Test 5: Dynamics

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom

Total Marks: 75

1) The velocity of a particle along and perpendicular to the radius vector are $\lambda r$ and $\mu \theta$ respectively. The acceleration perpendicular to the radius vector is given by:

a) ${\lambda }^{2}r-{\mu }^2{\theta }^2/r$
b) $\mu \theta (\lambda +{\displaystyle \frac{\mu }{r}})$
c) ${\lambda }^{2}r+{\mu }^2{\theta }^2/r$
d) $\mu \theta (\lambda -{\displaystyle \frac{\mu }{r}})$

2) A small bead slides with constant speed $v$ on a smooth wire in the shape of the cardioid $r=a(1+\cos \theta )$. The, the value of ${\displaystyle \frac{d\theta }{dt}}$ is given by:

a) $\left({\displaystyle \frac{v}{2a}}\right)\sec {\displaystyle \frac{\theta }{24}}$
b) $\left({\displaystyle \frac{v}{2a}}\right)\sec {\displaystyle \frac{\theta }{3}}$
c) $\left({\displaystyle \frac{v}{2a}}\right)\sec {\displaystyle \frac{\theta }{2}}$
d) 0

3) 8. An insect crawls at a constant rate u along the spoke of a cart wheel of radius a, the cart moving with velocity $v$. The acceleration perpendicular to the spoke is given by:

a) ${\displaystyle \frac{2uv}{a}}$
b) ${\displaystyle \frac{2uv}{3a}}$
c) ${\displaystyle \frac{uv}{a}}$
d) ${\displaystyle \frac{uv}{2a}}$

4) A point describes the cycloid $s=4a$ sin $\psi$ with uniform speed $v$. Its acceleration at any point is given by:

a) ${\displaystyle \frac{v^2}{\sqrt{\left[(a^2-s^2)\right]}}}$
b) ${\displaystyle \frac{v^2}{\sqrt{\left[(8a^2-s^2)\right]}}}$
c) ${\displaystyle \frac{v^2}{\sqrt{\left[(4a^2-s^2)\right]}}}$
d) ${\displaystyle \frac{v^2}{\sqrt{\left[(16a^2-s^2)\right]}}}$

5) A particle is describibng a plane curve. If the tangential and normal accelerations are each constant throughout the motion, then the angle $\psi ,$ through which the direction of motion turns in time $t$ is given by:

a) $\psi =A\log (1+Bt)$
b) $\psi =B\log (1+At)$
c) $\psi =A\log (1-Bt)$
d) $\psi =B\log (1-At)$

6) A particle moves in a catenary $s=c$ tan $\psi$. The direction of its accleration at any point makes equal angles with the tangent and the normal to the path at that point. If the speed at the vertex (where $\psi =0$ ) be $u$, then the acceleration at any other point $\psi$ is given by:

a) $\{\sqrt{(2/c)}\}{u}^2{e}^{2\psi }{\cos }^2\psi$
b) $\{\sqrt{(2/c)}\}{u}^2{e}^{2\psi }{\sin }^2\psi$
c) $\{\sqrt{(2/c)}\}{u}^2{e}^{2\psi }{\sec }^2\psi$
d) $\{\sqrt{(2/c)}\}{u}^2{e}^{2\psi }{\tan }^2\psi$

7) At the ends of three successive seconds, the distances of a point moving with S.H.M. from its mean position measured in the same direction are 1, 5 and 5. Then the period of a complete oscillation is given by:

a) ${\displaystyle \frac{2\pi }{{\cos }^{-1}\left({\displaystyle \frac{2}{5}}\right)}}$
b) ${\displaystyle \frac{2\pi }{{\cos }^{-1}\left({\displaystyle \frac{3}{5}}\right)}}$
c) ${\displaystyle \frac{2\pi }{{\cos }^{-1}\left({\displaystyle \frac{4}{5}}\right)}}$
d) ${\displaystyle \frac{2\pi }{{\cos }^{-1}\left({\displaystyle \frac{1}{5}}\right)}}$

8) 3. A particle is performing $SHM$ in the line joining two points $A$ and $B$ on a smooth plane and is connected with these points by elastic strings of natural lengths $a$ and $a^{\mathrm{\prime }}$, the moduli of elasticity being $\lambda$ and ${\lambda }^{\mathrm{\prime }}$ respectively. Then the periodic time is given by:

a) $4\pi \sqrt{\left[\left({\displaystyle \frac{m}{(\lambda /a)+\left({\lambda }^{\mathrm{\prime }}/a^{\mathrm{\prime }}\right)}}\right)\right]}$
b) $3\pi \sqrt{\left[\left({\displaystyle \frac{m}{(\lambda /a)+\left({\lambda }^{\mathrm{\prime }}/a^{\mathrm{\prime }}\right)}}\right)\right]}$
c) $\pi \sqrt{\left[\left({\displaystyle \frac{m}{(\lambda /a)+\left({\lambda }^{\mathrm{\prime }}/a^{\mathrm{\prime }}\right)}}\right)\right]}$
d) $2\pi \sqrt{\left[\left({\displaystyle \frac{m}{(\lambda /a)+\left({\lambda }^{\mathrm{\prime }}/a^{\mathrm{\prime }}\right)}}\right)\right]}$

