Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(4 x-5)}{\left(2 x^{2}-5 x+1\right)} d x$ Solution:...
Read More →A small block oscillates back and forth on a smooth concave surface of radius R.
Question: A small block oscillates back and forth on a smooth concave surface of radius R. Find the time period of small oscillation. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(x+1)}{\left(x^{2}+2 x-3\right)} d x$ Solution:...
Read More →The maximum tension in the spring of an oscillating pendulum
Question: The maximum tension in the spring of an oscillating pendulum is double of the minimum tension. Find the angular amplitude. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{4 x}{\left(2 x^{2}+3\right)} d x$ Solution:...
Read More →A simple pendulum is constructed by hanging a heavy ball by
Question: A simple pendulum is constructed by hanging a heavy ball by a $5.0 \mathrm{~m}$ long string. It undergoes small oscillations. (a) How many oscillations does it make per second? (b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon $1.67 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left(\frac{2 \cos x-3 \sin x}{3 \cos x+2 \sin x}\right) d x$ Solution:...
Read More →A pendulum clock giving correct time at a place
Question: A pendulum clock giving correct time at a place where $g=9.800 \mathrm{~m} / \mathrm{s}^{2}$ is taken to another place where it loses 24 seconds during 24 hours. Find the value of $g$ at this new place. Solution:...
Read More →The pendulum of a certain clock has time period
Question: The pendulum of a certain clock has time period $2.04 \mathrm{~s}$. How fast or slow does the clock run during 24 hours? Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin 2 x}{\left(a^{2} \cos ^{2} x+b^{2} \sin ^{2} x\right)} d x$ Solution:...
Read More →The angle made by the string of a simple pendulum with
Question: The angle made by the string of a simple pendulum with the vertical depends on the time as $\theta=\frac{\pi}{90} \sin \left[\left(\pi s^{-1}\right) t\right]$. Find the length of the pendulum if $g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A pendulum having time period equal to two seconds is called a seconds pendulum.
Question: A pendulum having time period equal to two seconds is called a seconds pendulum. Those used in pendulum clocks are of this type. Find the length of a seconds pendulum at a place where $g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin 2 x}{\left(a^{2}+b^{2} \sin ^{2} x\right)} d x$ Solution:...
Read More →A uniform plate of mass M stays horizontally and symmetrically
Question: A uniform plate of mass M stays horizontally and symmetrically on two wheels rotating in opposite directions. The separation between the wheels is $L$. The friction is co-efficient between each wheel and the plate is $\mu$. Find the time period of oscillation of the plate if it is slightly displaced along its length and released. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: i. $\int \frac{(1+\tan x)}{(x+\log \sec x)} d x$ ii. $\int \frac{(1-\sin 2 x)}{\left(x+\cos ^{2} x\right)} d x$ Solution:...
Read More →All the surfaces shown in figure are frictionless.
Question: All the surfaces shown in figure are frictionless. The mass of the car is $\mathrm{M}$, that of the block is $\mathrm{m}$ and the spring has spring constant $k$. Initially, the car and the block are at rest and the spring is stretched through a length $x_{0}$ when the system is released. (a) Find the amplitudes of the simple harmonic motion of the block and of the car as seen from the road. (b) Find the time period(s) of the two simple harmonic motions. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left(\frac{1+\tan x}{1-\tan x}\right) d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin x}{(1+\cos x)} d x$ Solution:...
Read More →Find the time period of the motion of the particle shown in figure.
Question: Find the time period of the motion of the particle shown in figure. Neglect the small effect of the bend near the bottom. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sec ^{2} x}{(1+\tan x)} d x$ Solution:...
Read More →The left in figure moves at a speed v towards the right block
Question: The left in figure moves at a speed v towards the right block placed in equilibrium. All collisions to take place are elastic and the surfaces are frictionless. Show that the motions of the two blocks are periodic. Find the time period of those periodic motions. Neglect the widths of the block. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sqrt{(2+\log x)}}{x} d x$ Solution:...
Read More →A 1kg block is executing simple harmonic motion of amplitude 0.1m
Question: A $1 \mathrm{~kg}$ block is executing simple harmonic motion of amplitude $0.1 \mathrm{~m}$ on a smooth horizontal surface under the restoring force of a spring of spring constant $100 \mathrm{~N} / \mathrm{m}$. A block of mass $3 \mathrm{~kg}$ is gently placed on it at instant it passes through the mean position. Assuming that the two blocks moves together, find the frequency and amplitude of the motion. Solution:...
Read More →A rectangular plate of sides a and b is suspended from
Question: A rectangular plate of sides $a$ and $b$ is suspended from a ceiling by two parallel strings of length $L$ each. The separation between the strings is $d$. The plate is displaced slightly in its plane keeping the strings tight. Show that it will execute simple harmonic motion. Find the time period. Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period. Solution:...
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