Gravitational Potential of a Ring

Q. A point $\text{P}$ lies on the axis of a fixed ring of mass $M$ and radius $R$, at a distance $2R$ from its centre $\text{O}$. A small particle which is at rest starts from $\text{P}$ and reaches $\text{O}$ under the influence of gravitational attraction only. Its speed at $\text{O}$ will be

1. $\sqrt{\frac{2GM}{R}(1-\frac{1}{\sqrt{5}})}$
2. $\sqrt{\frac{2GM}{R}}$
3. $\sqrt{\frac{2GM}{R}(\sqrt{5}-1)}$
4. Zero

Q. A thin uniform annular disc of mass $M$ has an outer radius $4R$ and an inner radius $3R$ as shown in the figure. The work required to take a unit mass from point $\text{P}$ on its axis to infinity is

1. $\frac{2GM}{7R}(4\sqrt{2}-5)$
2. $\frac{2GM}{R}(4\sqrt{2}-5)$
3. $\frac{GM}{4R}$
4. $\frac{4GM}{7R}(4\sqrt{2}-5)$

Q. Two rings having masses $M$ and $2M$, respectively, having the same radius are placed coaxially as shown in the figure. If the mass distribution on both the rings is non-uniform, then gravitational potential at point $P$ is

1. $-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$
2. $-\frac{GM}{R}[1+\frac{1}{\sqrt{2}}]$
3. $-\frac{GM}{R}[\frac{1}{\sqrt{2}}-\frac{2}{\sqrt{5}}]$
4. Zero

Q. A thin rod of length $L$ is bent to form a semicircle. The mass of rod is $M$. What will be the gravitational potential at the centre of the circle?

1. $-\frac{GM}{\pi L}$
2. $-\frac{GM}{2\pi L}$
3. $-\frac{\pi GM}{2L}$
4. $-\frac{\pi GM}{L}$

Q. A thin rod of length $L$ is bent to form a semicircle. The mass of the rod is $M$. What will be the gravitational potential at the centre of the semicircle?

1. $-\frac{GM}{L}$
2. $-\frac{GM}{2\pi L}$
3. $-\frac{\pi GM}{2L}$
4. $-\frac{\pi GM}{L}$

Q. A ring has radius $R$ and mass $m_1=m\text{kg}$ which is distributed uniformly over its circumference. A highly dense particle of mass, $m_2=2m\text{kg}$ is placed at rest on the axis of the ring at a distance $x_0=\sqrt{3}R$ from the centre. Neglecting all other forces, except mutual gravitational interaction between the ring and particle. Calculate the speed of the ring at the instant when the particle is at the centre of the ring.

1. $\sqrt{\frac{2G}{3R}}$
2. $\sqrt{\frac{G}{3R}}$
3. $\sqrt{\frac{G}{R}}$
4. $\sqrt{\frac{4G}{3R}}$

Q. A ring of radius $R=4\text{m}$ has mass of $m_1=5.4$$\times {10}^9\text{kg}$, which is distributed uniformly over its circumference. A highly dense particle of mass $m_2=5.4$$\times {10}^9\text{kg}$ is placed along the axis of the ring at a distance $x_0=3\text{m}$ from the centre. Neglecting all other forces, except mutual gravitational interaction between the ring and particle. Calculate displacement of the ring when particle is at the centre of the ring.

1. $1\text{m}$
2. $1.5\text{m}$
3. $2\text{m}$
4. $3\text{m}$

Q. A point $\text{P}$ lies on the axis of a fixed ring of mass $M$ and radius $R$, at a distance $2R$ from its centre $\text{O}$. A small particle which is at rest starts from $\text{P}$ and reaches $\text{O}$ under the influence of gravitational attraction only. Its speed at $\text{O}$ will be

1. Zero
2. $\sqrt{\frac{2GM}{R}(1-\frac{1}{\sqrt{5}})}$
3. $\sqrt{\frac{2GM}{R}(\sqrt{5}-1)}$
4. $\sqrt{\frac{2GM}{R}}$

Q. A point $P$ lies on the axis of a ring of mass $M$ and radius $a$, at a distance $a$ from its centre $C$. A small particle starts at rest from $P$ and reaches $C$ under gravitational attraction only. Its speed at $C$ will be

1. zero
2. $\sqrt{\frac{2GM}{a}(1-\frac{1}{\sqrt{2}})}$
3. $\sqrt{\frac{2GM}{a}(\sqrt{2}-1)}$
4. $\sqrt{\frac{2GM}{a}}$

Q. Two identical thin rings each of radius $R$ are coaxially placed at a separation of $R$. If the rings have a uniform mass distribution and masses $m_1$ and $m_2$ respectively, then the work done in moving a mass $m$ from centre of the ring having mass $m_1$ to center of the other ring is

1. $\frac{Gm(2-\sqrt{2})(m_1+m_2)}{2R}$
   
2. $\frac{Gm(2-\sqrt{2})(m_2-m_1)}{2R}$
3. $0$
4. $\frac{Gm(2-\sqrt{2})(m_1-m_2)}{2R}$