A body of mass m is situated in a potential
Question: A body of mass m is situated in a potential field U(x) = U0 (1 cos x) when U0 and are constants. Find the time period of small oscillations. Solution: The time period of the small oscillation is T = 2/ m/U0...
Read More →Find the displacement of a simple
Question: Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator. Solution: The mean position of the oscillator at x with PE = 1/2 kx2 PE = -m2x2 When PE is maximum, KE = 0 at x = A E = 1/2 m2A2 PE = 1/2 total energy x2= 1/2 A2 x = A/2...
Read More →Find a point on the curve
Question: Find a point on the curve $y=x^{3}-2 x^{2}-2 x$ at which the tangent lines are parallel to the line $y=2 x-3$. Solution: Given: The curve $y=x^{3}-2 x^{2}-2 x$ and a line $y=2 x-3$ First, we will find The Slope of tangent $y=x^{3}-2 x^{2}-2 x$ $\frac{d y}{d x}=\frac{d}{d x}\left(x^{3}\right)-\frac{d}{d x}\left(2 x^{2}\right)-\frac{d}{d x}(2 x)$ $\Rightarrow \frac{d y}{d x}=3 x^{3}-1-2 \times 2\left(x^{2}-1\right)-2 \times x^{1-1}$ $\Rightarrow \frac{d y}{d x}=3 x^{2}-4 x-2 \ldots(1)$ $...
Read More →Show that the motion of a particle
Question: Show that the motion of a particle represented by y = sin t cos t is simple harmonic with a period of 2/. Solution: A function will represent SHM and is written as sin (2/T t + ϕ) y = sin t cos t y = 2 sin (t /4) The standard SHM has y = a s sin (2/T t + ϕ) T = 2/...
Read More →The length of a second’s pendulum
Question: The length of a seconds pendulum on the surface of the earth is 1 m. What will be the length of a seconds pendulum on the moon? Solution: The time period of a simple pendulum, T = 2l/g Where l is the length of the pendulum and g is the acceleration due to gravity Therefore, lmoon/gmoon= learth/gearth = 1/6 m...
Read More →Show that for a particle executing SHM,
Question: Show that for a particle executing SHM, velocity, and displacement have a phase difference of /2. Solution: The displacement equation of SHM, x = a cos ꞷt Velocity, v = dx/dt = -aꞷ sin ꞷt The phase displacement, ϕ1 = ꞷt Phase velocity, ϕ2 = /2 + ꞷt Therefore, difference in phase of displacement = ϕ2 ϕ1 = /2...
Read More →What is the ratio between the distance
Question: What is the ratio between the distance travelled by the oscillator in one time period and amplitude? Solution: Distance travelled by the oscillator in one time period = 4A Where A is the amplitude of the oscillation Therefore, required ration = 4A/A = 4/1 = 4:1...
Read More →What is the ratio of maximum acceleration
Question: What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic oscillator? Solution: The displacement equation of SHM is x = a sin (ꞷt + ϕ) The velocity of the particle, v = dx/dt = d a sin (ꞷt + ϕ)/dt Maximum velocity, |v|max = aꞷ Acceleration, A = dv/dt = aꞷ2sin (ꞷt + ϕ) Maximum acceleration, |A|max = ꞷ2a |v|max/|A|max = ꞷ...
Read More →A body is performing SHM. Then its
Question: A body is performing SHM. Then its (a) average total energy per cycle is equal to its maximum kinetic energy (b) average kinetic energy per cycle is equal to half of its maximum kinetic energy (c) mean velocity over a complete cycle is equal to 2/ times of its maximum velocity (d) root mean square velocity is 1/2 times of its maximum velocity Solution: The correct answers are (a) average total energy per cycle is equal to its maximum kinetic energy (b) average kinetic energy per cycle ...
Read More →Which of the following statements
Question: Which of the following statements is/are true for a simple harmonic oscillator? (a) force acting is directly proportional to the displacement from the mean position and opposite to it (b) motion is periodic (c) acceleration of the oscillator is constant (d) the velocity is periodic Solution: The correct answers are (a) force acting is directly proportional to the displacement from the mean position and opposite to it (b) motion is periodic (d) the velocity is periodic...
Read More →Find a point on the curve
Question: Find a point on the curve $y=x^{3}-3 x$ where the tangent is parallel to the chord joining $(1,-2)$ and $(2,2)$. Solution: Given: The curve $y=x^{3}-3 x$ First, we will find the Slope of the tangent $y=x^{3}-3 x$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{3}\right)-\frac{\mathrm{d}}{\mathrm{dx}}(3 \mathrm{x})$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \mathrm{x}^{3}-1-3\left(\frac{\mathrm{dx}}{\mathrm{dx}}\right)$ $\Rightarrow \frac{\mathrm{dy...
Read More →The motion of a ball bearing inside a
Question: The motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is (a) simple harmonic motion (b) non-periodic motion (c) periodic motion (d) periodic but not SHM Solution: The correct answers are (a) simple harmonic motion (b) non-periodic motion...
Read More →The rotation of the earth about
Question: The rotation of the earth about its axis is (a) periodic motion (b) simple harmonic motion (c) periodic but not simple harmonic motion (d) non-periodic motion Solution: The correct answers are (a) periodic motion (c) periodic but not simple harmonic motion...
Read More →A particle executing SHM has
Question: A particle executing SHM has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s2. The period of oscillation is (a) s (b) /2 s (c) 2 s (d) /t s Solution: The correct answer (a) s...
