There have been suggestions that the value
Question: There have been suggestions that the value of the gravitational constant G becomes smaller when considered over a very large time period in the future. If that happens for our earth, (a) nothing will change (b) we will become hotter after billions of years (c) we will be going around but not strictly in closed orbits (d) after a sufficiently long time we will leave the solar system Solution: The correct answers are (c) we will be going around but not strictly in closed orbits (d) after...
Read More →If the sun and the planets carried
Question: If the sun and the planets carried huge amounts of opposite charges, (a) all three of Keplers laws would still be valid (b) only the third law will be valid (c) the second law will not change (d) the first law will still be valid Solution: The correct answers are (a) all three of Keplers laws would still be valid (c) the second law will not change (d) the first law will still be valid...
Read More →Prove that
Question: Prove that $\frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x}=\frac{1}{2}(2+\sin 2 x)$ Solution: To Prove: $\frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x}=\frac{1}{2}(2+\sin 2 x)$ Taking LHS, $=\frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x} \ldots$ (i) We know that, $a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)$ So, $\cos ^{3} x-\sin ^{3} x=(\cos x-\sin x)\left(\cos ^{2} x+\cos x \sin x+\sin ^{2} x\right)$ So, eq. (i) becomes $=\frac{(\cos x-\sin x)\left(\cos ^{2} x+\cos x \sin x+\sin ^{2} ...
Read More →If the mass of sun were ten times smaller
Question: If the mass of sun were ten times smaller and gravitational constant G were ten times larger in magnitudes (a) walking on ground would become more difficult (b) the acceleration due to gravity on earth will not change (c) raindrops will fall much faster (d) aeroplanes will have to travel much faster Solution: The correct answers are (a) walking on ground would become more difficult (c) raindrops will fall much faster (d) aeroplanes will have to travel much faster...
Read More →If the law of gravitation,
Question: If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube-law (a) planets will not have elliptic orbits (b) circular orbits of planets is not possible (c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic (d) there will be no gravitational force inside a spherical shell of uniform density Solution: The correct answers are (a) planets will not have elliptic orbits (b) circular orbits of planets is not po...
Read More →Which of the following options is correct?
Question: Which of the following options is correct? (a) acceleration due to gravity decreases with increasing altitude (b) acceleration due to gravity increases with increasing depth (c) acceleration due to gravity increases with increasing latitude (d) acceleration due to gravity is independent of the mass of the earth Solution: The correct answers are (a) acceleration due to gravity decreases with increasing altitude (c) acceleration due to gravity increases with increasing latitude...
Read More →Choose the wrong option.
Question: Choose the wrong option. (a) inertial mass is a measure of the difficulty of accelerating a body by an external force whereas the gravitational mass is relevant in determining the gravitational force on it by an external mass (b) that the gravitational mass and inertial mass are equal is an experimental result (c) that the acceleration due to gravity on earth is the same for all bodies is due to the equality of gravitational mass and inertial mass (d) gravitational mass of a particle-l...
Read More →In our solar system, the inter-planetary
Question: In our solar system, the inter-planetary region has chunks of matter called asteroids. They (a) will not move around the sun since they have very small masses compared to the sun (b) will move in an irregular way because of their small masses and will drift away outer space (c) will move around the sun in closed orbits but not obey Keplers laws (d) will move in orbits like planets and obey Keplers laws Solution: The correct answer is (d) will move in orbits like planets and obey Kepler...
Read More →Both earth and moon are subject to
Question: Both earth and moon are subject to the gravitational force of the sun. as observed from the sun, the orbit of the moon (a) will be elliptical (b) will not be strictly elliptical because the total gravitational force on it is not central (c) is not elliptical but will necessarily be a closed curve (d) deviates considerably from being elliptical due to the influence of planets other than earth Solution: The correct answer is (b) will not be strictly elliptical because the total gravitati...
Read More →Satellites orbiting the earth have a finite
Question: Satellites orbiting the earth have a finite life and sometimes debris of satellites fall to the earth. This is because (a) the solar cells and batteries in satellites run out (b) the laws of gravitation predict a trajectory spiralling inwards (c) of viscous forces causing the speed of the satellite and hence height to gradually decrease (d) of collisions with other satellites Solution: The correct answer is (c) of viscous forces causing the speed of the satellite and hence height to gr...
