Question 766: Centre of Mass and Rotational Motion

Question

The position vector $$\overrightarrow r $$ of a particle of mass m is given by the following equation
$$$\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,$$$where $$\alpha  = {{10} \over 3}m{s^{ - 3}}$$, $$\beta  = 5\,m{s^{ - 2}}$$ and m = 0.1 kg. At t = 1 s, which of the following
statement(s) is(are) true about the particle?

Accepted Answer

$$r = \alpha {t^3}\widehat i + \beta {t^2}\widehat j$$
$$v = {{dr} \over {dt}} = 3\alpha {t^2}\widehat i + 2\beta {t^2}\widehat j$$
$$a = {{{d^2}r} \over {d{t^2}}} = 6\alpha {t^2}\widehat i + 2\beta {t^2}\widehat j$$
At t = 1 s,
(a) $$v = 3 	imes {{10} \over 3} 	imes 1\widehat i + 2 	imes 5 	imes 1\widehat j$$
$$ = (10\widehat i + 10\widehat j)$$ m/s
(b) $$\widehat L = \widehat r 	imes \widehat p$$
$$ = \left( {{{10} \over 3} 	imes 1\widehat i + 5 	imes 1\widehat j} ight) 	imes 0.1(10\widehat i + 10\widehat j)$$
$$ = \left( { - {5 \over 3}\widehat k} ight)$$ N-ms
(c) $$F = ma$$
$$ = m 	imes \left( {6 	imes {{10} \over 3} 	imes 1\widehat i + 2 	imes 5\widehat j} ight) = (2\widehat i + \widehat j)N$$
(d) $$	au  = r 	imes F = \left( {{{10} \over 3}\widehat i + 5\widehat j} ight) 	imes (2\widehat i + \widehat j)$$
$$ =  + {{10} \over 3}\widehat k + 10( - \widehat k) = \left( { - {{20} \over 3}\widehat k} ight)$$ N-m

Answer

The velocity $$\overrightarrow v $$ is given by $$\overrightarrow v  = \left( {10\widehat i + 10\widehat j} \right)$$ ms
-1

Answer

The angular momentum $$\overrightarrow L $$
with respect to the origin is given by $$\overrightarrow L  =  - \left( {{5 \over 3}} \right)\widehat k\,N\,m\,s$$

Answer

The force $$\overrightarrow F $$
is given by $$\overrightarrow F  = \left( {\widehat i + 2\widehat j} \right)N$$

Answer

The torque $$\overrightarrow 	au  $$ with respect to the origin is given by $$\overrightarrow 	au   =  - \left( {{{20} \over 3}} ight)\widehat k\,N\,m$$

Centre of Mass and Rotational Motion

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