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) \(\mu \theta\left(\lambda+\dfrac{\mu}{r}\right)\)
c) \(\lambda^{2} r+\mu^{2} \theta^{2} / r\)
d) \(\mu \theta\left(\lambda-\dfrac{\mu}{r}\right)\)

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 \(\dfrac{d \theta}{d t}\) is given by:

a) \(\left(\dfrac{v}{2 a}\right) \sec \dfrac{\theta}{24}\)
b) \(\left(\dfrac{v}{2 a}\right) \sec \dfrac{\theta}{3}\)
c) \(\left(\dfrac{v}{2 a}\right) \sec \dfrac{\theta}{2}\)
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) \(\dfrac{2 u v}{a}\)
b) \(\dfrac{2 u v}{3a}\)
c) \(\dfrac{u v}{a}\)
d) \(\dfrac{u v}{2a}\)

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

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

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) \(\psi=A \log (1+B t)\)
b) \(\psi=B \log (1+A t)\)
c) \(\psi=A \log (1-B t)\)
d) \(\psi=B \log (1-A t)\)

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) \(\dfrac{2 \pi}{\cos ^{-1}\left(\dfrac{2}{5}\right)}\)
b) \(\dfrac{2 \pi}{\cos ^{-1}\left(\dfrac{3}{5}\right)}\)
c) \(\dfrac{2 \pi}{\cos ^{-1}\left(\dfrac{4}{5}\right)}\)
d) \(\dfrac{2 \pi}{\cos ^{-1}\left(\dfrac{1}{5}\right)}\)

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 \(a^{\prime}\), the moduli of elasticity being \(\lambda\) and \(\lambda^{\prime}\) respectively. Then the periodic time is given by:

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

9) A particle is projected from the vertex of a smooth parabolic tube of latus rectum \(4 a\) along the tube and is acted upon by a repulsive force \(\dfrac{m g r}{c}\) 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) \(2 \sqrt{\left(\dfrac{c}{g}\right)} \log \tan \left(\dfrac{\pi+\theta}{3}\right)\)
b) \(2 \sqrt{\left(\dfrac{c}{g}\right)} \log \tan \left(\dfrac{\pi+\theta}{2}\right)\)
c) \(2 \sqrt{\left(\dfrac{c}{g}\right)} \log \tan \left(\dfrac{\pi+\theta}{4}\right)\)
d) \(\sqrt{\left(\dfrac{c}{g}\right)} \log \tan \left(\dfrac{\pi+\theta}{4}\right)\)

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) \(F \propto \dfrac{1}{r^{2}}\)
b) \(F \propto \dfrac{1}{r^{3}}\)
c) \(F \propto \dfrac{1}{r^{4}}\)
d) \(F \propto \dfrac{1}{r^{5}}\)

11) 1. A particle moves with a central acceleration \(\mu\left\{r+\dfrac{a}{r^{3}}\right\}\) 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) \(r^{2}(2+\sin \sqrt{3} \theta)=3 a^{2}\)
b) \(r^{2}(2+\cos \sqrt{5} \theta)=3 a^{2}\)
c) \(r^{2}(2+\cos \sqrt{3} \theta)=3 a^{2}\)
d) \(r^{2}(2+\cos \sqrt{3} \theta)=4 a^{2}\)

12) A particle subject to a force producing an acceleration \(\mu(r+2 a) / r^{5}\) 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 \(\mu\left(r^{5}-c^{4} r\right)\) being projected from an apse at a distance \(c\) with velocity \(\sqrt{\dfrac{2 \mu}{3}} c^{3}\), 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 \(\dfrac{\mu}{[\text {distance}]^{2}}\) 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) \(\dfrac{ \pi}{\sqrt{\mu}}\left\{\dfrac{2}{r}-\dfrac{V^{2}}{\mu}\right\}^{-3 / 2}\)
b) \(\dfrac{2 \pi}{\sqrt{\mu}}\left\{\dfrac{2}{r}-\dfrac{V^{2}}{\mu}\right\}^{-3 / 2}\)
c) \(\dfrac{3 \pi}{\sqrt{\mu}}\left\{\dfrac{2}{r}-\dfrac{V^{2}}{\mu}\right\}^{-3 / 2}\)
d) \(\dfrac{4 \pi}{\sqrt{\mu}}\left\{\dfrac{2}{r}-\dfrac{V^{2}}{\mu}\right\}^{-3 / 2}\)

15) A particle is moving in ellipse of eccentricity \(e,\) under the acceleration \(\dfrac{\mu}{r^{2}}\) 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) \(\mu=\mu^{\prime}\left(1+e^{2}\right) a^{2}\)
b) \(\mu=2\mu^{\prime}\left(1-e^{2}\right) a^{2}\)
c) \(\mu=\mu^{\prime}\left(1-e^{2}\right) a^{2}\)
d) \(\mu=3\mu^{\prime}\left(1-e^{2}\right) a^{2}\)