Read More →Different points in the earth are at slightly
Question: Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the cm causing translation and a net torque at the cm causing translation and a net torque at the cm causing rotation around an axis through the cm. For the earth-sun system (a) the torque is zero (b) the torque causes the ea...
Read More →It is given that the Rolle's theorem holds for the function
Question: It is given that the Rolle's theorem holds for the function $f(x)=x^{3}+b x^{2}+c x, x \in[1,2]$ at the point $x=\frac{4}{3}$. Find the values of $b$ and $c$. Solution: As, the Rolle's theorem holds for the function $f(x)=x^{3}+b x^{2}+c x, x \in[1,2]$ at the point $x=\frac{4}{3}$ So, $f(1)=f(2)$ $\Rightarrow(1)^{3}+b(1)^{2}+c(1)=(2)^{3}+b(2)^{2}+c(2)$ $\Rightarrow 1+b+c=8+4 b+2 c$ $\Rightarrow 3 b+c+7=0$ ......(1) And $f^{\prime}\left(\frac{4}{3}\right)=0$ $\Rightarrow 3\left(\frac{4}...
Read More →Prove that
Question: Prove that $\left(\cos ^{4} x+\sin ^{4} x\right)=\frac{1}{2}\left(2-\sin ^{2} 2 x\right)$ Solution: To Prove: $\cos ^{4} x+\sin ^{4} x=\frac{1}{2}\left(2-\sin ^{2} 2 x\right)$ Taking LHS, $=\cos ^{4} x+\sin ^{4} x$ Adding and subtracting $2 \sin ^{2} x \cos ^{2} x$, we get $=\cos ^{4} x+\sin ^{4} x+2 \sin ^{2} x \cos ^{2} x-2 \sin ^{2} x \cos ^{2} x$ We know that, $a^{2}+b^{2}+2 a b=(a+b)^{2}$ $=\left(\cos ^{2} x+\sin ^{2} x\right)-2 \sin ^{2} x \cos ^{2} x$ $=(1)-2 \sin ^{2} x \cos ^{...
Read More →As observed from earth,
Question: As observed from earth, the sun appears to move in an approximately circular orbit. For the motion of another planet like mercury as observed from earth, this would (a) be similarly true (b) not be true because the force between earth and mercury is not inverse square law (c) not be true because the major gravitational force on mercury is due to sun (d) not be true because mercury is influenced by forces other than gravitational forces Solution: The correct answer is (c) not be true be...
Read More →The earth is an approximate sphere.
Question: The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity (a) will be directed towards the centre but not the same everywhere (b) will have the same value everywhere but not directed towards the centre (c) will be same everywhere in magnitude directed towards the centre (d) cannot be zero at any point Solution: The correct answer is (d) cannot be zero at any point...
Read More →Prove that
Question: Prove that cot x 2cot 2x = tan x Solution: To Prove: cot x 2cot 2x = tan x Taking LHS, $=\cot x-2 \cot 2 x \ldots$ (i) We know that $\cot x=\frac{\cos x}{\sin x}$ Replacing x by 2x, we get $\cot 2 x=\frac{\cos 2 x}{\sin 2 x}$ So, eq. (i) becomes $=\frac{\cos x}{\sin x}-2\left(\frac{\cos 2 x}{\sin 2 x}\right)$ $=\frac{\cos x}{\sin x}-2\left(\frac{\cos 2 x}{2 \sin x \cos x}\right)[\because \sin 2 x=2 \sin x \cos x]$ $=\frac{\cos x}{\sin x}-\left(\frac{\cos 2 x}{\sin x \cos x}\right)$ $=\...
Read More →Examine if Rolle's theorem is applicable to any one of the following functions.
Question: Examine if Rolle's theorem is applicable to any one of the following functions. (i) $f(x)=[x]$ for $x \in[5,9]$ (ii) $f(x)=[x]$ for $x \in[-2,2]$ Can you say something about the converse of Rolle's Theorem from these functions? Solution: By Rolle's theorem, for a function $f:[a, \mathrm{~b}] \rightarrow \mathbf{R}$, if (a) $f$ is continuous on $[a, b]$, (b) $f$ is differentiable on $(a, b)$ and (c) $f(a)=f(b)$, then there exists some $c \in(a, b)$ such that $f^{\prime}(c)=0$ Therefore,...
