A particle moves in a circle of radius 25, cm at two revolutions per second. acceleration of particl

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3-2.Motion in Plane

medium

A particle moves in a circle of radius $25\, cm$ at two revolutions per second. The acceleration of the particle in $meter/second^2$ is

A${\pi ^2}$

B$8\,{\pi ^2}$

C$4\,{\pi ^2}$

D$2\,{\pi ^2}$

(AIIMS-2012)

Solution

$\begin{array}{l}
Here\,T = \frac{1}{2}\sec \,the\,required\,centripetal\\
acceleration\,for\,moving\,in\,a\,circle\,is\\
{a_C} = \frac{{{v^2}}}{r} = \frac{{{{\left( {r\omega } \right)}^2}}}{r} = r{\omega ^2}r \times {\left( {2\pi /T} \right)^2}\\
so\,{a_c} = 0.25 \times {\left( {2\pi /0.5} \right)^2}\\
\,\,\,\,\,\,\,\,\,\,\, = 16{\pi ^2} \times .25 = 4.0{\pi ^2}
\end{array}$

Standard 11

Physics

A particle moves in a circular path of radius $r$ with speed $v.$ It then increases its speed to $2\,v$ while travelling along the same circular path. The centripetal acceleration of the particle has changed by a factor of

easy

Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian co-ordinates $A=A_{x} \hat{i}+A_{y} \hat{j},$ where $\hat{i}$ and $\hat{\jmath}$ are unit vector along $x$ and $y$ – directions, respectively and $A_{x}$ and $A_{y}$ are corresponding components of $A$. Motion can also be studied by expressing vectors in circular polar co-ordinates as $\overrightarrow A \, = \,{A_r}\widehat r\,\, + \,{A_\theta }\hat \theta $ where $\hat{r}=\frac{r}{r}=\cos \theta \hat{i}+\sin \theta \hat{\jmath}$ and $\hat{\theta}=-\sin \theta \hat{i}+\cos \theta \hat{j}$ are unit vectors along direction in which $\hat{r}$ and $\hat{\theta}$ are increasing.
$(a)$ Express ${\widehat {i\,}}$ and ${\widehat {j\,}}$ in terms of ${\widehat {r\,}}$ and ${\widehat {\theta }}$ .
$(b)$ Show that both $\widehat r$ and $\widehat \theta $ are unit vectors and are perpendicular to each other.
$(c)$ Show that $\frac{d}{{dr}}(\widehat r)\, = \,\omega \hat \theta \,$, where $\omega \, = \,\frac{{d\theta }}{{dt}}$ and $\frac{d}{{dt}}(\widehat \theta )\, = \, – \theta \widehat r\,$.
$(d)$ For a particle moving along a spiral given by $\overrightarrow r \, = \,a\theta \widehat r$, where $a = 1$ (unit), find dimensions of $a$.
$(e)$ Find velocity and acceleration in polar vector representation for particle moving along spiral described in $(d)$ above.

hard

A body is whirled in a horizontal circle of radius $20 \,cm$. It has angular velocity of $10\, rad/s$. What is its linear velocity at any point on circular path ……. $m/s$

easy

(AIPMT-1996)

A particle $B$ is moving in a circle of radius $a$ with a uniform speed $u$. $C$ is the centre of the circle and $AB$ is diameter. The angular velocity of $B$ about $A$ and $C$ are in the ratio

medium

Two point masses of mass $m_1 = fM$ and $m_2 = (1 -f)\,M\,(f < 1 )$ are in outer space ( far from gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular vaocities $\omega _1$ for $m_1$ and $\omega _2$ for $m_2.$ In that case

medium

(AIEEE-2012)

A particle moves in a circle of radius $25\, cm$ at two revolutions per second. The acceleration of the particle in $meter/second^2$ is

Solution

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