An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h . Compare i

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3-2.Motion in Plane

medium

An aircraft executes a horizontal loop of radius $1.00\; km$ with a steady speed of $900 \;km/h$. Compare its centripetal acceleration with the acceleration due to gravity.

Option A

Option B

Option C

Option D

Solution

Radius of the loop, $r=1 \,km =1000\, m$
Speed of the aircraft, $v=900\, km / h =900 \times \frac{5}{18}=250\, m / s$
Centripetal acceleration, $\quad a_{e}=\frac{v^{2}}{r}$
$=\frac{(250)^{2}}{1000}=62.5 \,m / s ^{2}$
Acceleration due to gravity, $g=9.8\, m / s ^{2}$
$\frac{a_{c}}{g}=\frac{62.5}{9.8}=6.38$
$a_{c}=6.38\, g$

Standard 11

Physics

The acceleration of a train travelling with speed of $400 \,m/s$ as it goes round a curve of radius $160\,m$, is

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 moving with constant speed, in a circle of radius $10 m$. The body completes one revolution in $4 s$. At the end of $3 rd$ second, the displacement of body (in $m$ ) from its starting point is:

medium

(JEE MAIN-2023)

Velocity of two particles $A$ and $B$ separated by distance $20\ m$ apart is as shown in figure then …….. $rad/s$ will be the angular velocity of $B$ with respect to $A$

medium

The angular speed of a fly wheel making $120$ revolutions/minute is

easy

(AIPMT-1995)

An aircraft executes a horizontal loop of radius $1.00\; km$ with a steady speed of $900 \;km/h$. Compare its centripetal acceleration with the acceleration due to gravity.

Solution

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