Read More →If the tangent to the curve
Question: If the tangent to the curve $y=x^{3}+a x+b$ at $(1,-6)$ is parallel to the line $x-y+5=0$, find $a$ and $b$ Solution: Given: The Slope of the tangent to the curve $y=x^{3}+a x+b$ at $(1,-6)$ First, we will find The Slope of tangent $y=x^{3}+a x+b$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{3}\right)+\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{ax})+\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{b})$ $\Rightarrow \frac{d y}{d x}=3 x^{3}-1+a\left(\frac{d x}{d x}\r...
Read More →The equation of motion of
Question: The equation of motion of a particle is x = a cos (t)2. The motion is (a) periodic but not oscillatory (b) periodic and oscillatory (c) oscillatory but not periodic (d) neither periodic nor oscillatory Solution: The correct answer (c) oscillator but not periodic...
Read More →If A + B + C = π, prove that
Question: If A + B + C = , prove that $\tan 2 A+\tan 2 B+\tan 2 C=\tan 2 A \tan 2 B \tan 2 C$ Solution: = tan 2A + tan 2B + tan 2C Since $A+B+C=\pi$ $A+B=\pi-C$ $2 A+2 B=2 \pi-2 C$ $\operatorname{Tan}(2 A+2 B)=\tan (2 \pi-2 C)$ Since $\tan (2 \pi-C)=-\tan C$ $\operatorname{Tan}(2 A+2 B)=-\tan 2 C$ Now using formula, $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$ $\frac{\tan 2 \mathrm{~A}+\tan 2 \mathrm{~B}}{1-\tan 2 \mathrm{~A} \tan 2 \mathrm{~B}}=-\tan 2 \mathrm{C}$ Tan $2 A+\tan 2 B=-\tan ...
Read More →The displacement of a particle varies
Question: The displacement of a particle varies with time according to the relation y = a sin t + b cos t (a) the motion is oscillatory but not SHM (b) the motion is SHM with amplitude a + b (c) the motion is SHM with amplitude a2+ b2 (d) the motion is SHM with amplitude a2+ b2 Solution: The correct answer is (d) the motion is SHM with amplitude a2+ b2...
Read More →A particle is acted simultaneously by mutually
Question: A particle is acted simultaneously by mutually perpendicular simple harmonic motions x = a cos t and y = a sin t. The trajectory of motion of the particle will be (a) an ellipse (b) a parabola (c) a circle (d) a straight line Solution: The correct answer is (c) a circle...
Read More →If A + B + C = π, prove that
Question: If A + B + C = , prove that $\sin ^{2} \frac{A}{2}+\sin ^{2} \frac{B}{2}+\sin ^{2} \frac{C}{2}=1-2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ Solution: $=\sin ^{2} \frac{A}{2}+\sin ^{2} \frac{B}{2}+\sin ^{2} \frac{C}{2}$ Using formula , $\frac{1-\cos 2 \mathrm{~A}}{2}=\sin ^{2} \mathrm{~A}$ $=\frac{1-\cos A}{2}+\frac{1-\cos B}{2}+\frac{1-\cos C}{2}$ $=\frac{1-\cos A+1-\cos B+1-\cos C}{2}$ $=\frac{3-\cos A-\cos B-\cos C}{2}$ Using, $\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\righ...
Read More →Motion of an oscillating liquid
Question: Motion of an oscillating liquid column in a U-tube is (a) periodic but not simple harmonic (b) non-periodic (c) simple harmonic and time period is independent of the density of the liquid (d) simple harmonic and time period is directly proportional to the density of the liquid Solution: The correct answer is (c) simple harmonic and time period is independent of the density of the liquid...
Read More →If A + B + C = π, prove that
Question: If A + B + C = , prove that $\sin ^{2} A-\sin ^{2} B+\sin ^{2} C=2 \sin A \cos B \sin C$ Solution: $=\sin ^{2} A-\sin ^{2} B+\sin ^{2} C$ Using formula, $\frac{1-\cos 2 A}{2}=\sin ^{2} A$ $=\frac{1-\cos 2 \mathrm{~A}}{2}-\frac{1-\cos 2 \mathrm{~B}}{2}+\frac{1-\cos 2 \mathrm{C}}{2}$ $=\frac{1-\cos 2 \mathrm{~A}-1+\cos 2 \mathrm{~B}+1-\cos 2 \mathrm{C}}{2}$ $=\frac{1-\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}-\cos 2 \mathrm{C}}{2}$ Using , $\cos A-\cos B=2 \sin \left(\frac{A+B}{2}\right) \sin...
Read More →The relation between acceleration
Question: The relation between acceleration and displacement of four particles are given below: (a) ax= +2x (b) ax= +2x2 (c) ax= -2x2 (d) ax= -2x Which one of the particles is executing simple harmonic motion? Solution: The correct answer is (d) ax= -2x...
Read More →The displacement of a particle is represented
Question: The displacement of a particle is represented by the equation y = sin3t. The motion is (a) non-periodic (b) periodic but not simple harmonic (c) simple harmonic with period 2/ (d) simple harmonic with period / Solution: The correct answer is (c) simple harmonic with period 2/...
Read More →Find the values of a and b
Question: Find the values of $a$ and $b$ if the The Slope of the tangent to the curve $x y+a x+b y=2$ at $(1,1)$ is 2 . Solution: Given: The Slope of the tangent to the curve $x y+a x+b y=2$ at $(1,1)$ is 2 First, we will find The Slope of tangent we use product rule here, $\therefore \frac{\mathrm{d}}{\mathrm{dx}}(U V)=U \times \frac{\mathrm{dV}}{\mathrm{dx}}+\mathrm{V} \times \frac{\mathrm{dU}}{\mathrm{dx}}$ $\Rightarrow x y+a x+b y=2$ $\Rightarrow x \times \frac{d}{d x}(y)+y \times \frac{d}{d...
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