(A) The gravitational force is given by $F \propto \frac{1}{R^n} = R^{-n}$.For a planet in a circular orbit,the centripetal force is provided by the g

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$R^{\left( \frac{n+1}{2} \right)}$

(A) The gravitational force is given by $F \propto \frac{1}{R^n} = R^{-n}$.For a planet in a circular orbit,the centripetal force is provided by the gravitational force:$M R \omega^2 = F \propto R^{-n}$.Since $\omega = \frac{2\pi}{T}$,we have $\omega^2 \propto \frac{1}{T^2}$.Substituting this into the force equation:$R \cdot \frac{1}{T^2} \propto R^{-n}$.$\frac{1}{T^2} \propto R^{-n-1} = R^{-(n+1)}$.$T^2 \propto R^{n+1}$.Taking the square root of both sides:$T \propto R^{\frac{n+1}{2}}$.

Class 11 Chemistry Data Entry - Chemistry Level 0 of all question bank

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

AIEEE 2004 All AIEEE

A
$R^{\left( \frac{n+1}{2} \right)}$
B
$R^{\left( \frac{n-1}{2} \right)}$
C
$R^n$
D
$R^{\left( \frac{n-2}{2} \right)}$

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The principal section of a glass prism is an isosceles triangle $ABC$ with $AB=AC$. The face $AC$ is silvered. $A$ ray of light is incident normally on the face $AB$ and after two reflections,it emerges from the base $BC$ perpendicular to the base. Angle $BAC$ of the prism is (in $^{\circ}$)

TS EAMCET 2004

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Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the midpoints of the chords of the circle $C$ that subtend an angle of $\frac{2\pi}{3}$ at its center is:

AIEEE 2006

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The horizontal and vertical displacements of a projectile at time $t$ are $x=36 t$ and $y=48 t-4.9 t^2$,respectively. Initial velocity of the projectile in $m/s$ is

AP EAMCET 2002

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The correct match of the group reagents in List$-I$ for precipitating the metal ion given in List$-II$ from solutions, is

List$-I$ List$-II$

$(P)$ Passing $H _2 S$ in the presence of $NH _4 OH$ $(1) \ Cu ^{2+}$

$(Q) \ \left( NH _4\right)_2 CO _3$ in the presence of $NH _4 OH$ $(2) \ Al ^{3+}$

$(R) \ NH _4 OH$ in the presence of $NH _4 Cl$ $(3) \ Mn ^{2+}$

$(S)$ Passing $H _2 S$ in the presence of dilute $\text{HCl}$ $(4) \ Ba ^{2+}$

$(5) \ Mg ^{2+}$

IIT 2025

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$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$,then $(\tan \alpha, \tan \beta)$ is equal to

AP EAMCET 2012

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List$-I$	List$-II$
$(P)$ Passing $H _2 S$ in the presence of $NH _4 OH$	$(1) \ Cu ^{2+}$
$(Q) \ \left( NH _4\right)_2 CO _3$ in the presence of $NH _4 OH$	$(2) \ Al ^{3+}$
$(R) \ NH _4 OH$ in the presence of $NH _4 Cl$	$(3) \ Mn ^{2+}$
$(S)$ Passing $H _2 S$ in the presence of dilute $\text{HCl}$	$(4) \ Ba ^{2+}$
	$(5) \ Mg ^{2+}$

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

Similar Questions

The principal section of a glass prism is an isosceles triangle $ABC$ with $AB=AC$. The face $AC$ is silvered. $A$ ray of light is incident normally on the face $AB$ and after two reflections,it emerges from the base $BC$ perpendicular to the base. Angle $BAC$ of the prism is (in $^{\circ}$)

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the midpoints of the chords of the circle $C$ that subtend an angle of $\frac{2\pi}{3}$ at its center is:

The horizontal and vertical displacements of a projectile at time $t$ are $x=36 t$ and $y=48 t-4.9 t^2$,respectively. Initial velocity of the projectile in $m/s$ is

$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$,then $(\tan \alpha, \tan \beta)$ is equal to

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