The centre of gravity of a body on the earth
Question: The centre of gravity of a body on the earth coincides with its centre of mass for a small object whereas for an extended object it may not. What is the qualitative meaning of small and extended in this regard? For which of the following the two coincides? A building, a pond, a lake, a mountain? Solution: The centre of gravity is the geometric centre whereas the centre of mass is the mass of the point where the entire mass of the body is considered. The object is said to be small when ...
Read More →The net external torque on a system
Question: The net external torque on a system of particle about any axis is zero. Which of the following are compatible with it? (a) the forces may be acting radially from a point on the axis (b) the forces may be acting on the axis of rotation (c) the forces may be acting parallel to the axis of rotation (d) the torque caused by some forces may be equal and opposite to that caused by other forces Solution: The correct is all the four options...
Read More →If the solve the problem
Question: If $f:[-5,5] \rightarrow R$ is differentiable and if $f(x)$ doesnot vanish anywhere, then prove that $f(-5) \pm f(5)$ Solution: It is given that $f:[-5,5] \rightarrow \mathbf{R}$ is a differentiable function. Every differentiable function is a continuous function. Thus, (a) $f$ is continuous in $[-5,5]$. (b) $f$ is differentiable in $(-5,5)$. Therefore, by the Mean Value Theorem, there exists $c \in(-5,5)$ such that $f^{\prime}(c)=\frac{f(5)-f(-5)}{5-(-5)}$ $\Rightarrow 10 f^{\prime}(c...
Read More →Prove that
Question: Prove that cosec 2x + cot 2x = cot x Solution: To Prove: cosec 2x + cot 2x = cot x Taking LHS, = cosec 2x + cot 2x (i) We know that, $\operatorname{cosec} x=\frac{1}{\sin x} \ \cot x=\frac{\cos x}{\sin x}$ Replacing x by 2x, we get $\operatorname{cosec} 2 x=\frac{1}{\sin 2 x} \ \cot 2 x=\frac{\cos 2 x}{\sin 2 x}$ So, eq. (i) becomes $=\frac{1}{\sin 2 x}+\frac{\cos 2 x}{\sin 2 x}$ $=\frac{1+\cos 2 x}{\sin 2 x}$ $=\frac{2 \cos ^{2} x}{\sin 2 x}\left[\because 1+\cos 2 x=2 \cos ^{2} x\righ...
Read More →Choose the correct alternatives:
Question: Choose the correct alternatives: (a) for a general rotational motion, angular momentum L and angular velocity need not be parallel (b) for a rotational motion about a fixed axis, angular momentum L and angular velocity are always parallel (c) for a general translational motion, momentum p and velocity v is always parallel (d) for a general translational motion, acceleration a and velocity v are always parallel Solution: The correct answer is (a) for a general rotational motion, angular...
Read More →At what points on the following curves, is the tangent parallel to x-axis?
Question: At what points on the following curves, is the tangent parallel tox-axis? (i) $y=x^{2}$ on $[-2,2]$ (ii) $y=e^{1-x^{2}}$ on $[-1,1]$ (iii) $y=12(x+1)(x-2)$ on $[-1,2]$. Solution: (i) Let $f(x)=x^{2}$ Since $f(x)$ is a polynomial function, it is continuous on $[-2,2]$ and differentiable on $(-2,2)$. Also, $f(2)=f(-2)=4$ Thus, all the conditions of Rolle's theorem are satisfied. Consequently, there exists at least one point $c \in(-2,2)$ for which $f^{\prime}(c)=0$. But $f^{\prime}(c)=0 ...
Read More →Prove that
Question: Prove that $\sin 2 x(\tan x+\cot x)=2$ Solution: To Prove: $\sin 2 x(\tan x+\cot x)=2$ Taking LHS, sin 2x(tan x + cot x) We know that, $\tan \theta=\frac{\sin \theta}{\cos \theta} \ \cot \theta=\frac{\cos \theta}{\sin \theta}$ $=\sin 2 x\left(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\right)$ $=\sin 2 x\left(\frac{\sin x(\sin x)+\cos x(\cos x)}{\cos x \sin x}\right)$ $=\sin 2 x\left(\frac{\sin ^{2} x+\cos ^{2} x}{\cos x \sin x}\right)$ We know that, $\sin 2 x=2 \sin x \cos x$ $=2 \sin...
Read More →At what points on the following curves, is the tangent parallel to x-axis?