Read More →Two discs of moments of inertia I1 and I2
Question: Two discs of moments of inertia I1 and I2 about their respective axes and rotating with angular speed 1 and 2 are brought into contact face to face with their axes of rotation coincident. (a) does the law of conservation of angular momentum apply to the situation? why? (b) find the angular speed of the two-disc system (c) calculate the loss in kinetic energy of the system in the process (d) account for this loss Solution: (a) The law of conservation of angular momentum can be applied a...
Read More →Examine if Rolle's theorem is applicable to any one of the following functions.
Question: Examine if Rolle's theorem is applicable to any one of the following functions. (i) $f(x)=[x]$ for $x \in[5,9]$ (ii) $f(x)=[x]$ for $x \in[-2,2]$ Can you say something about the converse of Rolle's Theorem from these functions? Solution: By Rolle's theorem, for a function $f:[a, \mathrm{~b}] \rightarrow \mathbf{R}$, if (a) $f$ is continuous on $[a, b]$, (b) $f$ is differentiable on $(a, b)$ and (c) $f(a)=f(b)$, then there exists some $c \in(a, b)$ such that $f^{\prime}(c)=0$ Therefore,...
Read More →(n-1) equal point masses each of mass m are placed
Question: (n-1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector concerning the centre of the polygon. Find the position vector of the centre of mass. Solution: Given, $r_{c m}=\frac{(n-1) m b+m a}{(n-1) m+m}$ Where, rcmis the place where mass m is placed at the nth vertex rcm= 0 $\frac{(n-1) m b+m a}{(n-1) m+m}=0$ (n-1)mb + ma = 0 b = -ma/(n-1)m $\vec{b}=-\frac{\vec{a}}{n-1}$ Where, $\vec{b}$ is the position vector. Als...
Read More →A wheel in uniform motion about an axis passing
Question: A wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical equilibrium because no net external force or torque is required to sustain its motion. However, the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre. How do you reconcile this fact with the wheel being in equilibrium? How would you set a half-wheel into uniform motion about an axis passing through the...
Read More →Prove that
Question: Prove that $(\sin x-\cos x)^{2}=1-\sin 2 x$ Solution: To Prove: $(\sin x-\cos x)^{2}=1-\sin 2 x$ Taking LHS, $=(\sin x-\cos x)^{2}$ Using, $(a-b)^{2}=\left(a^{2}+b^{2}-2 a b\right)$ $=\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x$ $=\left(\sin ^{2} x+\cos ^{2} x\right)-2 \sin x \cos x$ $=1-2 \sin x \cos x\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\right]$ $=1-\sin 2 x[\because \sin 2 x=2 \sin x \cos x]$ = RHS LHS = RHS Hence Proved...
Read More →The vector sum of a system of non-collinear
Question: The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point? Solution: The vector sum of the torques is zero. But the resultant force is not zero. The mathematical explanation is given as: $G_{i} \sum_{i=1}^{n} F_{t} \neq 0$ $\tau=\tau_{1}+\tau_{2}+\ldots+\tau_{n}=\sum_{i=1...
Read More →Why does a solid sphere have a smaller moment
Question: Why does a solid sphere have a smaller moment of inertia than a hollow cylinder of same mass and radius, about an axis passing through their axes of symmetry? Solution: The moment of inertia is directly proportional to the square of the distance of the mass from the axis of the rotation. In a solid sphere, the distribution of mass takes place from the centre to the radius of the sphere. Whereas in a hollow cylinder, the mass is concentrated on the peripheral surface. Therefore, the mom...
Read More →Prove that
Question: Prove that $\cos 2 x+2 \sin ^{2} x=1$ Solution: To Prove: $\cos 2 x+2 \sin ^{2} x=1$ Taking LHS, $=\cos 2 x+2 \sin ^{2} x$ $=\left(2 \cos ^{2} x-1\right)+2 \sin ^{2} x\left[\because 1+\cos 2 x=2 \cos ^{2} x\right]$ $=2\left(\cos ^{2} x+\sin ^{2} x\right)-1$ $=2(1)-1\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\right]$ $=2-1$ $=1$ = RHS LHS = RHS Hence Proved...
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