Equations to be remembered in respect of the section on gravitation were given in the post dated 9^{th} May 2008. A few multiple choice practice questions were discussed in the post dated 12^{th} May 2008 followed by a free response practice question on gravitation in the post dated 15^{th} May 2008. You can access all posts on gravitation on this site by clicking on the

Today we will discuss a few more multiple choice practice questions on gravitation. The following questions are meant for AP Physics B as well as C:

**(1)** The speed of an artificial satellite moving in an orbit of radius *r *around* *the earth is increased by 41.4%. The satellite will

(a) continue to move in the same orbit

(b) move in an orbit of radius 1.414 *r*

(c) move in an orbit of radius 0.414 *r*

(d) move in an orbit of radius 0.586 *r*

(e) escape into the outer space

The orbital speed *v *of any satellite is given by

*v =*√(*gr*)* *where *g* is the acceleration due to gravity *at the orbit* of radius *r*. * *

[You will get this by equating the centripetal force to the gravitational pull:

*mv*^{2}/*r* = *mg*]

The escape velocity (*v*_{escape})* *of a body at a point distant *r* from the centre of the earth where the acceleration due to gravity is equal to *g* is given by

*v*_{escape} *=*√(2*gr*)* *

Thus the escape velocity is √2 times the orbital speed.

The speed *v *of the satellite becomes 1.414 times the initial value on increasing the speed by 41.4%. Now, 1.414 *v = *√2 *v*. Therefore the new speed is the escape speed so that the satellite will escape into the outer space.

**(2)** A launch vehicle carrying an artificial satellite of mass *m* is set for launch on the surface of the earth of mass *M* and radius *R*. What is the minimum energy to be spent by the launch vehicle on the satellite so that the satellite will move in a circular orbit of radius 7*R*? (Gravitational constant = *G*)

(a) *GMm /*7*R*

(b) 6*GMm /*7*R*

(c) 6*GMm /*14*R*

(d) 13*GMm /*14*R*

(e) *GMm /*14*R*

The satellite of mass *m* is initially at rest and therefore its initial energy (*E*_{i})* *is its gravitational potential energy given by

*E*_{i} = – *GMm/R*

When the satellite is in its orbit of radius 7*R*, its final energy (*E*_{f}) is given by

*E*_{f} = – *GMm/*(2×7*R*) = – *GMm/*(14*R*)

[Note that the total energy of a satellite in an orbit of radius *r* is – *GMm/*2*r*]

The minimum energy required for placing the satellite in its orbit is *E*_{f }–* E*_{i} given by

* E*_{f }–* E*_{i} = – [*GMm/*(14*R*)] – [– *GMm/R*]

= (*GMm/R*) – [*GMm/*(14*R*)]

= 13*GMm /*14*R*

**(3)** A small body is projected vertically up with a speed equal to half the escape speed from the earth’s surface. If the radius of the earth is *R*, what is the maximum height (*h*) reached by the body? (Neglect air resistance).

(a) *R/*3* *

(b)* R/*2

(c) *R *

(d) 2*R*

(e) 5*R*

When the body projected vertically reaches the maximum height, its speed will be zero since its initial kinetic energy us used up in increasing its gravitational potential energy. Therefore we have

½ *m* (*v*_{e}/2)^{2} = [–*GMm /*(*R+h*)] – [–*GMm /*R] where *m* is the mass of the body,* M* is the mass of the earth,* v*_{e} is the escape speed from the earth’s surface and *G* is the gravitational constant.

[The first term on the right hand side of the above equation is the gravitational potential energy of the body at height *h. *The second term is the gravitational potential energy of the body on the surface of the earth].* *

Since *v*_{e} = √(2*GM/R*), the above equation becomes

*GM /*4*R* = (*GM /*R) –[*GM /*(*R+h*)]

Or, *GM /*4*R* = *GMh /*[*R*(*R+h*)

Therefore, ¼ = *h/*(*R+h*) from which *h = R/*3

**(4)** The orbit of a planet moving around the sun is elliptical with the sun S at one focus of the ellipse. If the orbital speeds of the planet while at points A and B (Fig.) are *V*_{1} and *V*_{2} and the distances SA and SB are *d*_{1} and *d*_{2} respectively, *V*_{1}/* V*_{2} is equal to

(a) √(*d*_{1 }/*d*_{2})

(b) √(*d*_{2 }/*d*_{1})

(c) *d*_{1 }/*d*_{2}

(d) *d*_{2 }/*d*_{1}

(e)* *(*d*_{2 }+ *d*_{1}) /(*d*_{2 }– *d*_{1})

To obtain the answer you may apply Kepler’s law of areas (in respect of plnetary motion), which says that the straight line joining the sun to the planet sweeps equal areas in equal intervals of time. The adjoining figure shows path lengths CD and EF traced by the planet in equal intervals (say, *∆t*) and hence the areas CADS and EBFS must be equal. If *∆t* is sufficiently small, CD and EF will become straight line segments and the areas will become triangular. Equating the areas of these triangles, we have

½ ×CD×AS = ½ ×EF×BS

Or, ½ ×* V*_{1 }*∆t *×* d*_{1} = ½ ×* V*_{2 }*∆t *×* d*_{2}

Therefore, *V*_{1}/* V*_{2} = *d*_{2 }/*d*_{1}

[You can easily get the answer from angular momentum conservation:

*m* *V*_{1}* d*_{1} = *m* *V*_{2}* d*_{2}

where *m* is the mass of the planet. This gives *V*_{1}/* V*_{2} = *d*_{2 }/*d*_{1}]

**(5)** We know that the gravitational force between two point masses is inversely proportional to the square of the distance between them. If the gravitational force between two point masses were (let us imagine) inversely proportional to the *n*^{th} power of the distance between them, what would be the relation between the orbital period (*T*)* *of a planet around the sun and the mean distance (*r*) of the planet from the sun?

(a) *T*^{2 }α * r*^{2n}

(b) *T*^{2 }α * r*^{n}

(c) *T*^{2 }α * r*^{2n+1} ^{}

(d) *T*^{2 }α * r*^{(n+1)}

(e) *T*^{2 }α * r*^{(n – 1)}

For simplicity, let us assume that the orbit is circular. (Circular orbit is only a special case of the general elliptical orbit and the relation between *T* and *r *will be unchanged). The centripetal force required for the circular motion is supplied by the gravitational pull so that we have

*mr**ω*^{2} = *GMm/r*^{n} where *m* is the mass of the planet, *ω* is its orbital angular velocity, *M* is the mass of the sun and *G* is the gravitational constant.

The above equation gives *ω*^{2} = *GM/r*^{(n+1)}

Since *ω = T/*2π, we have

(*T/*2π)^{2} = *GM/r*^{(n+1)}

Therefore, *T*^{2 }α * r*^{(n+1)}

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