Question: At what points on the following curves, is the tangent parallel tox-axis? (i) $y=x^{2}$ on $[-2,2]$ (ii) $y=e^{1-x^{2}}$ on $[-1,1]$ (iii) $y=12(x+1)(x-2)$ on $[-1,2]$. Solution: (i) Let $f(x)=x^{2}$ Since $f(x)$ is a polynomial function, it is continuous on $[-2,2]$ and differentiable on $(-2,2)$. Also, $f(2)=f(-2)=4$ Thus, all the conditions of Rolle's theorem are satisfied. Consequently, there exists at least one point $c \in(-2,2)$ for which $f^{\prime}(c)=0$. But $f^{\prime}(c)=0 ...
Read More →A merry-go-round, made of a ring-like platform
Question: A merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed . A person of mass M is standing on it. At one instant, the person jumps off the ground, radially away from the centre of the round. The speed of the round afterwards is (a) 2 (b) (c) /2 (d) 0 Solution: The correct answer is (a) 2...
Read More →The density of a non-uniform rod of length 1 m is given by
Question: The density of a non-uniform rod of length 1 m is given by $\rho(x)=a\left(1+b x^{2}\right)$ where $\mathrm{a}$ and $\mathrm{b}$ are constant and $0 \leq \mathrm{x} \leq 1$. The centre of mass of the rod will be at (a) $\frac{3(2+b)}{4(3+b)}$ (b) $\frac{4(2+b)}{3(3+b)}$ (c) $\frac{3(3+b)}{4(2+b)}$ (d) $\frac{4(3+b)}{3(2+b)}$ Solution: The correct answer is (a) $\frac{3(2+b)}{4(3+b)}$...
Read More →Prove that
Question: Prove that $\frac{\tan 2 x}{1+\sec 2 x}=\tan x$ Solution: To Prove: $\frac{\tan 2 x}{1+\sec 2 x}=\tan x$ Taking LHS, $=\frac{\frac{\sin 2 x}{\cos 2 x}}{1+\frac{1}{\cos 2 x}}\left[\because \tan \theta=\frac{\sin \theta}{\cos \theta} \ \sec \theta=\frac{1}{\cos \theta}\right]$ $=\frac{\sin 2 x}{\cos 2 x\left(\frac{\cos 2 x+1}{\cos 2 x}\right)}$ $=\frac{\sin 2 x}{1+\cos 2 x}$ $=\frac{2 \sin x \cos x}{1+\cos 2 x}[\because \sin 2 x=2 \sin x \cos x]$ $=\frac{2 \sin x \cos x}{2 \cos ^{2} x}\l...
Read More →In problem 7.5. the CM of the plate
Question: In problem 7.5. the CM of the plate is now in the following quadrant of the x-y plane (a) I (b) II (c) III (d) IV Solution: The correct answer is (c) III...
Read More →When a disc rotates with uniform angular velocity,
Question: When a disc rotates with uniform angular velocity, which of the following is not true? (a) the sense of rotation remains the same (b) the orientation of the axis of rotation remains the same (c) the speed of rotation is non-zero and remains the same (d) the angular acceleration is non-zero and remains the same Solution: The correct answer is (d) the angular acceleration is non-zero and remains same...
Read More →For which of the following does
Question: For which of the following does the centre of mass lie outside the body? (a) a pencil (b) a shotput (c) a dice (d) a bangle Solution: The correct answer is (d) a bangle...
Read More →Prove that
Question: Prove that $\frac{\sin 2 x}{1-\cos 2 x}=\cot x$ Solution: To Prove: $\frac{\sin 2 x}{1-\cos 2 x}=\tan x$ Taking LHS, $=\frac{\sin 2 x}{1-\cos 2 x}$ $=\frac{2 \sin x \cos x}{1-\cos 2 x}[\because \sin 2 x=2 \sin x \cos x]$ $=\frac{2 \sin x \cos x}{2 \sin ^{2} x}\left[\because 1-\cos 2 x=2 \sin ^{2} x\right]$ $=\frac{\cos x}{\sin x}$ $=\cot x\left[\because \cot \theta=\frac{\cos \theta}{\sin \theta}\right]$ = RHS LHS = RHS Hence Proved...
