Subjective Type

Several planets ( Jupiter, Saturn, Uranus ) are encircled by rings, perhaps composed of material that faild to from a satellite. In addition, many galaxies con tain ring- like structures. Consider a ho-tain ring-like structures. Consider a homogeneous thin ring of mass M outer radius R (a) What gravitational attraction does it exert on a particle of mass m located on the ring's central axis a distance x from the ring center? (b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which is passes through the center of the ring?

Solution

x(a) All points on the ring are the same distance $$ (r=\sqrt{x^{2}+R^{2}}) $$ from the particle,
so the gravitational potential energy is simply $$ U=-G M m / \sqrt{x^{2}+R^{2}}, $$
The corresponding force (by symmetry) is expected to be along the $$ x $$ axis, so we take a (negative) derivative of $$ U $$ (with respect to $$ x $$ ) to obtain it.
The result for the magnitude of the force is $$ G M m x\left(x^{2}+R^{2}\right)^{-3 / 2} $$
(b) Using our expression for $$ U $$, the change in potential energy as the particle falls to the center is

$$\Delta U=-G M m\left(\dfrac{1}{R}-\dfrac{1}{\sqrt{x^{2}+R^{2}}}\right)$$

By conservation of mechanical energy, this must "turn into" kinetic energy,
$$ \Delta K=-\Delta U=m v^{2} / 2 . $$
We solve for the speed and obtain
$$\dfrac{1}{2} m v^{2}=G M m\left(\dfrac{1}{R}-\dfrac{1}{\sqrt{x^{2}+R^{2}}}\right) \Rightarrow v=\sqrt{2 G M\left(\dfrac{1}{R}-\dfrac{1}{\sqrt{x^{2}+R^{2}}}\right)}$$

SIMILAR QUESTIONS

Gravitation

Consider two solid spheres of radii $$R_1 = 1m, \, R_2 = 2m$$ and masses $$M_1$$ and $$M_2$$, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $$\dfrac{M_1}{M_2}$$ is:

Gravitation

Let $$V$$ and $$E$$ be the gravitational potential and gravitational field at a distance $$r$$ from the centre of a hollow sphere. Consider the following statements : $$(A)$$ the $$V-r$$ graph is continuous $$(B)$$ the $$E-r$$ graph is discontinuous

Gravitation

In Fig 13-37 a, particle A is fixed in placed at x= -0.20 m on the x aixs and particle B with a mass 1.0 kg is fixed in place at the origin particle C are ( not shown ) can be moved along the x axis , between particle B and x= $$ \infty $$ Figure 13-37b shows the x component $$ F_{net x} $$ of the net gravitational force on particle B due to practices A and C, as a functional of position x particle c. The plot actually extend to the right approaching an asymptote of $$ -4.17\times 10^{-10} $$ N as $$ x\rightarrow \infty $$ what are the masses of (a) particles A and (b) particles (C)

Gravitation

If a mass $$m$$ is placed in the vicinity of another mass $$M$$, It experiences a gravitational force of attraction. However, if these two masses are at a very large distance, the force of attraction is negligible. The gravitation potential is the amount of work done per unit mass in bringing it slowly from infinity to some finite distance from $$M$$. Hence, gravitational potential is a state function rather than a path function. One application of this potential is in finding the escape velocity of a body from earth. As we know the gravitation potential energy associated with mass $$m$$ on earth surface is $$-\dfrac {GMm}{R}$$. The mechanical energy conservation gives $$\underline {V_{escape} = \sqrt {\dfrac {2GM}{R}}}$$ or $$V_{escape} = \sqrt {2gR}$$. An inquisitive mind decides to get the result by kinematics, considering a particle following curvilinear path from surface to infinity, gravity changes with height as $$g' = g\left (1 + \dfrac {h}{R}\right )^{-2}$$ or $$v\dfrac {dv}{dh} = -g\left (1 + \dfrac {h}{R}\right )^{-2}$$. Solution of the equation within limits $$h = 0$$ to $$h = \infty$$ gives $$\underline {V_{escape} = \sqrt {2gR}}$$. For most of the objects, either point masses or objects of finite dimension, the variation of potential in space exhibits symmetric behaviour. In case of spherical objects of uniform density, the locus of equipotential points is a spherical shell of any given radius. Hence, this potential field is symmetric about all three axes passing through the centre of spherical object. Consider a somewhat complicated objects as shown in the adjacent figure. The figure shows a solid cube of edge length $$10\ cm$$. The origin is the centre of cube as shown. Eight spherical cavities are formed in this cube, each having a radius of $$1\ cm$$ and centers at $$(\pm 2\ cm, \pm 2\ cm,\pm 2\ cm$$). This figure shows wide range of equipotential surfaces / curves. In which of the following ways, will, potential vary from the centre to surface of any spherical cavity?

Gravitation

A tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

Gravitation

Is the velocity of escape of a particle at rest on the surface of the earth the same as that of a particle just orbiting the earth? Explain.

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