Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 2

Show that the most general movement of a rigid body is a helical movement, i.e., the composition of a rotation through angle \(\varphi\) around some axis and a translation by \(h\) along it.

Short Answer

Expert verified
Answer: The most general movement of a rigid body is a helical movement, which can be represented as a composition of a rotation through angle \(\varphi\) around the axis \(\vec{n}\) and a translation by \(h\) along it.

Step by step solution

01

Initial position of a point in the rigid body

Consider a point P with position vector \(\vec{r}\) in the rigid body initially.
02

Final position of point P after the movement

Let the point P move to a new position P' after the movement with position vector \(\vec{r'}\).
03

Displacement of point P due to movement

The displacement of point P due to the movement can be given as \(\vec{r'} - \vec{r}\).
04

Expressing displacement in terms of rotation and translation

Express the displacement of point P as the sum of a rotation, represented by a rotation matrix \(\textbf{R}(\varphi)\), and a translation vector, \(\vec{t}\). So, we have \(\vec{r'} = \textbf{R}(\varphi) \vec{r} + \vec{t}\).
05

Finding the axis of rotation

The axis of rotation can be found by considering the cross product of the position vectors, \(\vec{n} = \vec{r} \times \vec{r'}\). This will give us the direction of the axis around which the body has rotated.
06

Projecting the displacement onto the axis of rotation

Project the displacement vector \(\vec{r'} - \vec{r}\) onto the axis of rotation \(\vec{n}\), giving us the translation component, \(\vec{h}\). We can find this projection by taking the dot product of the displacement vector and the unit vector along the axis of rotation, i.e., \(\vec{h} = ((\vec{r'} - \vec{r}) \cdot \hat{n}) \hat{n}\).
07

Decomposing the displacement into rotation and translation components

Now, we have the rotation component represented by the rotation matrix \(\textbf{R}(\varphi)\) and the translation component represented by the vector \(\vec{h}\). We can then write the new position of point P after the movement as \(\vec{r'} = \textbf{R}(\varphi) \vec{r} + \vec{h}\).
08

Helical movement representation

From Step 7, we can see that the most general movement of the rigid body can be represented as a helical movement, which is a composition of a rotation through angle \(\varphi\) around the axis \(\vec{n}\) and a translation by \(h\) along it. Thus, we have demonstrated that the most general movement of a rigid body is indeed a helical movement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove Steiner's theorem: The moments of incrtia of any rigid body relative to two parallel axes, one of which passes through the center of mass, are related by the equation $$ I-I_{0}+m r^{2}, $$ where \(m\) is the mass of the body, \(r\) is the distance between the axes, and \(I_{0}\) is the moment of inertia relative to the axis passing through the center of mass. Thus the moment of inertia relative to an axis passing through the center of mass is less than the moment of inertia relative to any parallel axis.

Find the axes and moments of inertia of a homogeneous ellipsoid of mass \(m\) with semiaxes \(a, b\), and c relative to the center 0 . Hint. First look at the sphere.

Are stationary rotations of the body around the largest and smallest principal axes Liapunov stable?

Show that the moments of inertia of any body satisfy the triangle inequalities $$ I_{3} \leq I_{2}+I_{1} \quad I_{2} \leq I_{1}+I_{3} \text { and } I_{1} \leq I_{2}+I_{3} \text {. } $$ and that equality holds only for a planar body.

A river flows with velocity \(3 \mathrm{~km} / \mathrm{hr}\). For what radius of curvature of a river bend is the Coriolis force from the earth's rotation greater than the centrifugal force determined by the flow
See all solutions

Recommended explanations on Physics Textbooks

Translational Dynamics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Modern Physics

Read Explanation

Wave Optics

Read Explanation

Measurements

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free