A block suspended from a vertical spring is in equilibrium.
Question: A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum i.e., a pendulum having frequency same as that of the block. Solution:...
Read More →The pendulum of a clock is replaced by a spring-mass system
Question: The pendulum of a clock is replaced by a spring-mass system with the spring having spring constant $0.1 \mathrm{~N} / \mathrm{m}$. What mass should be attached to the spring? Solution:...
Read More →Consider a simple harmonic motion of time period T.
Question: Consider a simple harmonic motion of time period $T$. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude. Solution:...
Read More →Consider a particle moving in simple harmonic motion according to the equation.
Question: Consider a particle moving in simple harmonic motion according to the equation. $x=2.0 \cos \left(50 \pi t+\tan ^{-1} 0.75\right)$ where $x$ is in centimeter and $t$ in second. The motion is started at $t=0$. (a) When does the particle come to rest for the first time? (b) When does the acceleration have its maximum magnitude for the first time? (c) When does the particle come to rest for the second time? Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x^{2}} e^{-1 / x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cos ^{3} x d x$ Solution:...
Read More →The equation of motion of a particle started at
Question: The equation of motion of a particle started at $\mathrm{t}=0$ is given by $x=5 \sin (20 t+\pi / 3)$ where $\mathrm{x}$ is in centimeter and t in second. What does the particle (a) first come to rest. (b) First have zero acceleration. (c) First have maximum speed? Solution:...
Read More →A particle having mass 10g oscillates according to the equation
Question: A particle having mass $10 \mathrm{~g}$ oscillates according to the equation $x=(2.0 \mathrm{~cm}) \sin \left[\left(100 \mathrm{~s}^{-1}\right) t+\frac{\pi}{6}\right]$. Find (a) the amplitude, the time period and the spring constant (b) the position, the velocity and the acceleration at $t=0$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin (a x+b) \cos (a x+b) d x$ Solution:...
Read More →The maximum speed and acceleration of a particle executing simple harmonic motion
Question: The maximum speed and acceleration of a particle executing simple harmonic motion are $10 \mathrm{~cm} / \mathrm{s}$ and $50^{\mathrm{cm}} / \mathrm{s}^{2}$. Find the position(s) of the particle when the speed is $8 \mathrm{~cm} / \mathrm{s}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int e^{\cos ^{2} x} \sin 2 x d x$ Solution:...
Read More →A particle executes simple harmonic motion with an amplitude of
Question: A particle executes simple harmonic motion with an amplitude of $10 \mathrm{~cm}$. At what distance from the mean position are the kinetic and potential energies are equal? Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int e^{\tan x} \sec ^{2} x d x$ Solution: $=e^{\tan x}+c$...
Read More →The position, velocity and acceleration of a particle executing simple
Question: The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes $2 \mathrm{~cm}$, $1 \mathrm{~m} / \mathrm{s}$ and $10 \mathrm{~m} / \mathrm{s}^{2}$ at a certain instant. Find the amplitude and the time period of the motion. Solution:...
Read More →A particle executes simple harmonic motion with an amplitude of
Question: A particle executes simple harmonic motion with an amplitude of $10 \mathrm{~cm}$ and time period $6 \mathrm{~s}$. At $t=0$ it is at position $x=5 \mathrm{~m}$ going towards positive $x$-direction. Write the equation for the displacement $x$ at time $t$. Find the magnitude of the acceleration of the particle at $t=4 \mathrm{~s}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(\log x)^{2}}{x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(x+1)(x+\log x)^{2}}{x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x \log x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\operatorname{cosec}^{2}(\log x)}{x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos (\log x)}{x} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin \left(2 \tan ^{-1} x\right)}{\left(1+x^{2}\right)} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(\sqrt{\cos x}) \sin x d x$ Solution:...
Read More →Solve this following
Question: Evaluate the following integrals: $\int \sin ^{3} x \cos x d x$ Solution:...
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