# Questions 59 - 61#

## Q59 Cross product#

The differential of a cross product is $$\displaystyle \frac{d}{dt}(\vec A \times \vec B)=\vec A \times \frac{d\vec B}{dt}+\frac{d\vec A}{dt}\times \vec B$$. Calculate $$d\vec L/dt$$ if $$\displaystyle \vec L=m\vec v \times \frac{d\vec r}{dt}$$.

## Q60 Centripetal acceleration#

A particle with orbital motion has a centripetal acceleration of $$\vec a = \vec \omega\times (\vec \omega \times \vec r)$$, where $$\vec \omega$$ is angular velocity and $$\vec r$$ a positional vector. Find $$|\vec a|$$ if the motion is in a circle where $$\vec r$$ and $$\vec \omega$$ are perpendicular.

## Q61 Rotating disc#

Derive the equation for the linear velocity of a disc, and by inference any rigid body, using the $$(i, j, k)$$ basis set of vectors. Although these are fixed in space they may be used by allowing the amplitude of the vector’s components to change with rate $$\alpha$$ which is also the magnitude of the angular velocity $$\vec \omega$$,

$\displaystyle \vec r(t) = \vec r\cos(\alpha t)\;\boldsymbol i + \vec r\sin(\alpha t)\;\boldsymbol j$

Calculate the cross product $$\vec \omega \times\vec r$$ and the position vector $$\vec r$$. Next, differentiate $$\vec r$$ to obtain the linear velocity $$\vec \upsilon$$ and equate the two equations. Figure 55 shows the definition of vectors. Let $$r$$ be the magnitude of vector $$\vec r$$.

Figure 55 Angular velocity $$\vec \omega$$, linear velocity $$\vec \upsilon$$ of a disc of radius $$r$$ with radius vector $$\vec r$$.