Solve the previous problem if the pulley has a moment of inertia
Question: Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\left(\sqrt{1-x^{2}}\right) \sin ^{-1} x} d x$ Solution:...
Read More →The spring, the spring and the pulley shown in figure are in light.
Question: The spring, the spring and the pulley shown in figure are in light. Find the time period of the mass $m$. Solution:...
Read More →Find the elastic potential energy stored in each spring shown in figure,
Question: Find the elastic potential energy stored in each spring shown in figure, when the block is in equilibrium. Also find the time period of vertical oscillation of the block. Solution:...
Read More →The springs shown in the figure are all unstretched in the beginning
Question: The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} d x$ Solution:...
Read More →Repeat the exercise if the angle between each
Question: Repeat the exercise if the angle between each pair of springs is $120^{\circ}$ initially. Solution:...
Read More →Solve this following
Question: Evaluate the following integrals: $\int \frac{x \tan ^{-1} x^{2}}{\left(1+x^{4}\right)} d x$ Solution:...
Read More →A particle of mass m is attached to three springs A,
Question: A particle of mass $\mathrm{m}$ is attached to three springs $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of equal force constants $\mathrm{k}$ as shown in figure. If the particle is pushed slightly against the spring $C$ and released, find the time period of oscillation. Solution:...
Read More →The spring shown in figure is unstretched when a man starts pulling on the cord.
Question: The spring shown in figure is unstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block, (b) the energy stored in the spring when the block passes through the equilibrium and (c) the kinetic energy of the block at this position. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\tan \sqrt{x} \sec ^{2} \sqrt{x}}{\sqrt{x}} d x$ Solution:...
Read More →Find the time period of the oscillation of mass
Question: Find the time period of the oscillation of mass $\mathrm{m}$ in figures. What is the equivalent spring constant of the pair of strings in each case? Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sec x \log (\sec x+\tan x) d x$ Solution:...
Read More →In figure k=100N/m, M=1kg and F=10N.
Question: In figure $\mathrm{k}=100 \mathrm{~N} / \mathrm{m}, \mathrm{M}=1 \mathrm{~kg}$ and $\mathrm{F}=10 \mathrm{~N}$. (a) Find the compression of the spring in the equilibrium position. (b) A sharp blow by some external agent imparts a speed of $2 \mathrm{~m} / \mathrm{s}$ to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion. (d) Find the amplitude. (...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int 2 x \sin \left(x^{2}+1\right) d x$ Solution:...
Read More →Solve this following
Question: Evaluate the following integrals: $\int \frac{\cot x}{\log (\sin x)} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot x \log (\sin x) d x$ Solution:...
Read More →Solve this following
Question: Evaluate the following integrals: $\int \frac{e^{2 x}}{\left(e^{2 x}-2\right)} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{\left(e^{x}+e^{-x}\right)}$ Solution:...
Read More →The block of mass m1 shown in figure is fastened to the spring and the block of
Question: The block of mass $m_{1}$ shown in figure is fastened to the spring and the block of mass $m_{2}$ is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance $\left(\frac{2}{k}\right)\left(m_{1}+m_{2}\right) g \sin \theta$ against the spring and released. Find the position where the two blocks separate. (c) What is common speed of blocks at the time of separation? Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x^{2}} \cos \left(\frac{1}{x}\right) d x$ Solution:...
Read More →A small block of mass m is kept on a bigger block of mass M
Question: A small block of mass $m$ is kept on a bigger block of mass $M$ which is attached to a vertical spring of spring constant $k$ as shown in figure. The system oscillates vertically. (a) Find the resultant force on the smaller block when it is displaced through a distance $x$ above its equilibrium position. (b) Find the normal force on the smaller block at this position. What is the force smallest in the magnitude? (c) What can be the maximum amplitude with which the two blocks may oscill...
Read More →A spring stores 5J of energy when stretched by 25cm.
Question: A spring stores $5 \mathrm{~J}$ of energy when stretched by $25 \mathrm{~cm}$. It is kept vertical with the lower end fixed. A block fastened to its other end is made to undergo small oscillations. If the block makes 5 oscillations each second, what is the mass of the block? Solution:...
Read More →A body of mass 2kg suspended through a vertical spring executes simple
Question: A body of mass $2 \mathrm{~kg}$ suspended through a vertical spring executes simple harmonic motion of periods. If the oscillations are stopped and the body hangs in equilibrium, find the potential energy stored in the spring. Solution:...
Read More →A block of mass 0.5kg hanging from a vertical spring executes simple harmonic
Question: A block of mass $0.5 \mathrm{~kg}$ hanging from a vertical spring executes simple harmonic motion of amplitude $0.1 \mathrm{~m}$ and time period $0.314 \mathrm{~s}$. Find the maximum force exerted by the spring on the block. Solution:...
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