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3.2 , motion in plane — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysics3.2 , motion in plane1 Mark
MCQ
A projectile is projected from ground with initial velocity $\vec u\, = \,{u_0}\hat i\, + \,{v_0}\hat j\,$. If acceleration due to gravity $(g)$  is along the negative $y-$ direction then find maximum displacement in $x-$ direction
  • A
    $\frac {u_0^2}{2g}$
  • ✓
    $\frac {2u_0v_0}{g}$
  • C
    $\frac {v_0^2}{2g}$
  • D
    $\frac {4u_0v_0}{g}$
✓

Answer

Correct option: B.
$\frac {2u_0v_0}{g}$
b
Given,

$\overrightarrow{ U }=u_0 \hat{1}+v_0 \hat{j}$

$u_0=u \cos \theta \ldots \ldots .1$

$v_0=v \sin \theta \ldots \ldots . .2$

Since,

$R =\frac{ u ^2 2 \sin 2 \theta}{ g }$

$R =\frac{ u ^2 2(\sin \theta \cdot \cos \theta)}{ g } \ldots \ldots$

Substituting the value of equation 1,2 in eqyuation 3 then we get,

$R =\frac{2 u _0 v _0}{ g }$

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The general motion of a rigid body can be considered to be a combination of $(i)$ a motioon --- centre of mass about an axis, and $(ii)$ its motion about an instantanneous axis passing through center of mass. These axes need not be stationary. Consider, for example, a thin uniform welded (rigidly fixed) horizontally at its rim to a massless stick, as shown in the figure. Where disc-stick system is rotated about the origin ona horizontal frictionless plane with angular sp--- $\omega$, the motion at any instant can be taken as a combination of $(i)$ a rotation of the centre of mass the disc about the $z$-axis, and $(ii)$ a rotation of the disc through an instantaneous vertical axis pass through its centre of mass (as is seen from the changed orientation of points $P$ and $Q$). Both the motions have the same angular speed $\omega$ in the case. $Image$  Now consider two similar systems as shown in the figure: case $(a)$ the disc with its face ver--- and parallel to $x - z$ plane; Case $(b)$ the disc with its face making an angle of $45^{\circ}$ with $x$-y plane its horizontal diameter parallel to $x$-axis. In both the cases, the disc is weleded at point $P$, and systems are rotated with constant angular speed $\omega$ about the $z$-axis.$Image$

$1.$ Which of the following statement regarding the angular speed about the istantaneous axis (passing through the centre of mass) is correct?

$(A)$ It is $\sqrt{2} \omega$ for boht the cases

$(B)$ it is $\omega$ for case $(a)$; and $\frac{w}{\sqrt{2}}$ for case $(b)$.

$(C)$ It is $\omega$ for case $(a)$; and $\sqrt{2} \omega$ for case $(b)$.

$(D)$ It is $\omega$ for both the cases

$2.$ Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

$(A)$ It is vertical for both the cases $(a)$ and $(b)$.

$(B)$ It is verticle for case $(a)$; and is at $45^{\circ}$ to the $x-z$ plane and lies in the plane of the disc for case $(b)$

$(C)$ It is horizontal ofr case $(a)$; and is at $45^{\circ}$ to the $x - z$ plane and is normal to the plane of the disc for case $(b)$.

$(D)$ It is vertical of case $(a)$; and is at $45^{\circ}$ to the $x - z$ plane and is normal to the plane of the disc for case $(b)$.

Give the answer question $1$ and $2.$

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