2.3 Exercises from the Feyman Lectures on Physics

[sysprog]

It seems that you have adjusted it’s height too!

I used \dots instead of typing period marks in -- oh, and, 'its', in the possessive sense, doesn't take the apostrophe -- that's for a contraction of 'it is'; not for a possession of something that belongs to something that we call 'it'.

 

[Adesh]

I used \dots instead of typing period marks in -- oh, and, 'its', in the possessive sense, doesn't take the apostrophe -- that's for a contraction of 'it is'; not for a possession of something that belongs to something that we call 'it'.

Thank you for teaching me that. Something like that occurs in Shakespearen Sonnets too, “ ‘Tis “ , Shakespeare did it for his iambic feet (In my Opinion).

 

[wrobel]

Assume that a rigid body consists of $N$ particles and i-th particle has a mss $m_i$ and a radius-vector $\boldsymbol r_i$.
Principle of virtual work for equilibrium is written as follows $$\sum_{i=1}^N(\boldsymbol F_i,\delta\boldsymbol r_i)=0,\qquad (1)$$ here $\delta \boldsymbol r_i$ is a virtual displacement of the i-th particle.

Fix any point $A$ in the rigid body. Then $$\delta\boldsymbol r_i=\delta\boldsymbol r_A+\delta\boldsymbol\psi\times\boldsymbol\rho_i,\quad \boldsymbol\rho_i=\boldsymbol r_i-\boldsymbol r_A.\qquad (2)$$
Here $\delta\boldsymbol\psi$ is a virtual rotation of the rigid body (perhaps someone prefers another term).

Now plug formula (2) in (1). After some transformations we get:
$$(\boldsymbol F,\delta \boldsymbol r_A)+(\boldsymbol\tau_A,\delta\boldsymbol\psi)=0,$$
here $\boldsymbol F=\sum_i\boldsymbol F_i$ is the net force and $\boldsymbol\tau_A=\sum_i\boldsymbol\rho_i\times \boldsymbol F_i$ is the total torque about the point $A$.

If $\delta \boldsymbol r_A$ and $\delta\boldsymbol\psi$
are independent then we obtain the standard condition of equilibrium: $\boldsymbol F=0,\quad \boldsymbol \tau_A=0$.

 

[sysprog]

Thank you for teaching me that. Something like that occurs in Shakespearen Sonnets too, “ ‘Tis “ , Shakespeare did it for his iambic feet (In my Opinion).

Using 'tis as an abbreviation for 'it is' is different from (incorrectly) using 'it's' for something possessed by an 'it'. Yes, a lot of of Shakespeare is written in iambic pentameter.

Friends, Romans, countrymen, lend me your ears.
I come to bury Caesar, not to praise him.
The evil that men do lives after them;
The good is oft interrèd with their bones.
So let it be with Caesar. The noble Brutus
Hath told you Caesar was ambitious.
If it were so, it was a grievous fault,
And grievously hath Caesar answered it.
Here, under leave of Brutus and the rest—
For Brutus is an honorable man;
So are they all, all honorable men—
Come I to speak in Caesar’s funeral.
He was my friend, faithful and just to me.