9) A particle is projected from the vertex of a smooth parabolic tube of latus rectum $4a$ along the tube and is acted upon by a repulsive force ${\displaystyle \frac{mgr}{c}}$ from the focus. If the velocity of projection is that which would be acquired in moving from focus to the vertex, then the time of describing angle $\theta$ about the focus is given by:

a) $2\sqrt{\left({\displaystyle \frac{c}{g}}\right)}\log \tan \left({\displaystyle \frac{\pi +\theta }{3}}\right)$
b) $2\sqrt{\left({\displaystyle \frac{c}{g}}\right)}\log \tan \left({\displaystyle \frac{\pi +\theta }{2}}\right)$
c) $2\sqrt{\left({\displaystyle \frac{c}{g}}\right)}\log \tan \left({\displaystyle \frac{\pi +\theta }{4}}\right)$
d) $\sqrt{\left({\displaystyle \frac{c}{g}}\right)}\log \tan \left({\displaystyle \frac{\pi +\theta }{4}}\right)$

10) A particle describes the curve $p^2=$ ar under a force F to the pole, then the law of force is given by:

a) $F\propto {\displaystyle \frac{1}{r^2}}$
b) $F\propto {\displaystyle \frac{1}{r^3}}$
c) $F\propto {\displaystyle \frac{1}{r^4}}$
d) $F\propto {\displaystyle \frac{1}{r^5}}$

11) 1. A particle moves with a central acceleration $\mu \{r+{\displaystyle \frac{a}{r^3}}\}$ being projected from an apse at a distance a with a velocity $2\sqrt{\mu }a$. The curve described by the particle is given by:

a) $r^2(2+\sin \sqrt{3}\theta )=3a^2$
b) $r^2(2+\cos \sqrt{5}\theta )=3a^2$
c) $r^2(2+\cos \sqrt{3}\theta )=3a^2$
d) $r^2(2+\cos \sqrt{3}\theta )=4a^2$

12) A particle subject to a force producing an acceleration $\mu (r+2a)/r^5$ towards the origin is projected from the point $(a,0)$ with $a$ velocity equal to the velocity from infinity at an angle ${\cot }^{-1}2$ with the initial line. The equation to the path is given by:

a) $r=2a(1+\sin \theta )$
b) $r=a(1+\sin \theta )$
c) $r=a(1+2\cos \theta )$
d) $r=a(1+2\sin \theta )$

13) 7. A particle is moving with central acceleration $\mu (r^5-c^{4}r)$ being projected from an apse at a distance $c$ with velocity $\sqrt{{\displaystyle \frac{2\mu }{3}}}{c}^3$, then its path is given by the curve:

a) $x^4+y^4=c^4$
b) $x^2+y^2=c^2$
c) $x^3+y^3=c^3$
d) $x^5+y^5=c^5$

14) A particle describes an ellipse under a force ${\displaystyle \frac{\mu }{[\text{distance}{]}^2}}$ towards the focus. If it was projected with velocity $V$ from a point distant $r$ from the centre of force, then its periodic time is given by:

a) ${\displaystyle \frac{\pi }{\sqrt{\mu }}}{\{{\displaystyle \frac{2}{r}}-{\displaystyle \frac{V^2}{\mu }}\}}^{-3/2}$
b) ${\displaystyle \frac{2\pi }{\sqrt{\mu }}}{\{{\displaystyle \frac{2}{r}}-{\displaystyle \frac{V^2}{\mu }}\}}^{-3/2}$
c) ${\displaystyle \frac{3\pi }{\sqrt{\mu }}}{\{{\displaystyle \frac{2}{r}}-{\displaystyle \frac{V^2}{\mu }}\}}^{-3/2}$
d) ${\displaystyle \frac{4\pi }{\sqrt{\mu }}}{\{{\displaystyle \frac{2}{r}}-{\displaystyle \frac{V^2}{\mu }}\}}^{-3/2}$

15) A particle is moving in ellipse of eccentricity $e,$ under the acceleration ${\displaystyle \frac{\mu }{r^2}}$ to a focus; when the particle is nearest to a focus, the acceleration is suddenly replaced by an acceleration $\mu r$ towards the centre of the ellipse. If the particle continues to move in the same ellipse, then:

a) $\mu ={\mu }^{\mathrm{\prime }}(1+e^2)a^2$
b) $\mu =2{\mu }^{\mathrm{\prime }}(1-e^2)a^2$
c) $\mu ={\mu }^{\mathrm{\prime }}(1-e^2)a^2$
d) $\mu =3{\mu }^{\mathrm{\prime }}(1-e^2)a^2$