Two small balls, each of mass m are connected by a light rigid rod of length L.
Question: Two small balls, each of mass $m$ are connected by a light rigid rod of length $L$. The system is suspended from its center by a thin wire of torsional constant $k$. The rod is rotated about the wire through an angle $\theta_{0}$ and released. Find the tension in the rod as the system passes through the mean position. Solution:...
Read More →A uniform of disc of mass m and radius r is suspended through a
Question: A uniform of disc of mass $m$ and radius $r$ is suspended through a wire attached to its center. If the time period of the torsional constant of the wire. Solution:...
Read More →A closed circular wire hung on a nail in a wall undergoes small oscillations
Question: A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude $2^{\circ}$ and time period 2 s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspension as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an extreme position. Take $g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A hollow sphere of radius 2cm is attached to an 18cm
Question: A hollow sphere of radius $2 \mathrm{~cm}$ is attached to an $18 \mathrm{~cm}$ long thread to make a pendulum. Find the time period of oscillation of this pendulum. How does it differ from the time period calculated using the formula for a simple pendulum? Solution:...
Read More →A uniform disc of radius r is to be suspended through a small
Question: A uniform disc of radius $r$ is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the center for it to have minimum time period? Solution:...
Read More →A uniform rod of length I is suspended by an end and is made
Question: A uniform rod of length $I$ is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the rod. Solution:...
Read More →Find the time period of small oscillations of the following systems.
Question: Find the time period of small oscillations of the following systems. (a) A meter stick suspended through the $20 \mathrm{~cm}$ mark. (b) A ring of mass $m$ and radius $r$ suspended through a point on its periphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass $m$ and radius $r$ suspended through a point $r / 2$ away from the center. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}}{\left(1+x^{6}\right)} d x$ Solution:...
Read More →The ear ring of a lady shown in figure has a
Question: The ear ring of a lady shown in figure has a $3 \mathrm{~cm}$ long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at $4 \mathrm{~m} / \mathrm{s}$ in a circle of radius $2 \mathrm{~cm}$. Find the time period of small oscillations of the ear-ring. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{(\sqrt{1-3 x}-\sqrt{5-3 x})}$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{(\sqrt{x+a}+\sqrt{x+b})}$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(2 x-1)}{\sqrt{x^{2}-x-1}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(2 x+3)}{\sqrt{x^{2}+3 x-2}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin x}{(1+\cos x)^{2}} d x$ Solution:...
Read More →A simple pendulum of length I is suspended from
Question: A simple pendulum of length $I$ is suspended from the ceiling of the car moving with a speed $v$ on a circular horizontal road of radius $\mathrm{r}$. (a) Find the tension in the string when it is at rest with respect to car. (b) Find the time period of small oscillation. Solution:...
Read More →A simple pendulum fixed in a car has a time period of 4 seconds
Question: A simple pendulum fixed in a car has a time period of 4 seconds when the car is moving uniformly on a horizontal road. When the accelerator is passed, the time period changes to $3.99$ seconds. Making an approximation analysis, find the acceleration of the car. Solution:...
Read More →A simple pendulum of length 1 feet suspended from the ceiling of
Question: A simple pendulum of length 1 feet suspended from the ceiling of an elevator takes $\pi / 3$ seconds to complete one oscillation. Find the acceleration of the elevator. Solution:...
Read More →A simple pendulum of length l is suspended through the ceiling of an elevator.
Question: A simple pendulum of length $\mathrm{I}$ is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with an acceleration $a_{0}(\mathrm{~b})$ is going down with an acceleration $a_{0}$ and (c) is moving with a uniform velocity. Solution:...
Read More →Assume that a tunnel is dug along a cord of the earth,
Question: Assume that a tunnel is dug along a cord of the earth, at a perpendicular distance $R / 2$ from the earth's center where $R$ is the radius of the earth. The wall of the tunnel is frictionless. (a) Find the gravitational force exerted by the earth on a particle of mass $m$ placed in the tunnel at a distance $x$ from the center of the tunnel. (b) Find the component of this force along the tunnel and perpendicular to the tunnel. (c) Find the normal force exerted by the wall on the particl...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(1+\cos x)}{(x+\sin x)^{3}} d x$ Solution:...
Read More →Assume that a tunnel is dug across the earth (radius=R) passing through its center.
Question: Assume that a tunnel is dug across the earth (radius=R) passing through its center. Find the time a particle takes to cover the length of the tunnel if (a) it is projected into the tunnel with a speed of $\sqrt{g R}$ (b) it is released from a height $\mathrm{R}$ above the tunnel (c) it is thrown vertically upward along the length of tunnel with a speed of $\sqrt{g R}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sec x \operatorname{cosec} x}{\log (\tan x)} d x$ Solution:...
Read More →A simple pendulum of length 40cm is taken inside a deep mine.
Question: A simple pendulum of length $40 \mathrm{~cm}$ is taken inside a deep mine. Assume for the time being that the mine is $1600 \mathrm{~km}$ deep. Calculate the time period of the pendulum there. Radius of the earth $=6400 \mathrm{~km}$. Solution:...
Read More →A spherical ball of mass m and radius r rolls without
Question: A spherical ball of mass $m$ and radius $r$ rolls without slipping on a rough concave surface of large radius $R$. It makes small oscillations about the lowest point. Find the time period. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\left(9 x^{2}-4 x+5\right)}{\left(3 x^{3}-2 x^{2}+5 x+1\right)} d x$ Solution:...
Read More →