Conical Pendulum & Funnel

A string of length 'L' fixed at one end carries a mass 'm' at the other end. The string makes $\frac{3}{\pi}$ r.p.s. $\pi$ around the vertical axis through fixed end. The tension in the string is

A particle describes a horizontal circle of radius 'r' in conical funnel with smooth inner surface with a speed of $0.5$ m/s. The height of the plane of the circle from the vertex of the funnel is (acceleration due to gravity, $g = 10$ m/s${}^{2}$)

An inextensible string of length ' $l$ ' fixed at one end, carries a mass ' $m$ ' at the other end. If the string makes $\frac{1}{\pi}$ revolutions per second around the vertical axis through the fixed end, the tension in the string is [The string makes an angle $\theta$ with the vertical]

A particle describes a horizontal circle on smooth inner surface of a cone as shown in figure. If the height of the circle above the vertex is 10 cm. The speed of the particle is $(g$, acceleration due to gravity $= 10\text{ m/s}^2)$

A string of length $L$ is fixed at one end and carries a mass of $M$ at the other end. The mass makes $\left(\frac{3}{\pi}\right)$ rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is __________ ML.

A ball of mass $0.5\text{ kg}$ is attached to a string of length $50\text{ cm}$. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $400\text{ N}$. The maximum possible value of angular velocity of the ball in $\text{rad/s}$ is:

A particle rotates in a horizontal circle of radius 'R' in a conical funnel with constant speed 'V'. The inner surface of the funnel is smooth. The height of the plane of the circle from the vertex of the funnel is (g-acceleration due to gravity)

In a conical pendulum the bob of mass '$m$' moves in a horizontal circle of radius '$r$' with uniform speed '$V$'. The string of length '$L$' describes a cone of semi vertical angle '$\theta$'. The centripetal force acting on the bob is ( $g =$ acceleration due to gravity)

A ball of mass '$m$' is attached to the free end of a string of length '$l$'. The ball is moving in horizontal circular path about the vertical axis as shown in the diagram.

The angular velocity '$\omega$' of the ball will be [ $T =$ Tension in the string.]

A stone of mass $1\text{ kg}$ is tied to end of a massless string of length $1\text{ m}$. If the breaking tension of the string is $400\text{ N}$, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is:

A string of length '$L$' fixed at one end carries a body of mass '$m$' at the other end. The mass is revolved in a circle in the horizontal plane about a vertical axis passing through the fixed end of the string. The string makes angle '$\theta$' with the vertical. The angular frequency of the body is '$\omega$'. The tension in the string is

A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N. If the maximum speed with which the stone can revolve is $\frac{K}{\pi}$ rev./min. The value of K is:

(Assume the string is massless and unstretchable)

A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r such that $r = \frac{L}{\sqrt{2}}$. The speed of particle will be:

A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4Cx2 and rotating with angular speed $\omega$ (see figure). The value of $\omega$ is (neglect friction):

A smooth wire of length 2$\pi$r is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of $\omega$2 is equal to -

A conical pendulum of length 1 m makes an angle $\theta$ = 45o w.r.t. Z-axis and moves in a circle in the XY plane. The radius of the circle is 0.4 m and its center is vertically below O. The speed of the pendulum, in its circular path, will be: (Take g = 10 ms−2)