(a) A uniform, thin rod of length L and mass M has a frictionlesspivot at one end. Starting from the definition of moment of inertiaas I = | r dm, show that the rod has a moment of inertia given byI3(b) The rod is initially vertical (with the pivot at the bottom) andfalls under gravity. By considering that gravity acts on the rod'scenter of mass, find the torque acting on the rod when it is at anangle 0 to the vertical, and hence show that the angular accelerationis given by3g sin 02Lwhere g is the acceleration due to gravity.(c) Sketch a as a function of 0.(d) Hence find the linear acceleration of the center of mass whenthe rod is horizontal. Show also that when the rod is horizontal thelinear acceleration of the tip of the rod is greater than that of anobject in free-fall.(e) Make a rough sketch of the distance travelled by the tip as afunction of time as the rod moves through one full revolution.

(a) A uniform, thin rod of length L and mass M has a frictionless pivot at one end. Starting from the definition of moment of inertia as I = | r² dm, show that the rod has a moment of inertia given by 1 -ML². 3 (b) The rod is initially vertical (with the pivot at the bottom) and falls under gravity. By considering that gravity acts on the rod's center of mass, find the torque acting on the rod when it is at an angle 0 to the vertical, and hence show that the angular acceleration

(a) A uniform, thin rod of length L and mass M has a frictionless pivot at one end. Starting from the definition of moment of inertia as I = | r² dm, show that the rod has a moment of inertia given by 1 -ML². 3 (b) The rod is initially vertical (with the pivot at the bottom) and falls under gravity. By considering that gravity acts on the rod's center of mass, find the torque acting on the rod when it is at an angle 0 to the vertical, and hence show that the angular acceleration

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)...

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Concept explainers

Rigid Body

A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.

Rigid Body Dynamics

Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).

Question

(a) A uniform, thin rod of length L and mass M has a frictionless
pivot at one end. Starting from the definition of moment of inertia
as I = | r dm, show that the rod has a moment of inertia given by
I
3
(b) The rod is initially vertical (with the pivot at the bottom) and
falls under gravity. By considering that gravity acts on the rod's
center of mass, find the torque acting on the rod when it is at an
angle 0 to the vertical, and hence show that the angular acceleration
is given by
3g sin 0
2L
where g is the acceleration due to gravity.
(c) Sketch a as a function of 0.
(d) Hence find the linear acceleration of the center of mass when
the rod is horizontal. Show also that when the rod is horizontal the
linear acceleration of the tip of the rod is greater than that of an
object in free-fall.
(e) Make a rough sketch of the distance travelled by the tip as a
function of time as the rod moves through one full revolution.

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