draw a free body diagram of the ball assuming that it is in contact with the cylinder and the arm OA. Write the equations of motion in the (r, θ) coordinate system. Find the normal force acting on the ball by the cylinder for Φ = Φ0. Find the normal force acting on the ball by the bar for Φ = Φ0. Determine the angle Φ at which the ball loses contact with the cylinder. Take m = 1 kg, R = 1.4 m, ω = 0.5 rad/s, and Φ = 60°
Let B be the solid bounded by the surfaces z = x^2+2y^2, x^2+y^2=16, and the xy-plane(distances in cm). If B has a constant mass density of 5g/cm^3, find the moment of inertia of B about the axis through(4,3,2) that is perpendicular to the yz-plane
Prove the invariance of the angular velocity vector in a rigid body
A particle of mass m is located at x = 1, y = 0,2 = 2. Find the tensor of inertia for the particle relative to the origin. The particle rotates about the z axis through a small angle a <<1 as shown below. Show that the moments of inertia are unchanged to second order in a but the products of inertia can change linearly with a.
(a) What does the quadrupole formula (P) = = = (Qij Q³ ³) compute? Reason the answer. (b) A point mass m undergoes a harmonic motion along the z-axis with frequency w and amplitude L, x(t) = y(t) = 0, z(t) = L cos(wt). Show that the only non-vanishing component of the quadrupole moment tensor is = Im L² cos² (wt). (c) Use the quadrupole formula to compute the power radiated by the emission of gravitational waves. (Hint: recall that (cos(t)) = (sin(t)) = 0 and (cos² (t)) = (sin² (t)) = ½½ for a given frequency 2.)