Read More →Prove that
Question: Prove that $\frac{\sin 2 x}{1+\cos 2 x}=\tan x$ Solution: To Prove: $\frac{\sin 2 x}{1+\cos 2 x}=\tan x$ Taking LHS, $=\frac{\sin 2 x}{1+\cos 2 x}$ $=\frac{2 \sin x \cos x}{1+\cos 2 x}[\because \sin 2 x=2 \sin x \cos x]$ $=\frac{2 \sin x \cos x}{2 \cos ^{2} x}\left[\because 1+\cos 2 x=2 \cos ^{2} x\right]$ $=\frac{\sin x}{\cos x}$ $=\tan \mathrm{x}\left[\because \tan \theta=\frac{\sin \theta}{\cos \theta}\right]$ = RHS LHS = RHS Hence Proved...
Read More →A balloon filled with helium rises against
Question: A balloon filled with helium rises against gravity increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect the viscous drag of air and assume that the density of air is constant. Solution: When the dragging viscous force of the air on the balloon is neglected, then the net buoyant force = vpg Where v is the volume of air displaced p is the net density upward After ...
Read More →Using Rolle's theorem, find points on the curve
Question: Using Rolle's theorem, find points on the curve $y=16-x^{2}, x \in[-1,1]$, where tangent is parallel to $x$-axis. Solution: The equation of the curve is $y=16-x^{2}$ ....(1) Let $\mathrm{P}\left(x_{1}, y_{1}\right)$ be a point on it where the tangent is parallel to the $x$-axis. Then, $\left(\frac{d y}{d x}\right)_{P}=0$ ......(2) Differentiating (1) with respect to $x$, we get $\frac{d y}{d x}=-2 x$ $\Rightarrow\left(\frac{d y}{d x}\right)_{P}=-2 x_{1}$ $\Rightarrow-2 x_{1}=0 \quad($ ...
Read More →Two identical steel cubes collide head-on face
Question: Two identical steel cubes collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Youngs modulus for steel = Y = 2 1011N/m2. Solution: Y = stress/strain Y = FL/A∆L WD = F∆L KE = 5 10-4J WD = KE ∆L = 5 10-7m...
Read More →Using Rolle's theorem, find points on the curve
Question: Using Rolle's theorem, find points on the curve $y=16-x^{2}, x \in[-1,1]$, where tangent is parallel to $x$-axis. Solution: The equation of the curve is $y=16-x^{2}$ ....(1) Let $\mathrm{P}\left(x_{1}, y_{1}\right)$ be a point on it where the tangent is parallel to the $x$-axis. Then, $\left(\frac{d y}{d x}\right)_{P}=0$ ......(2) Differentiating (1) with respect to $x$, we get $\frac{d y}{d x}=-2 x$ $\Rightarrow\left(\frac{d y}{d x}\right)_{P}=-2 x_{1}$ $\Rightarrow-2 x_{1}=0 \quad($ ...
Read More →Prove that
Question: Prove that $\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x$ Solution: To Prove: $\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x$ Taking LHS, $=\frac{\cos 2 x}{\cos x-\sin x}$ $=\frac{\cos ^{2} x-\sin ^{2} x}{\cos x-\sin x}\left[\because \cos 2 x=\cos ^{2} x-\sin ^{2} x\right]$ Using, $\left(a^{2}-b^{2}\right)=(a-b)(a+b)$ $=\frac{(\cos x-\sin x)(\cos x+\sin x)}{(\cos x-\sin x)}$ $=\cos x+\sin x$ = RHS LHS = RHS...
Read More →A rocket accelerates straight up by ejecting
Question: A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = (1/2) ∆m u2in this time interval. Solution: M is the mass of the rocket at any time t v is the velocity of the rocket ∆m is the mass of the gas that is ejected during the time interval ∆t Therefore, K = 1/2 u2∆m...
Read More →A curved surface as shown in the figure.
Question: A curved surface as shown in the figure. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from one by one from A which is at a slightly greater height than C. with the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction. (a) for which balls is total mechanical energy conserved? b) which ball can reach D? c) for balls whi...
Read More →On complete combustion, a litre of petrol gives
Question: On complete combustion, a litre of petrol gives off heat equivalent to 3 107J. In a test drive a car weighing 1200 kg, including the mass of driver, runs 15 km per litre while moving with a uniform speed on a surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were 0.5. Solution: Efficiency of the car engine = 0.5 Energy given by the car with 1 litre of petrol = 1.5 107 WD = 1.5 107 f = 103N...
Read More →An adult weighing 600 N raises the centre
Question: An adult weighing 600 N raises the centre of gravity of his body by 0.25 m while taking each step of 1 m length in jogging. If he jogs for 6 km, calculate the energy utilized by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting 10% of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilized for jogging. Solution: The ...
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