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Offline Centres

Consider a shell rolling purely on a horizontal fixed surface. The velocity of the centre of Shell is $v_{0} \hat{i}$ and acceleration of the centre of shell is $a_{0} \hat{i}$ .

Following representative point are chosen on the shell
P (on z-axis)
Q and R (on x-axis)
S (on y-axis)
Following symbolic representation has been used
$\vec{r} =$ position vector of a point with respect to $o$ on ground
$\vec{v}$ =velocity vector of point with respect to ground
$\vec{ω} =$ angular velocity of the body
$\vec{a} =$ acceleration of a point with respect to ground
Match the conditions given in column-I with points given in column-II that obey the given condition at the instant shown.
COLUMN -I COLUMN-II
a $\vec{r} . \vec{a} = 0$ p P
b $\vec{r} . \vec{v} = 0$ q Q
c $\vec{ω} . \vec{r} = 0$ r R
d $\vec{α} . \vec{r} = 0$ s S

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a-p,q,s; b-p,q,r; c-s; d-p

a-p,q,s,; b-p,q,s; c-r,s; d-q

a-p; b-p,q,r,s; c-q,r,s; d-q,r,s

a-p,q,s; b-p,q; c-r; d-p

answer is C.

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Detailed Solution

$[a \to p; b \to p, q, r, s; c \to q, r, s]$
Question Image
$\vec{r_{s}} . \vec{a_{s}} = - ω^{2} R . (2 R) \vec{r_{p}} . \vec{a_{Q}} \neq 0, \vec{r_{R}} . \vec{a_{R}} \neq 0 \vec{r_{p}} . {\vec{a}}_{p} \neq 0$

$v = ω \times r$

$∴ \vec{r} . \vec{v} = 0$

For any point
${\vec{r}}_{p} . \vec{ω} \neq 0$ see figure
For other ${\vec{r}}_{p} a n d \vec{ω}$ are 1st

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